In this paper we study the function , for z∈C. We derive a functional equation that relates G(z) and G(1−z) for all z∈C, and we prove: 1) that G and the Riemann zeta function ζ have exactly the same zeros in the cr...In this paper we study the function , for z∈C. We derive a functional equation that relates G(z) and G(1−z) for all z∈C, and we prove: 1) that G and the Riemann zeta function ζ have exactly the same zeros in the critical region D:= {z∈C:ℜz∈(0,1)};2) the Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line := {z∈D:ℜz =1/2};and that 3) all the zeros of the Riemann zeta function located on the critical line are simple.展开更多
The Riemann hypothesis is intimately connected to the counting functions for the primes. In particular, Perron’s explicit formula relates the prime counting function to fixed points of iterations of the explicit form...The Riemann hypothesis is intimately connected to the counting functions for the primes. In particular, Perron’s explicit formula relates the prime counting function to fixed points of iterations of the explicit formula with particular relations involving the trivial and non-trivial roots of the Riemann Zeta function and the Primes. The aim of the paper is to demonstrate this relation at the fixed points of iterations of explicit formula, defined by functions of the form limT∈Ν→∞fT(zw)=zw,where, zwis a real number.展开更多
This paper indicates the problem of the famous Riemann hypothesis (RH), which has been well-verified by a definite answering method using a Bose-Einstein Condensate (BEC) phase. We adopt mathematical induction, mappin...This paper indicates the problem of the famous Riemann hypothesis (RH), which has been well-verified by a definite answering method using a Bose-Einstein Condensate (BEC) phase. We adopt mathematical induction, mappings, and laser photons governed by electromagnetically induced transparency (EIT) to examine the existence of the RH. In considering the well-developed as Riemann zeta function, we find that the existence of RH has a corrected and self-consistent solution. Specifically, there is the only one pole at s = 1 on the complex plane for Riemann’s functions, which generalizes to all non-trivial zeros while s > 1. The essential solution is based on the BEC phases and on the nature of the laser photon(s). This work also incorporates Heisenberg commutators [ x^,p^]=1/2in the field of quantum mechanics. We found that a satisfactory solution for the RH would be incomplete without the formalism of Heisenberg commutators, BEC phases, and EIT effects. Ultimately, we propose the application of qubits in connection with the RH.展开更多
By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functionswith an integral representation of the form with a real-valued function which is non-increasing a...By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functionswith an integral representation of the form with a real-valued function which is non-increasing and decreases in infinity more rapidly than any exponential functions , possesses zeros only on the imaginary axis. The Riemann zeta function as it is known can be related to an entire functionwith the same non-trivial zeros as . Then after a trivial argument displacement we relate it to a function with a representation of the form where is rapidly decreasing in infinity and satisfies all requirements necessary for the given proof of the position of its zeros on the imaginary axis z=iy by the second mean-value theorem. Besides this theorem we apply the Cauchy-Riemann differential equation in an integrated operator form derived in the Appendix B. All this means that we prove a theorem for zeros of on the imaginary axis z=iy for a whole class of function which includes in this way the proof of the Riemann hypothesis. This whole class includes, in particular, also the modified Bessel functions for which it is known that their zeros lie on the imaginary axis and which affirms our conclusions that we intend to publish at another place. In the same way a class of almost-periodic functions to piece-wise constant non-increasing functions belong also to this case. At the end we give shortly an equivalent way of a more formal description of the obtained results using the Mellin transform of functions with its variable substituted by an operator.展开更多
In the following pages I will try to give a solution to this very known unsolved problem of theory of numbers. The solution is given here with an important analysis of the proof of formula (4.18), with the introductio...In the following pages I will try to give a solution to this very known unsolved problem of theory of numbers. The solution is given here with an important analysis of the proof of formula (4.18), with the introduction of special intervals between square of prime numbers that I call silver intervals . And I make introduction of another also new mathematic phenomenon of logical proposition “In mathematics nothing happens without reason” for which I use the ancient Greek term “catholic information”. From the theorem of prime numbers we know that the expected multitude of prime numbers in an interval is given by formula ?considering that interval as a continuous distribution of real numbers that represents an elementary natural numbers interval. From that we find that in the elementary interval around of a natural number ν we easily get by dx=1 the probability that has the ν to be a prime number. From the last formula one can see that the second part of formula (4.18) is absolutely in agreement with the above theorem of prime numbers. But the benefit of the (4.18) is that this formula enables correct calculations in set N on finding the multitude of twin prime numbers, in contrary of the above logarithmic relation which is an approximation and must tend to be correct as ν tends to infinity. Using the relationship (4.18) we calculate here the multitude of twins in N, concluding that this multitude tends to infinite. But for the validity of the computation, the distribution of the primes in a random silver interval is examined, proving on the basis of catholic information that the density of primes in the same random silver interval is statistically constant. Below, in introduction, we will define this concept of “catholic information” stems of “information theory” [1] and it is defined to use only general forms in set N, because these represent the set N and not finite parts of it. This concept must be correlated to Riemann Hypothesis.展开更多
