Mathematical physicists have announced progress on a 150-year-old theorem for which the Clay Mathematical Institute is offering a million-dollar reward. The scientists presented an operator that satisfies the Hilbert-Polya conjecture, which states that there exists a differential operator whose eigenvalues ​​correspond exactly to the non-trivial zeros of the Riemann zeta function. The article was published in the journal Physical Review Letters.

The Riemann Hypothesis is one of the Millennium Problems for which the American Clay Institute of Mathematics awards a million dollar prize. The Poincaré hypothesis (the Poincaré-Perelman theorem), which our compatriot proved, was included in this list. The Riemann Hypothesis, formulated in 1859, states that all non-trivial zeros of the Riemann zeta function (that is, the values ​​of the complex-valued argument that vanishes the function) lie on the line ½ + it, that is, their real part is equal to ½. The zeta function itself appears in many branches of mathematics, for example, in number theory, it is related to the number of primes less than a given one.

Function theory predicts that the set of non-trivial zeros of the zeta function should be similar to the set of eigenvalues ​​("solutions" for matrix equations) of some other function from the class of differential operators that are often used in physics. The idea of ​​the existence of a specific operator with such properties is called the Hilbert-Polya conjecture, although neither of them has published papers on this topic. “Since there are no publications by ‘authors’ on this topic, the formulation of the hypothesis changes depending on the interpretation,” explains one of the authors of the article, Dorje Brody from Brunel University in London. - However, two points must be met: a) one must find an operator whose eigenvalues ​​correspond to non-trivial zeros of the zeta function, and b) determine that the eigenvalues ​​are real numbers. The main goal of our work was point a). Further work is needed to prove part b).

Another important conjecture in this area is the idea of ​​Berry and Keating that if the desired operator exists, it will theoretically correspond to some quantum system with certain properties. “We determined the quantization conditions for the Berry-Keating Hamiltonian, thus proving the conjecture of their name,” adds Brody. - It may be disappointing, but the resulting Hamiltonian does not seem to correspond to any physical system in an obvious way; at least we didn't find such a match."

The greatest difficulty is the proof of the validity of eigenvalues. The authors are optimistic about this, the article contains a supporting argument based on PT-symmetry. This idea from particle physics means that if all four-dimensional space-time directions are reversed, the system will look the same. Nature is generally not PT-symmetric, however, the resulting operator has this property. As shown in the article, if we prove the violation of this symmetry for the imaginary part of the operator, then all eigenvalues ​​will be real, thus completing the proof of the Riemann hypothesis.

Logical proof of the Riemann hypothesis. SE VIEW OF THE WORLD.

The logical proof of the Riemann hypothesis is also a proof of God.
The Riemann hypothesis is an assumption about the existence of regularities in the distribution of prime numbers. The logical proof of the Riemann Hypothesis is, strictly speaking, the essence of what is known under the name "logic". From now on, this entity is known as it is in itself, in its own form of the Science of Rhetoric.

Information for thought:
“Prime numbers will “bury” cryptography” (NG-TELECOM, October 5, 04): “Mathematicians are close to proving the so-called “Riemann hypothesis”, recognized as one of the unsolved problems of mathematics. If the hypothesis that there are patterns in the nature of the “distribution” of prime numbers is proved, there will be a need to revise the fundamental principles of all modern cryptography, which underlies many e-commerce mechanisms.
The "Riemann Hypothesis" was formulated by the German mathematician G. F. B. Riemann in 1859. According to her, the nature of the distribution of prime numbers may differ significantly from what is currently assumed. The fact is that mathematicians have not yet been able to detect any system in the nature of the distribution of prime numbers. So, it is believed that in the neighborhood of an integer x, the average distance between successive prime numbers is proportional to the logarithm of x. Nevertheless, the so-called twin prime numbers have long been known, the difference between which is 2: 11 and 13, 29 and 31, 59 and 61. Sometimes they form whole clusters, for example 101, 103, 107, 109 and 113. mathematicians have long suspected that such clusters exist in the region of very large prime numbers, but so far they have not been able to prove or disprove this assertion. If such "clusters" are found, the strength of the cryptographic keys currently in use may suddenly become a big question mark.
According to a number of publications, the other day the American mathematician Louis de Brange from Purdue University said that he was able to prove the Riemann hypothesis. Earlier, in 2003, mathematicians Dan Goldston from the University of San Jose (California) and Kem Ildirim from Bogazici University in Istanbul already announced the existence of a proof of this theorem.
The proof of a seemingly abstract mathematical problem can fundamentally change the concepts underlying modern cryptographic systems - in particular, the RSA system. The discovery of a system in the distribution of prime numbers, says Oxford University professor Marcus du Satoy, would lead not only to a decrease in the strength of cryptographic keys, but also to the complete inability to ensure the security of electronic transactions using encryption. The implications of this cannot be overestimated given the role that cryptography plays in today's society, from guarding government secrets to enabling online financial and trading systems."

CALCULATION OF SIMPLE NUMBERS. THE ESSENCE OF MATHEMATICAL
01/16/2003 HTTP://LIB.RU/POLITOLOG/SHILOW_S/CHISLA.TXT

1. The phenomenon of development is calculus.

2. Universal calculus is fundamentally different from differential,
integral and other analytical calculus.

3. Universal calculus proceeds from the concept (formula) of a unit.

4. The idea of ​​an infinitesimal quantity, which underlies modern partial calculus, the idea of ​​Newton-Leibniz flux, is subject to fundamental
reflections.

5. Lorentz transformations, first used by Einstein as
project of a new, synthetic calculus, represent in practice the strategy
search for the foundations of number theory.

6. Set theory is a description, a description of number theory, which is not
is identical to the explication of the foundations of number theory.

7. Einstein's theory of relativity actually reveals numerical foundations
physical processes.

8. The idea of ​​an observer is a lexical description of the project of a synthetic
calculus.

9. In synthetic calculus, measurability is identical to calculus,
meaning is identical to the process, the meaning forms the process, which before
there was no meaning in "nature", in reality a series of numbers.

10. The problem of modern scientific knowledge, therefore, is
the problem of creating a synthetic calculus.

11. The main operation of synthetic calculus is the representation of a number
number.

12. The representation of a number by a digit is the result of the reflection of a number. Like
how the representation of a word by a concept (image) is the result of reflection
the words.

13. Reflection of the word is carried out by reading the letter. Reflection
numbers is carried out through the mathematization of physics.

14. The book of nature (physics) is written in the language of mathematics (read
mathematics). "The Book of Nature", Science is thus an idea,
presentation, description of numbers by numbers. Just like a book is
representation, formalization of words by letters, lexical and grammatical
forms.

15. Thus, the theory of numbers is, properly speaking, the universal theory of nature.

16. Calculus is thus the universal process of nature.
(nature as a process), Development, a process presented in digital form.

17. Representing a number as a digit is a fundamental technology
calculus, the essence of the phenomenology of development, the foundation of Technique as such.
So the representation of a word by an image (concept) is a fundamental technology
thinking is, strictly speaking, reflection.

18. Let's reveal the essence, the phenomenon of representing a number by a figure. Such and
there will be a technology of synthetic calculus.

19. The phenomenon of number representation in true number theory is revealed
as a phenomenon of fundamental difference between numbers in modern number theory.

20. The fundamental difference between numbers in modern number theory is
explication of the set of prime numbers. So the fundamental difference between words in
rhetoric is, first of all, an explication of the primary concepts of rhetoric.

21. A prime number is the possibility of representing a number as a digit, and
represented as a figure, it is the realization, the result of the representation
number as a digit, since there are numbers that cannot be represented as a finite
sign digits.

22. The fundamental position of synthetic calculus is, in the very
unconditional and necessary sense, the formula of unity.

23. An infinitely small value of analytic calculus is, in fact,
speaking, also a unit, as something one fixed by means of analysis.

24. The formula of a unit is the definition of a unit, since the concept itself
unit formulas are the result of the reflection of a number.

25. Since the unit formula is the concept of the language of science, the way
representation of a number by a digit, then the unit is nothing but a set,
set of prime numbers:

26. Sets of prime numbers in reality of a number series are, strictly speaking, phenomena of nature, the measurability of which is identical to their existence in time and space as a synthetic calculus,
calculus that produces numbers.

27. A prime number is the true limit of analytic calculations,
fixed in the form of physical constants indirectly.

28. The essence of synthetic calculus, a single act of computability of synthetic calculus, which can be characterized as a measurement that produces a physical object, and so, the essence of synthetic calculus is such a difference between sets of primes per unit set, which is also a specific set of primes. So the essence of the formation of rhetoric in a dialogue is such a phenomenon of a new basic concept (a unit of meaning, meaningfulness), not included in the circle of used primary concepts, which (a new concept) is also a set of primary concepts.

29. Divisibility as a technology for determining a prime number forms the essence of analytical calculus, which has not been fully reflected today.

30. Division is the path of a digit, entropy as a formal representation of
the reality of the number series.

31. Thus, the direct rule for determining a prime number
through divisibility there is a formula of a formula, the genesis and structure of a physical formula as a result of the reflection of the representability of a number by a digit.

32. The rule for determining a prime number determines the mechanism
synthetic calculus.

33. The rule for determining a prime number is simultaneous divisibility
digital parts of the number to the divisor. In terms of integer divisibility, the number
forms two digital parts, the unity of which is due to its position
with respect to its (all) prime numbers. The divider is working -
simultaneous division "on both sides" (digital) numbers.

34. The transition from analytic to synthetic calculus looks like
most direct form as the simultaneity of two operations of one
divisor in the digital form of the number.

35. A sequence of integer divisors defines a number as prime,
or not simple, that is, it is calculated.

36. Number is calculated in calculus.

37. The calculation of a number is the determination of the quality of a number.

38. In a number engine, the number is calculated.

39. The operation of the numerical engine: there is a sequential determination
(calculation of) prime numbers.

40. The mechanism for determining the simplicity of a number based on divisibility: "we divide
the initially divisible (for the initial sequence of divisors) digital beginning of the number by the initial sequence of divisors, taken, multiplied by an integer up to the maximum integer value of the digital beginning of the number, and we look at whether the remaining digit of the number is divided by an integer (without a remainder) by the real divisor, while the digital the beginning of the number will not be less than the divisor."

41. The physical world thus has a digital form.

42. The measurements of time in the system of measuring the number are identical to the measurements
spaces and are presented as digital forms: the number of digits (and digit) of the first part of the number (initial digital form), the number of digits (and digit) of the second part of the number (middle digital form), the number of digits (and digit) of the third part of the number (final digital form ).

43. Measurability of the physical world - an expression of the initial sequence of divisors in the digital beginning of a number with the simultaneous setting of the ratio of the divisor to the digital continuation of the number (integer, non-integer).

44. The basis of analytic calculus is division as
fundamental operation of number theory.

45. Division is the structure of the representation of a number by a digit.

46. ​​The product is the genesis of the representation of a number in the form of a figure.

47. The work is the fourth dimension, the dimension of time as
the fourth operation of number theory in relation to the triad "division - sum -
subtraction", which forms a single rule for calculating a prime number
(proof of its simplicity).

48. A work is a definition-reflection of a triad of operations.

49. The product is the meaning of the genesis of a number.

50. Division - the meaning of the number structure.

51. 1. The number in the form of the Power of the number (the meaning of the number) is first of all a square
digits of a number (first product).
51. 2. On the other hand, a number as a unit is a set of primes
numbers: 1 = Sp.
51.3. A prime number is a divisor of an integer non-simple number.
Thus, the rule for determining a prime number is written as
Fermat's theorem, which in this case becomes proven:
xn + yn = zn , holds for integers
x, y, z only for integers n > 2, namely:
The square of the digit of a number is the unit set of prime numbers.

52. The essence of Fermat's theorem:
Determination of the power of a number by the power of a set of prime numbers.

53. On the other hand, the geometry of Fermat's theorem is the interconversion of space and time in solving the problem of squaring the circle: The problem of squaring the circle is thus reduced to the problem of interconverting the square of a number into a specific set of primes, which has the "appearance" of the famous Möbius strip. The geometry of Euclid (the lack of proof of the fifth postulate - as a direct consequence of the underdetermination of the point, the lack of reflection of the point) and the geometry of Lobachevsky (the geometrization of the digital form of a number outside the number) are overcome together in the geometry of Fermat's theorem. The central postulate of the geometry of Fermat's theorem is the point postulate, which is revealed by the unity formula.

54. Thus, the reflection of the following operations of number theory based on
unit formulas - raising to a power, extracting a root - will lead to the creation of a physical theory of time-space control.

55. There is a number, a number is a unit that has the strength of a number. Representative
numbers are a prime number. This is the universal structure of a physical object,
the incompleteness of the reflection of which led to the corpuscular-wave
dualism, to the difference between the physics of elementary particles and the physics of the macrocosm.

56. Quantum calculus must be re-reflected into synthetic
calculus, Planck's constant expresses the discovery in the digit of the strength of a number.
Radiation is a phenomenon of representation of a number by a digit, revealed in the formula of unity as a solution to the paradox of black body physics.

57. The unity formula is thus the universal field theory.

58. The formula of unity expresses the intellectual essence of the Universe,
is the basis of the concept of the Universe as the reality of real
series of real numbers.

59. Development The Universe is a synthetic calculus, a calculus of prime numbers, the significance of which forms the objectivity of the Universe.

60. The formula of the unit proves, shows the power of the Word. unit formula
there is a structure of the Universe in accordance with the principle of the Word, when the self-shaping of the word is a product of being, the Book of Genesis. So the self-shaping of a number is a product of nature, the Book of the Universe. Formula
units in the most unconditional and necessary sense is the formula of time.
Synthetic calculus is a form of rhetoric.

CONSEQUENCE OF THE LOGICAL PROOF OF THE RIEMANN HYPOTHESIS:

WHAT IS AN ELECTRON? BEGINNINGS OF ELECTRONIC ENERGY
06/15/2004 HTTP://LIB.RU/POLITOLOG/SHILOW_S/S_ELEKTRON.TXT

1. The 20th and 21st centuries - respectively the Atomic and Electronic Ages - form two successive steps, two essences of the transition from the History of Modern Times to the History of New Being.