This article B is almost autonomous because it can be read independently from the first published article A [1] using only a few parts of the article A. Be-low are given instructions so to need the reader study only o...This article B is almost autonomous because it can be read independently from the first published article A [1] using only a few parts of the article A. Be-low are given instructions so to need the reader study only on few places of the article A. Also, in the part A of Introduction, here, you will find simple and useful definitions and the strategy we are going to follow as well useful new theorems (also and in Section 5, which have been produced in this solution). So the published solution of twin’s problem can now be easily understood. The inequalities (4.17), (4.18) of Article A are proved here in Section 4 by a new clear method, without the possible ambiguity of the text between the relations (4.14), (4.16) of the Article A. Also we complete the proof for the twin’s distri-bution which we use. At the end here are presented the Conclusions, the No-menclatures and the numerical control of the proof, which is probably useful as well in coding methods. For a general and convincing picture is sufficient, a study from the beginning of this article B until the end of the part A of the In-troduction here as well a general glance on the Section 5 and on the Conclu-sions below.展开更多
The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in t...The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in the complex domain by pictures. It can be seen how the zeros in finite approximations approach to the genuine zeros in the transition to higher-order approximation and in case of the Gaussian (bell) function that they go with great uniformity to infinity in the complex plane. A limiting transition from the modified Bessel functions to a Gaussian function is discussed and represented in pictures. In an Appendix a new building stone to a full proof of the Riemann hypothesis using the Second mean-value theorem is presented.展开更多
In this research we are going to define two new concepts: a) “The Potential of Events” (EP) and b) “The Catholic Information” (CI). The term CI derives from the ancient Greek language and declares all the Catholic...In this research we are going to define two new concepts: a) “The Potential of Events” (EP) and b) “The Catholic Information” (CI). The term CI derives from the ancient Greek language and declares all the Catholic (general) Logical Propositions (<img src="Edit_5f13a4a5-abc6-4bc5-9e4c-4ff981627b2a.png" width="33" height="21" alt="" />) which will true for every element of a set A. We will study the Riemann Hypothesis in two stages: a) By using the EP we will prove that the distribution of events e (even) and o (odd) of Square Free Numbers (SFN) on the axis Ax(N) of naturals is Heads-Tails (H-T) type. b) By using the CI we will explain the way that the distribution of prime numbers can be correlated with the non-trivial zeros of the function <em>ζ</em>(<em>s</em>) of Riemann. The Introduction and the Chapter 2 are necessary for understanding the solution. In the Chapter 3 we will present a simple method of forecasting in many very useful applications (e.g. financial, technological, medical, social, etc) developing a generalization of this new, proven here, theory which we finally apply to the solution of RH. The following Introduction as well the Results with the Discussion at the end shed light about the possibility of the proof of all the above. The article consists of 9 chapters that are numbered by 1, 2, …, 9.展开更多
Short Retraction Notice? The paper does not meet the standards of 'Applied Mathematics'.? This article has been retracted to straighten the academic record. In making this decision the Editorial Board follows ...Short Retraction Notice? The paper does not meet the standards of 'Applied Mathematics'.? This article has been retracted to straighten the academic record. In making this decision the Editorial Board follows COPE's Retraction Guidelines. The aim is to promote the circulation of scientific research by offering an ideal research publication platform with due consideration of internationally accepted standards on publication ethics. The Editorial Board would like to extend its sincere apologies for any inconvenience this retraction may have caused.? Editor guiding this retraction: Editorial Board of AM.? Please see the article page for more details. The full retraction notice in PDF is preceding the original paper which is marked 'RETRACTED'.展开更多
A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppo...A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppose ξ(s) = ξ1(a,b) + iξ2(a,b) = 0 but ζ(s) = ζ1(a,b) + iζ2(a,b) ≠ 0 with s = a + ib at first. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is to suppose that ζ1(a,b) = 0 and ζ2(a,b) = 0. However, by using the compassion method of infinite series, it is proved that ζ1(a,b) ≠ 0 and ζ2(a,b) ≠ 0. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.展开更多
In this paper, we introduce the application of random matrices in mathe- matical physics including Riemann-Hilbert problem, nuclear physics, big data, image processing, compressed sensing and so on. We start with the ...In this paper, we introduce the application of random matrices in mathe- matical physics including Riemann-Hilbert problem, nuclear physics, big data, image processing, compressed sensing and so on. We start with the Riemann- Hilbert problem and state the relation between the probability distribution of nontrivial zeros and the eigenvalues of the random matrices. Through the random matrices theory, we derive the distribution of Neutron width and prob- ability density between energy levels. In addition, the application of random matrices in quantum chromo dynamics and two dimensional Einstein gravity equations is also present in this paper.展开更多