2. History, as having, having and the future to have a "place", - from the point of view of the Science of Philosophy, is the identity-difference of being and being. The place itself, as something that provides the possibility and reality for something to exist in time, is a phenomenon that results from the identity-difference of being and being.
Existing is the real, arising from being, existing Now and disappearing into non-being. Being is what creates Now, creates "here and now". As independent, existing in itself, separate from being, being is time. Being is what creates Time. Time tends to Being, as non-existence, as the objectivity of being, as being. Time enters Being, becomes being through the path of two essences of being. Aristotle considered this path from being to time and saw two essences as a descent from being to being, to time. Aristotle's metaphysics, as the beginning of European rationality, prescribes two essences of being, as what makes science possible. Science arises as the first division of being into two essences - into necessary and sufficient grounds, which together determine the being of being as a whole, as it is. Science, according to Aristotle, is the naming of the path (Logic) from being to being. We, in our historical position, consider this same path from the other side, as a path from time, from being - to being. Both Aristotle and I (we) see the same two essences (necessary and sufficient) of being, which connect being and being, but Aristotle sees them from the side of being, and we, on the other hand, from the side of being, from the side of time . Such is the nature of the "new Aristotelianism". Thus, between Being and Time, there are two essences - the necessary and sufficient grounds, which create everything that generally happens, that is really.

3. Being, necessary reason, sufficient reason, Time. Time, sufficient reason, necessary reason, Being. This is a description and presentation of a Mobius strip, which, according to "modern scientists", is impossible to imagine. We quote "modern scientists": "Lobachevsky's geometry is the geometry of a pseudosphere, i.e. surfaces of negative curvature, while the geometry of a sphere, i.e. surfaces of positive curvature, this is Riemannian geometry. Euclidean geometry, i.e. the geometry of a surface of zero curvature is considered to be its special case. These three geometries are useful only as geometries of two-dimensional surfaces defined in three-dimensional Euclidean space. Then it is possible for them to construct in parallel the whole huge edifice of axioms and theorems (which is also described in visible images), which we know from the geometry of Euclid. And it is really very remarkable that the fundamental difference between all these three completely different "structures" is only in one 5th axiom of Euclid. As for the Möbius strip, this geometric object cannot be inscribed in three-dimensional space, but only in no less than four-dimensional space, and even more so, it cannot be represented as a surface of constant curvature. Therefore, nothing similar to the previous one can be built on its surface. By the way, that’s why we can’t visually imagine it in all its glory.”
Speculation, discovered by Parmenides and Plato, as the vision of "eidos", is used by Aristotle directly, and by us, who think on the other side than Aristotle, it is used, achieved indirectly. From this side, which is different from that of Aristotle, we see the formula of that being with which Aristotle deals directly. We do not have a direct relationship with this being, but we can receive it through a certain formula, de-mediation. The Möbius strip is a representation of the movement from being to time and from time to being, that is, the point of the Möbius strip belongs to both time and being - it creates itself. The 5th “unproven” postulate of Euclid is also an indication that, in addition to being, there is also being that generates being, and that being is nothing other than time. The fifth postulate of Euclid arises as a consequence of the under-axiomatization of the point, as a sign-consequence of the absence of a substantial understanding of the point. In essence, the correct axiomatization of the point axiom is the only necessary axiom of universal geometry, the universal geometry of being, and other axioms (postulates) are not required, they are superfluous. In other words, in the geometry of Euclid, only the first necessary essence of the axiom of the point is fixed, which is subjected to problematization in other geometries, problematization from the point of view of an entity whose geometry is not reducible to the geometry of Euclid. The second, sufficient essence of the axiom of a point is that a POINT IS ALWAYS A POINT OF A MOBIUS STRAP (there is NO POINT THAT IS NOT A POINT OF A MOBIUS STRAP). This is the only axiom of Shilov's geometry, as the universal geometry of being. As you can see, this geometry coincides with the existent, as the being of the existent: the objects forbidden in this geometry are non-existent objects. Such is the primary idea of ​​geometry as the law of the formation of the real.

4. The substantial point is both the essence and the problematization of the law of identity. Here logic and geometry coincide in their common source, foundation. Here logic and geometry reveal themselves as two essences of being, as produced by the being of time. Geometry is the necessary essence of existence. Logic is the sufficient essence of being. This is how Aristotle founded European science. By founding it in this way, Aristotle directly owned the topic of the substantiality of the point, while we own this topic indirectly (more precisely, this topic owns us with such power that we no longer think about the substantiality of the point). We must thus return from logic to geometry, formalizing the immediate Aristotelian understanding of the substantiality of a point. How do we do it? We problematize the law of identity (A=A) as a process, becoming, an event of how A is, becomes A, how A is held, fixed, grasped, like A. In this problematization, the whole being of logic participates, and in this understanding the law of identity also becomes the only law of logic when all other laws (contradiction, excluded third, sufficient reason) become measurements, participants in the process of identity, the process of becoming, the feasibility of identity. Logic, as sufficient, and geometry, as necessary, coincide in one essential essence, in the name of a single law of identity - the law of the substantiality of a point.

5. What is a substantial point, as real? This is the main question of Science, in the answer to which it becomes a single science not only in the sphere of the foundations of science, but also externally, “eidetically”. What is the root of all "-logies" as "separate scientific disciplines"? In the logical-geometric unity, first of all. What does logico-geometric unity study? point substance. The logical-geometric unity, poorly reflected by modern sciences, is the theory of a substantial point. The theory of a substantial point is the basis of the genesis and structure of scientific knowledge, rationality. In the field theory, truth, like the truth of the theory of a substantive point, is hidden, eludes the scientist. "Field theory", field theory is a scientific myth. The myth of the actual existence of a substantial point.

6. The actual being of a substantial point is a NUMBER. THE TIME OF THE SUBSTANTIAL POINT, THE POINT OF THE MOBIUS STRAP, AND THERE IS THE ONLY POSSIBLE AND EXISTING TIME, THE TRUE MOMENT OF TIME. NO, THERE IS NO TIME WHICH WOULD NOT BE, AS THE TIME OF A SUBSTANTIAL POINT. The logical-geometric unity, which, on the one logical side, is the law of substantial identity, and on the other geometric side, is the law of the substantial point, in its only essential essence, a priori logic and geometry, is the LAW OF NUMBER. Being creates a being, real in the form of a number, in the space of a real number series, as a material being of time. Number is a place that is created between time and being, between being and time, is a being.

7. The true science of number is, therefore, the mechanics of time (Mathematics is the science of the number, of the representation of a number by a figure). This is what makes it possible to understand the new Aristotelianism, "exposing" the "field myth" of modern physics. The space of being reveals itself as the space of a real numerical series. Field theory, the notion of a field, is a myth regarding the logical-geometric unity and its true nature. The quantum-mechanical interpretation is a kind of myth regarding the mechanics of time. The quantum mechanical interpretation does not yet know "nature" as a real number series, does not yet know the universal (universal for interactions of any "level") physical object, as a number. Modern physics has not yet known "nature" as calculus. The quantum mechanical interpretation is stuck in a logical-geometric unity, as in an indefinite duality (Heisenberg's principle).

8. Thus, the possibility of a “non-field” definition and understanding of energy arises. Field understanding-representation of energy comes from the law of conservation of energy and the inviolability of the principles of thermodynamics. NUMERICAL UNDERSTANDING OF ENERGY IS UNDERSTANDING OF THE MECHANISM OF ACTION OF NUMBER AS THE REAL AND ONLY POSSIBLE MOMENT OF TIME. ENERGY IS THE ENERGY OF MOVEMENT (EXISTENCE) OF THE MOBIUS STRIP. MOBIUS TAPE IS A FORM OF EXISTENCE OF ENERGY. ENERGY IN THE MOST NECESSARY AND UNCONDITIONAL SENSE IS WHAT VIOLATES THE LAW OF CONSERVATION OF ENERGY AND THE ORIGIN OF THERMODYNAMICS, AND THIS VIOLATION FORMES THE PHYSICAL ESSENCE OF TIME, THE POSSIBILITY AND REALITY OF THE MOMENT OF TIME AS THE MOMENT OF REALITY.

9. Energy can be defined as the Force of the Unit (the Force of the number), the strength of which lies in the calculable violation of the law of conservation of energy (beginnings of thermodynamics). In essence, atomic energy advanced humanity to a numerical understanding of energy, but stopped in its scientific development, being unable to comprehend atomic energy as a necessary prerequisite for revising the principles of thermodynamics and the law of conservation of energy. Science found itself here in exactly the same position before the need to comprehend its own foundations, in which the church found itself in the face of the achievements of science. Just like the church, science has remained "loyal" to the law of conservation of energy (the principles of thermodynamics), despite the need to comprehend the essence of the foundations of atomic science INDEPENDENTLY, outside of thermodynamic coordination. Atomic science in the matter of using atomic energy came to the idea-representation of a substantial point. The use of atomic energy is, in essence, the self-disclosure of the substance of a point, as a number growing throughout the entire space of a real number series (the idea of ​​a "chain reaction"). Moreover, this idea is quite visible: that is why an atomic explosion is an atomic mushroom, there is GROWTH, metaphysical growth, the running of a number over its own space, the place of a number series.

10. Electronic science will define the face of the 21st century. And this science will arise from the true definition of WHAT THE ELECTRON IS. All previous thoughts, as well as the consideration of atomic science (atomic energy), as a pure phenomenon that has its own truth - the FIRST STAGE, THE FIRST NECESSARY ESSENCE OF DISCLOSURE OF THE NUMERICAL NATURE OF ENERGY, as a physical fixation of the force and existence of a number, contribute to understanding the electron already directly, as a number, as an object manifesting itself physically. It is no coincidence that they say that "the electron is the most mysterious particle in physics." The electron is the second step, the second SUFFICIENT ESSENCE OF THE NUMERICAL NATURE OF ENERGY. An atom, an electron are located between being and time (existing), as, respectively, the first necessary and second sufficient essence of the existing. The transition from being to time and the reverse transition from time to being is not the “divisibility of matter” of being, but a substantial point, Number, and, in this sense of Number as “indivisibility of matter”, ELECTRON IS A SIMPLE NUMBER (an indivisible number). A prime number is the physical essence of an electron as a space-time phenomenon of time.

11. Electronic science completes the transition from time to being, necessarily begun by atomic science. Electronic Science discovers the Unity Formula: One is the set of prime numbers. The Unit formula reveals the device, the essence of time, the mechanics of time. Electronic science gives a person access to ELECTRONIC ENERGY, DIRECT ENERGY OF THE NUMERICAL SERIES, ENERGY OF CREATION. Electronic science will solve the problems that atomic science has stopped in front of and thereby incredibly change energy, fixing a “fundamentally new”, and, in fact, a true source of mega-energy - a number, a number series. Understanding WHAT THERE IS AN ELECTRON, we will create ELECTRONIC ENERGY as the mechanics of time, first of all. The mathematical procedure will become a part of the physical-technical process, the part that will bring this process into a new superphysical, superphysical-constant quality.

12. The task of creating electronic energy is the main task of forming a new technotronic mode. This is the task of beginning the History of the New Being, completing the transitional period from the History of the New Time to the History of the New Being, the first necessary foundation, the first necessary step of which was the past 20th Atomic Age. The scientific revolution of the 20s of the 20th century, carried out by Einstein, created the necessary prerequisites for the Mega-Science Revolution of the early 21st century, the result of which will be electronic science, electronic energy. The emergence of electronic science, electronic energy is, first of all, the discovery of what an electron is. The discovery of the “mystery of the electron” is, first of all, understanding, comprehension, the path of which is presented in this sequence of theses as the path of the “new Aristotelianism”.

13. With what experience did Aristotle work when he comprehended the truth of the world as a transition from being to time, when he discovered that possibility that was realized as Logic? The idea of ​​what is known to man as the closest circle of his being, defining him as a proper human being, was the Möbius strip. Where did a person see and know the Mobius strip? Where did a person draw the experience of the substantiality of a point? After all, all this is knowledge, “innate ideas” that make a living being a man, after all, a person is made human by his human perception (a person, in the words of Goethe, “sees what he knows”). How did the “early-ancient” man know everything that modern science, armed with powerful means of technology, experiment, mathematical apparatus, comes only in the 21st century, despite the fact that a person always has this knowledge precisely as a person? Answer: from speech, from human speech, as the direct reality of thinking. Speech is that movement from being to time and from time to being (in the movement from time to being, speech becomes thinking), which is a person, as a kind of movement and experience of real movement. A point, as a substantial point, is known, known to a person, as a point of speech, as a moment of truth, as a judgment. Time, as objectivity, is given to man, as the objectivity of speech (thinking). The meaning of the modern historical moment in the development of science lies in the most important experiment - in verifying modern science with the experience of speech, in the path of a radical logical rethinking of science as scientific speech, in identifying the necessary and sufficient grounds for the truth of a scientific judgment. Speech contains a program of truth, the disclosure of which required all the power of modern science, directed outside of man, but requiring comprehension of the results obtained in the language of science. Speech for a person is not only “between” being and time, but also embraces being, as the being of a person, and time, as the time of a person, with a Mobius strip. Speech is something more than a philological set of words and rules, speech is a being that enters the world at such a time as a person, creates such a being as a person. Speech creates number as the essence of man, the number that is man.
Therefore, the mega-scientific revolution is a humanitarian-technotronic revolution, which begins with the disclosure of the secret of the essence of the electron, as a prime number, BY THE MEANS OF THINKING, THE MEANS OF THE LANGUAGE OF SCIENCE.

THE FIRST MENTION OF THE LOGICAL PROOF OF THE RIEMANN HYPOTHESIS
20.10.2000 HTTP://LIB.RU/POLITOLOG/SHILOW_S/MEGANAUKA.TXT
"CHRONICLE. DEFINITIONS OF MEGA SCIENCE»

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The unshakable and final foundation that Descartes was looking for at the beginning of the Modern Age is understood and revealed at the End of the History of the Modern Age. This base is a number. As being truly described by the language of science. At the End of the History of the Modern Age, this foundation is revealed and becomes visible as the "last" of the Modern Age. One can see the number through the "optics" of the reductionism of the soliptic (methodoritical) doctrine, as the highest form of the Cartesian "methodological" doubt. The number discovered in this way has characteristics characteristic not only of the arithmetic concept of “number”, but also of the philosophical concept of “foundation” (I will add - and the physical concept of “nature” (“matter”) - the concept of “atom” and the concept of “electron”), so that mathematicians (and physicists) will have to make room in the boat of numbers, sailing in the “borderless ocean of the unknown” (which Newton writes about in the Mathematical Principles of Natural Philosophy, treating himself not as “the discoverer of the laws of the universe”, but “like a boy throwing pebbles on the coast ”) and give a place in this boat also to philosophers. Strictly speaking, for the benefit of physico-mathematicians as well, the boat of number (Noah’s Ark of modern civilization) under the control of which, crowded on one of its sides, is already almost under water (for example, the collapse of the Hilbert-Goedel “formal-logical” formalization program) . The formalization program of the Science of Rhetoric deduces the notion of a true set theory, bound by the Unity formula, as a set of primes.

Mathematical science. Work on them had a tremendous impact on the development of this area of ​​human knowledge. 100 years later, the Clay Mathematical Institute presented a list of 7 problems known as the Millennium Problems. Each of them was offered a prize of $1 million.

The only problem that appeared among both lists of puzzles that have been haunting scientists for more than one century was the Riemann hypothesis. She is still waiting for her decision.

Brief biographical note

Georg Friedrich Bernhard Riemann was born in 1826 in Hannover, in a large family of a poor pastor, and lived only 39 years. He managed to publish 10 works. However, already during his lifetime, Riemann was considered the successor of his teacher Johann Gauss. At the age of 25, the young scientist defended his dissertation "Fundamentals of the theory of functions of a complex variable." Later he formulated his hypothesis, which became famous.

prime numbers

Mathematics appeared when man learned to count. At the same time, the first ideas about numbers arose, which they later tried to classify. Some of them have been observed to have common properties. In particular, among natural numbers, i.e., those that were used in counting (numbering) or designating the number of objects, a group was distinguished that were divisible only by one and by themselves. They are called simple. An elegant proof of the theorem of infinity of the set of such numbers was given by Euclid in his Elements. At the moment, their search continues. In particular, the largest of the already known is the number 2 74 207 281 - 1.

Euler formula

Along with the concept of the infinity of the set of primes, Euclid also defined the second theorem on the only possible decomposition into prime factors. According to it, any positive integer is the product of only one set of prime numbers. In 1737, the great German mathematician Leonhard Euler expressed Euclid's first infinity theorem in the form of the formula below.

It is called the zeta function, where s is a constant and p takes on all prime values. Euclid's statement about the uniqueness of the expansion directly followed from it.

Riemann zeta function

Euler's formula, on closer inspection, is absolutely amazing, as it defines the relationship between primes and integers. After all, on its left side, infinitely many expressions that depend only on primes are multiplied, and on the right side there is a sum associated with all positive integers.

Riemann went further than Euler. In order to find the key to the problem of the distribution of numbers, he proposed to define a formula for both real and complex variables. It was she who subsequently received the name of the Riemann zeta function. In 1859, the scientist published an article entitled "On the number of prime numbers that do not exceed a given value", where he summarized all his ideas.

Riemann suggested using the Euler series, which converges for any real s>1. If the same formula is used for complex s, then the series will converge for any value of this variable with a real part greater than 1. Riemann applied the analytic continuation procedure, extending the definition of zeta(s) to all complex numbers, but "thrown out" the unit. It was excluded because for s = 1 the zeta function increases to infinity.

practical meaning

A natural question arises: what is interesting and important about the zeta function, which is the key to Riemann's work on the null hypothesis? As you know, at the moment no simple pattern has been identified that would describe the distribution of prime numbers among natural numbers. Riemann was able to discover that the number pi(x) of primes that did not exceed x is expressed in terms of the distribution of non-trivial zeros of the zeta function. Moreover, the Riemann Hypothesis is a necessary condition for proving time estimates for the operation of some cryptographic algorithms.

Riemann hypothesis

One of the first formulations of this mathematical problem, which has not been proven to this day, sounds like this: non-trivial 0 zeta functions are complex numbers with a real part equal to ½. In other words, they are located on the line Re s = ½.

There is also a generalized Riemann hypothesis, which is the same statement, but for generalizations of zeta functions, which are usually called Dirichlet L-functions (see photo below).

In the formula χ(n) is some numerical character (modulo k).

The Riemannian assertion is considered the so-called null hypothesis, as it has been tested for consistency with existing sample data.

As Riemann argued

The remark of the German mathematician was initially formulated rather casually. The fact is that at that time the scientist was going to prove the theorem on the distribution of prime numbers, and in this context, this hypothesis did not have much meaning. However, its role in solving many other issues is enormous. That is why Riemann's assumption is currently recognized by many scientists as the most important of the unproven mathematical problems.

As already mentioned, to prove the distribution theorem, the full Riemann hypothesis is not needed, and it is enough to logically justify that the real part of any non-trivial zero of the zeta function is in the interval from 0 to 1. From this property it follows that the sum over all 0-th The zeta functions that appear in the exact formula above are a finite constant. For large values ​​of x, it may be lost altogether. The only member of the formula that remains the same even for very large x is x itself. The remaining complex terms vanish asymptotically in comparison with it. So the weighted sum tends to x. This circumstance can be considered a confirmation of the truth of the theorem on the distribution of prime numbers. Thus, the zeros of the Riemann zeta function have a special role. It lies in the fact that the values ​​cannot make a significant contribution to the expansion formula.

Followers of Riemann

The tragic death from tuberculosis did not allow this scientist to bring his program to its logical end. However, Sh-Zh took over from him. de la Vallée Poussin and Jacques Hadamard. Independently of each other, they deduced a theorem on the distribution of prime numbers. Hadamard and Poussin succeeded in proving that all nontrivial 0 zeta functions are within the critical band.

Thanks to the work of these scientists, a new direction in mathematics appeared - the analytic theory of numbers. Later, several more primitive proofs of the theorem that Riemann was working on were obtained by other researchers. In particular, Pal Erdős and Atle Selberg even discovered a very complex logical chain confirming it, which did not require the use of complex analysis. However, by this point, several important theorems had already been proved by means of Riemann's idea, including the approximation of many functions of number theory. In this regard, the new work of Erdős and Atle Selberg had practically no effect on anything.

One of the simplest and most beautiful proofs of the problem was found in 1980 by Donald Newman. It was based on the famous Cauchy theorem.

Does the Riemannian Hypothesis threaten the foundations of modern cryptography?

Data encryption arose along with the advent of hieroglyphs, more precisely, they themselves can be considered the first codes. At the moment, there is a whole area of ​​digital cryptography, which is developing

Prime and "semi-prime" numbers, i.e. those that are only divisible by 2 other numbers in the same class, form the basis of the public key system known as RSA. It has the widest application. In particular, it is used when generating an electronic signature. Speaking in terms accessible to dummies, the Riemann hypothesis asserts the existence of a system in the distribution of prime numbers. Thus, the strength of cryptographic keys, on which the security of online transactions in the field of e-commerce depends, is significantly reduced.

Other unresolved mathematical problems

It is worth finishing the article by devoting a few words to other millennium tasks. These include:

• Equality of classes P and NP. The problem is formulated as follows: if a positive answer to a particular question is checked in polynomial time, is it true that the answer to this question itself can be found quickly?
• Hodge hypothesis. In simple words, it can be formulated as follows: for some types of projective algebraic varieties (spaces), Hodge cycles are combinations of objects that have a geometric interpretation, i.e., algebraic cycles.
• The Poincaré hypothesis. This is the only Millennium Challenge that has been proven so far. According to it, any 3-dimensional object that has the specific properties of a 3-dimensional sphere must be a sphere up to deformation.
• Statement of the quantum theory of Yang-Mills. It is required to prove that the quantum theory put forward by these scientists for the space R 4 exists and has a 0th mass defect for any simple compact gauge group G.
• Birch-Swinnerton-Dyer hypothesis. This is another issue related to cryptography. It concerns elliptic curves.
• The problem of the existence and smoothness of solutions of the Navier-Stokes equations.

Now you know the Riemann hypothesis. In simple terms, we have formulated some of the other Millennium Challenges. That they will be solved or it will be proved that they have no solution is a matter of time. And it is unlikely that this will have to wait too long, since mathematics is increasingly using the computing capabilities of computers. However, not everything is subject to technology, and first of all, intuition and creativity are required to solve scientific problems.

The Riemann Hypothesis is one of the seven Millennium Problems, for its proof the Clay Mathematics Institute, Cambridge, Massachusetts will pay a $1 million prize. Solutions that were published in a well-known mathematical journal are accepted for consideration, and not earlier than 2 years after publication (for comprehensive consideration by the mathematical community) (http://www.claymath.org/millennium/).
I had my own ideas and approaches, as always, very different from those known. I wanted to write artistically about the Riemann hypothesis. In the process of my calculations and collecting material, I discovered a beautifully written book by John Derbyshire: John DERBYshire. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Astrel Publishing House, 2010
After reading this book, I had only to give this link.
“In August 1859, Bernhard Riemann became a corresponding member of the Berlin Academy of Sciences; it was a great honor for the thirty-two-year-old mathematician. In accordance with tradition, Riemann on this occasion presented to the academy a paper on the topic of research in which he was busy at that time. It was called "On the number of prime numbers not exceeding a given value." In it, Riemann explored a simple question from the realm of ordinary arithmetic. To understand this question, let's first find out how many prime numbers there are that do not exceed 20. There are eight of them: 2, 3, 5, 7, 11, 13, 17 and 19. And how many prime numbers that do not exceed a thousand? Million? Billion? Is there a general law or general formula that would save us from direct recalculation?
Riemann tackled this problem using the most advanced mathematical apparatus of his day, tools that even today are only taught in advanced college courses; in addition, for his own needs, he invented a mathematical object that combines power and elegance at the same time. At the end of the first third of his article, he makes some conjecture about this object, and then remarks:
“It would be desirable, of course, to have a rigorous proof of this fact, but after several short fruitless attempts, I postponed the search for such a proof, since this is not required for the immediate purposes of my research.”
This occasional speculation went largely unnoticed for decades. But then, for reasons I have set out to describe in this book, it gradually captured the imagination of mathematicians until it reached the status of an obsession, an irresistible obsession.
The Riemann Hypothesis, as this conjecture came to be called, remained an obsession throughout the 20th century and remains so to this day, reflecting by now every single attempt to prove or disprove it. This obsession with the Riemann Hypothesis has become stronger than ever after other great problems that have long remained open have been successfully solved in recent years: the Four Color Theorem (formulated in 1852, solved in 1976), Fermat's Last Theorem (formulated, apparently in 1637, proved in 1994), as well as many others less well known outside the world of professional mathematicians. The Riemann hypothesis absorbed the attention of mathematicians throughout the 20th century. Here is what David Hilbert, one of the most prominent mathematical minds of his time, said, addressing the Second International Congress of Mathematicians: “Recently, significant advances have been made in the theory of the distribution of prime numbers by Hadamard, de la Vallée Poussin, von Mangoldt and others. But for a complete solution of the problem posed in Riemann's study "On the number of primes not exceeding a given value", it is necessary first of all to prove the validity of Riemann's extremely important assertion<...>».
Further Hilbert gives the formulation of the Riemann Hypothesis. And here is what Philip A. Griffiths, director of the Institute for Advanced Study at Princeton and formerly professor of mathematics at Harvard University, said a hundred years later. In his article entitled "Challenge for 21st Century Researchers" in the January 2000 issue of the Journal of the American Mathematical Society, he writes:
“Despite the colossal achievements of the 20th century, dozens of outstanding problems still await their solution. Probably most of us will agree that the following three problems are among the most challenging and interesting.
The first of these is the Riemann Hypothesis, which has been teasing mathematicians for 150 years.<...>».
An interesting development in the United States in the last years of the 20th century was the emergence of private mathematical research institutes funded by wealthy math enthusiasts. Both the Clay Mathematical Institute (founded in 1998 by Boston financier Landon T. Clay) and the American Mathematical Institute (founded in 1994 by California entrepreneur John Fry) have focused their research on the Riemann Hypothesis. The Clay Institute set a million dollar prize for proving or refuting it. The American Mathematical Institute addressed the Hypothesis at three full-scale conferences (in 1996, 1998 and 2000) that brought together researchers from all over the world. Whether these new approaches and initiatives will eventually defeat the Riemann Hypothesis remains to be seen.
Unlike the Four Color Theorem or Fermat's Last Theorem, the Riemann Hypothesis is not easy to formulate in a way that makes it understandable to a non-mathematician, because it is the very essence of one difficult to understand mathematical theory. Here's how it sounds:
Riemann hypothesis.
All non-trivial zeros of the zeta function
have a real part equal to one second.
When you come into contact with the works around the Riemann hypothesis, a mystical idea comes not only about the evolution of ideas and thinking, not only about the laws of the development of mathematics, not only about the structure of the very plan for the unfolding of the universe, but also about primordial knowledge, absolute truth, logos as the program of the One.
Mathematical abstractions rule the world, control the behavior of elementary particles, high energies, mathematical operators generate and destroy anything. After a number of centuries of dominance of the material, worship of the material, the power of the world spirit began to manifest itself again in the form of mathematical abstractions, Pythagoreanism, Platonism became the methodological guidelines of modern science.
Since childhood, I have found errors in the works of great mathematicians. Not out of envy or mischief, but just wondering if I could surpass Pythagoras, Diophantus, Euclid, Fermat, Mersenne, Descartes, Gauss, Euler, Legendre, Riemann, Dirichlet, Dedekind, Klein, Poincaré. And oddly enough, he did. Formulated new problems, proved new theorems. But it turned out that the mathematical world is arranged, despite the requirements of accuracy and evidence, somehow bureaucratically. It turned out that your evidence is simply not believed. Contrary to logic and objectivity. And they believe the tales of the press, radio and television. At the same time, the media distort the actual state of affairs so much that you are surprised to find out how your phrases have been altered. So I started avoiding interviews.
I want to note the presence of many errors around the hypothesis and the Riemann zeta function, as well as in attempts to prove or disprove the hypothesis. Riemann did not attach much importance to finding the zeros of the zeta function. But the chorus of "prominent" followers has inflated the significance of the hypothesis beyond belief. I show even elementary calculations that the hypothesis is wrong, that there are other solutions. Firstly, the zeta function does not have the symmetry that is being talked about - a completely different function has the symmetry of solutions. Secondly, if you are not lazy and know how to calculate the roots of equations for functions with complex variables, you can see that the situation is actually somewhat different. Want to make sure? Read the formulas in the attached figure carefully. More exhaustive examples and calculations can be found in the note "The Riemann's Hypothesis Refutation Formulae" You can add your generalizations (especially the function itself) and the corresponding calculations. "And the chest just opened!"
I wish you success!

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✪ #170. THE RIEMANN HYPOTHESIS IS THE MILLENNIUM PROBLEM!

✪ Science show. Issue 30. Riemann Hypothesis

✪ Riemann hypothesis. The problem of the millennium is solved (but this is not accurate) | Trushin will answer #031 +

✪ Riemann hypothesis. The problem of the millennium is solved (but this is not accurate). Part II | Trushin will answer #032 +

✪ What did Grigory Perelman prove?

Subtitles

If a natural number has only two divisors - itself and one, then it is called prime. The smallest prime number is two, three is also divisible only by itself and by one, but twice or two is four, and this number is composite, five squares can only make a rectangle with sides 5 and 1, but six squares can be built not only in one row, but also in a 2x3 rectangle. Interest in prime numbers appeared in antiquity: the first records on the topic known to us date back to the second millennium BC - the ancient Egyptians knew a lot about mathematics. In ancient times, Euclid proved that there are infinitely many prime numbers, and, in addition, he had an idea of ​​\u200b\u200bthe fundamental theorem of arithmetic. Eratosthenes, in turn, came up with (or at least fixed) an algorithm for finding prime numbers. This is a very cool thing called the sieve of Eratosthenes, look: now we will quickly use it to determine all primes in the first hundred natural numbers. The one is not simple by definition, the two is the first simple: we cross out all the numbers that are multiples of it, because they are necessarily composite. Well, there are already half as many candidates! We take the next prime number - three, cross out all the numbers that are multiples of three. Note that the five knocks out not so many numbers, because many have already turned out to be a multiple of two or three. But what is most surprising is that our algorithm can be terminated at the number seven! Think why this is so! And if you guessed it, write in the comments on which number you can finish the procedure when working with the first ten thousand natural numbers! So, in total in the first hundred we have twenty-five prime numbers. Hmm… how many prime numbers are in the first thousand or, say, a million? This question disturbed the brightest minds of mankind in earnest, no one then needed the practical benefits of cryptography for nothing: mathematics is rather a conversation with God, or, in any case, one of the ways to hear him. Well, prime numbers are like atoms in chemistry and like the alphabet in literature. Okay, back to topic! Centuries later, the whole of Europe takes over the baton of ancient Greek scientists: Pierre Fermat develops the theory of numbers, Leonard Euler makes a huge contribution, and, of course, everyone does not compile huge tables of prime numbers. However, the regularity of the appearance of our special numbers among the composite numbers cannot be found. And only at the end of the 18th century, Gauss and Legendre put forward the assumption that the most wonderful function π(x), which would count the number of prime numbers less than or equal to the real number x, is arranged as follows π(x)=x/lnx. By the way, in the first hundred, how many numbers turned out to be prime? Twenty-five, right? Even for such small values, the function produces a result that is adequate to the truth. Although it is more about the limit of the ratio π(x) and x/lnx: at infinity it is equal to one. This statement is the theorem on the distribution of prime numbers. A significant contribution to its proof was made by our compatriot Pafnuty Lvovich Chebyshev, and it would be possible to finish the topic entirely by informing you in the end that this theorem was proved independently by Jacques Hadamard and de la Vallée-Poussin back in 1896. Yeah ... if not for one "but"! In their reasoning, they relied on the thesis of one predecessor colleague. And this scientist, given that Einstein was not yet born, was Bernhard Riemann. Here's a frame with Riemann's original manuscript. Do you know why he came up with this topic: the reason is as old as our educational system: prime numbers were studied by Riemann's supervisor - Carl Friedrich Gauss, the king of mathematics, by the way! Here is the old printed version of the report in German. I was lucky to find a Russian translation, but even dusting it off, some of the formulas are hard to see, so we'll use the English version. Look! Bernhard starts from Euler's results: on the right, with the capital Greek letter sigma, the sum of all natural numbers is written, and on the left, with the capital and at least Greek letter Pi, the product is denoted, moreover, the small letter p runs through all prime numbers. This is a very beautiful ratio - think about it! Next, the zeta function is introduced and ideas related to it are developed. And then the narrative, through the thorny road of mathematical analysis, goes to the stated theorem on the distribution of prime numbers, although from a slightly different angle. And now look here: an equation in which on the left is a xi function, closely related to zeta, and on the right is a zero. Riemann writes: "Probably all zeros of the x-function are real; in any case, it would be desirable to find a rigorous proof of this proposition." Then he adds that after several vain, not very persistent attempts to find one, he temporarily abandoned them, since there is no need for this for a further purpose. Well, this is how the Riemann hypothesis was born! In a modern way and with all the refinements, it sounds like this: all non-trivial zeros of the zeta function have a real part equal to ½. There are, of course, other equivalent formulations. In 1900, David Hilbert included the Riemann Hypothesis in his famous list of 23 unsolved problems. By the way, doesn't it seem strange to you that Hilbert worked in the same department at the University of Göttingen as Riemann did in his time. If this was a manifestation of fellowship, then with a clear conscience I once again add shots of a birch tree and Chebyshev here in sequence. Fine! We can move on. In 2000, the Clay Institute included the Riemann Hypothesis in the list of seven open problems of the millennium, and now 10⁶ ($) is required for its solution. Yes, I understand that you, as real mathematicians, are not very attracted by money, but still this is a good reason to realize the essence of the Riemann hypothesis. Go! Everything is very easy and understandable! At least it was for Riemann. Here is the explicit zeta function. As always, we would be able to see the zeros of the function if we drew its graph. Hmm... Okay, let's try it! If we take a two instead of the argument s, we get the famous Basel problem - we will need to calculate the sum of a series of inverse squares. But this is not a problem, Euler coped with the problem a long time ago: it immediately became obvious to him that this sum is equal to π² / 6. Well, then let's take s=4 - but, by the way, Euler calculated this too! Obviously, π⁴/90. In general, you already understood who calculated the values ​​of the zeta function, at points 6, 8, 10 and so on. So, what is this? Riemann zeta function from unity? Let's get a look! Ahh, so it's a harmonic series! So, what do you think the sum of such a series is equal to? The terms are small, small, but still more than in a series of inverse squares, right? Click pause, think a bit and give your estimate. Well, how many are here? Two? Or maybe three? Drum roll... the harmonic series diverges! This amount flies to infinity, you understand, no?! Look, we take a series in which each of the terms does not exceed the corresponding members of the harmonic series. And we see: ½, then another ½, again ½ and so on ad infinitum! What am I getting at? The zeta function from one is not defined! Well, now it seems to be clear what the Zeta chart looks like. One thing is not clear, where are the zeros of the zeta function? Well, show me where the non-trivial zeros of the zeta function are, and also the real part, equal to one second! After all, if we take the argument of the zeta function ½, then all members of the resulting series will be no less than harmonic, which means sadness, divergence, infinity. That is, in general, for any real s less than or equal to one, the series diverges. And, of course, with s=-1 zeta will appear as the sum of all natural numbers and will not be equal to any specific number. Yeah ... there is only one "but"! If my savvy friend is asked to calculate the zeta function at the point -1, then he, being a soulless piece of iron, will give the value -1/12. And in general, his zeta is defined for any arguments, except for one, moreover, zeros are also reached - in even negative values! Yes-ah-ah, we arrived, what could be the reason for this? Oh, it's good that there is a textbook on the theory of the function of a complex variable at hand: there will certainly be an answer here. So it is, so it is! It turns out that some functions have analytic continuation! We are talking about functions that are differentiated arbitrarily many times, expanded in a Taylor series, remember those? They have a continuation in the form of some other function, by the way, the only one. And in particular, our native zeta function for a real argument, as long as it fits all conditions, can be extended to the entire complex plane according to the principle of analytic continuation. And Riemann coped with it with a bang! I must say right away that all possible values ​​of the complex argument could be depicted only on a plane. But if the argument runs through the points of the plane, then how to represent the values ​​of the function? On the plane, you can limit yourself to zeros of the function, or you can take into service the third dimension, although in a good way four of them are needed for zeta. Well, you can also try using color. See for yourself! The real part of the argument is plotted along the abscissa axis, and the imaginary part is plotted along the ordinate axis. Well, now keep your ears open: all non-trivial zeros of the zeta function have a real part equal to ½. Here the fairy tale is over, and whoever listened - well done! Homework is to prove or disprove the Riemann Hypothesis, and don't try to copy from Atya! Think critically, do math, have fun! [Music is playing]

Wording

Equivalent formulations

Considerations about the truth of the hypothesis

Among the data that allow us to assume the truth of the conjecture, we can single out the successful proof of similar conjectures (in particular, the Riemann conjecture on manifolds over finite fields). This is the strongest theoretical argument that allows us to assume that the Riemann condition is satisfied for all zeta functions associated with automorphic mappings (English) Russian, which includes the classical Riemann hypothesis. The truth of a similar hypothesis has already been proven for the Selberg zeta function (English) Russian, similar in some respects to the Riemann function, and for the Goss zeta function (English) Russian(an analogue of the Riemann zeta function for function fields).

On the other hand, some of Epstein's zeta functions (English) Russian do not satisfy the Riemann condition, although they have an infinite number of zeros on the critical line. However, these functions are not expressed in terms of Euler series and are not directly related to automorphic mappings.

The "practical" arguments in favor of the truth of the Riemannian hypothesis include the computational verification of a large number of non-trivial zeros of the zeta function within the framework of the ZetaGrid project.

Related Issues

Two Hardy-Littlewood hypotheses

1. For anyone ε > 0 (\displaystyle \varepsilon >0) exist T 0 = T 0 (ε) > 0 (\displaystyle T_(0)=T_(0)(\varepsilon)>0), such that for and H = T 0 , 25 + ε (\displaystyle H=T^(0(,)25+\varepsilon )) the interval contains a zero of odd order of the function .
2. For anyone ε > 0 (\displaystyle \varepsilon >0) there are T 0 = T 0 (ε) > 0 (\displaystyle T_(0)=T_(0)(\varepsilon)>0) and c = c (ε) > 0 (\displaystyle c=c(\varepsilon)>0), which at T ⩾ T 0 (\displaystyle T\geqslant T_(0)) and the inequality N 0 (T + H) − N 0 (T) ⩾ c H (\displaystyle N_(0)(T+H)-N_(0)(T)\geqslant cH).

A. Selberg's hypothesis

In 1942, Atle Selberg investigated the Hardy-Littlewood problem 2 and proved that for any ε > 0 (\displaystyle \varepsilon >0) exist T 0 = T 0 (ε) > 0 (\displaystyle T_(0)=T_(0)(\varepsilon)>0) and c = c (ε) > 0 (\displaystyle c=c(\varepsilon)>0), such that for T ⩾ T 0 (\displaystyle T\geqslant T_(0)) and H = T 0 , 5 + ε (\displaystyle H=T^(0(,)5+\varepsilon )) the inequality N (T + H) − N (T) ⩾ c H log ⁡ T (\displaystyle N(T+H)-N(T)\geqslant cH\log T).

In turn, Atle Selberg hypothesized that it is possible to reduce the exponent a = 0 , 5 (\displaystyle a=0(,)5) for the quantity H = T 0 , 5 + ε (\displaystyle H=T^(0(,)5+\varepsilon )).

In 1984, A. A. Karatsuba proved that for a fixed condition 0 < ε < 0,001 {\displaystyle 0<\varepsilon <0{,}001} , large enough T (\displaystyle T) and H = T a + ε (\displaystyle H=T^(a+\varepsilon )), a = 27 82 = 1 3 − 1 246 (\displaystyle a=(\tfrac (27)(82))=(\tfrac (1)(3))-(\tfrac (1)(246))) interval (T , T + H) (\displaystyle (T,T+H)) contains at least c H ln ⁡ T (\displaystyle cH\ln T) real zeros of the Riemann zeta function ζ (1 2 + i t) (\displaystyle \zeta (\Bigl ()(\tfrac (1)(2))+it(\Bigr))). Thus, he confirmed Selberg's hypothesis.

The estimates by A. Selberg and A.A. Karatsuba are unimprovable in order of growth for T → + ∞ (\displaystyle T\to +\infty ).

In 1992, A. A. Karatsuba proved that the analogue Selberg's hypotheses valid for "almost all" intervals (T , T + H ] (\displaystyle (T,T+H]), H = T ε (\displaystyle H=T^(\varepsilon )), where ε (\displaystyle \varepsilon ) is an arbitrarily small fixed positive number. The method developed by Karatsuba makes it possible to investigate the zeros of the Riemann zeta function on "ultra-short" intervals of the critical line, that is, on the intervals (T , T + H ] (\displaystyle (T,T+H]), length H (\displaystyle H) which grows more slowly than any, even arbitrarily small, degree T (\displaystyle T). In particular, he proved that for any given numbers ε (\displaystyle \varepsilon ), ε 1 (\displaystyle \varepsilon _(1)) with the condition 0 < ε , ε 1 < 1 {\displaystyle 0<\varepsilon ,\varepsilon _{1}<1} almost all intervals (T , T + H ] (\displaystyle (T,T+H]) at H ⩾ exp ⁡ ( (ln ⁡ T) ε ) (\displaystyle H\geqslant \exp (\((\ln T)^(\varepsilon )\))) contain at least H (ln ⁡ T) 1 − ε 1 (\displaystyle H(\ln T)^(1-\varepsilon _(1))) function zeros ζ (1 2 + i t) (\displaystyle \zeta (\bigl ()(\tfrac (1)(2))+it(\bigr))). This estimate is very close to the one that follows from the Riemann hypothesis.

see also

Notes

1. Weisstein, Eric W. Riemann Hypothesis (English) on the Wolfram MathWorld website.
2. Rules for the Millennium Prizes
3. Which is somewhat unusual, since lim sup n → ∞ σ (n) n log ⁡ log ⁡ n = e γ . (\displaystyle \limsup _(n\rightarrow \infty )(\frac (\sigma (n))(n\ \log \log n))=e^(\gamma ).)
   The inequality is violated when n= 5040 and some smaller values, but Guy Robin in 1984 showed that it holds for all larger integers if and only if the Riemann Hypothesis is true.