The Riemann hypothesis is a well-known mathematical problem that has been in suspense for 162 years. Its difficulty lies in the fact that it is involved in an infinite integral which includes infinite series with comp...The Riemann hypothesis is a well-known mathematical problem that has been in suspense for 162 years. Its difficulty lies in the fact that it is involved in an infinite integral which includes infinite series with complex variables. To detour this is in vain, since all the messages are hid in it. To unscramble them, there is a totally new idea, that is, the “periodicity”! By investigating the numerical approximate values of zero points, an explicit distribution law on the critical line was found. To accord with this, a periodic form for the real part of Xi function was constructed and rigidly proved. The Riemann hypothesis can be divided into three progressive propositions. The first proposition (the number of zero points in the critical strip satisfies a certain estimation) had been proved in 1905. The second proposition (the number of zero points on the critical line satisfies the same estimation as in the critical strip) is ever in suspense. It can be solved perfectly with the newly found “periodicity”. The third proposition (all the nontrivial zero points are on the critical line), that is, the Riemann hypothesis, is also true. The proof is a combination of the symmetry, monotonicity, periodicity of the Xi function and the extremum principle of the harmonic functions. It is the moment to draw full stop for this suspending problem.展开更多
We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating t...We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other information which has been waiting for a rigorous proof. Such a task is partially fulfilled in this paper, where we succeeded among other things, to prove a theorem about general Dirichlet series having as corollary the Speiser’s theorem. We have also proved that those functions do not possess multiple zeros of order higher than 2 and the double zeros have very particular locations. Moreover, their derivatives have only simple zeros. With these results at hand, we revisited GRH for a simplified proof.展开更多
The Riemann hypothesis is part of Hilbert’s eighth problem in David Hilbert’s list of 23 unsolved problems. It is also one of the Clay Mathematics Institute’s Millennium Prize Problems. Some mathematicians consider...The Riemann hypothesis is part of Hilbert’s eighth problem in David Hilbert’s list of 23 unsolved problems. It is also one of the Clay Mathematics Institute’s Millennium Prize Problems. Some mathematicians consider it the most important unresolved problem in pure mathematics. Many mathematicians made a lot of efforts;they don’t have to prove the Riemann hypothesis. In this paper, I use the analytic methods to deny the Riemann Hypothesis;if there’s something wrong, please criticize and correct me.展开更多
文摘In this paper we study the function , for z∈C. We derive a functional equation that relates G(z) and G(1−z) for all z∈C, and we prove: 1) that G and the Riemann zeta function ζ have exactly the same zeros in the critical region D:= {z∈C:ℜz∈(0,1)};2) the Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line := {z∈D:ℜz =1/2};and that 3) all the zeros of the Riemann zeta function located on the critical line are simple.
文摘The Riemann hypothesis is intimately connected to the counting functions for the primes. In particular, Perron’s explicit formula relates the prime counting function to fixed points of iterations of the explicit formula with particular relations involving the trivial and non-trivial roots of the Riemann Zeta function and the Primes. The aim of the paper is to demonstrate this relation at the fixed points of iterations of explicit formula, defined by functions of the form limT∈Ν→∞fT(zw)=zw,where, zwis a real number.
文摘This paper indicates the problem of the famous Riemann hypothesis (RH), which has been well-verified by a definite answering method using a Bose-Einstein Condensate (BEC) phase. We adopt mathematical induction, mappings, and laser photons governed by electromagnetically induced transparency (EIT) to examine the existence of the RH. In considering the well-developed as Riemann zeta function, we find that the existence of RH has a corrected and self-consistent solution. Specifically, there is the only one pole at s = 1 on the complex plane for Riemann’s functions, which generalizes to all non-trivial zeros while s > 1. The essential solution is based on the BEC phases and on the nature of the laser photon(s). This work also incorporates Heisenberg commutators [ x^,p^]=1/2in the field of quantum mechanics. We found that a satisfactory solution for the RH would be incomplete without the formalism of Heisenberg commutators, BEC phases, and EIT effects. Ultimately, we propose the application of qubits in connection with the RH.
文摘By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functionswith an integral representation of the form with a real-valued function which is non-increasing and decreases in infinity more rapidly than any exponential functions , possesses zeros only on the imaginary axis. The Riemann zeta function as it is known can be related to an entire functionwith the same non-trivial zeros as . Then after a trivial argument displacement we relate it to a function with a representation of the form where is rapidly decreasing in infinity and satisfies all requirements necessary for the given proof of the position of its zeros on the imaginary axis z=iy by the second mean-value theorem. Besides this theorem we apply the Cauchy-Riemann differential equation in an integrated operator form derived in the Appendix B. All this means that we prove a theorem for zeros of on the imaginary axis z=iy for a whole class of function which includes in this way the proof of the Riemann hypothesis. This whole class includes, in particular, also the modified Bessel functions for which it is known that their zeros lie on the imaginary axis and which affirms our conclusions that we intend to publish at another place. In the same way a class of almost-periodic functions to piece-wise constant non-increasing functions belong also to this case. At the end we give shortly an equivalent way of a more formal description of the obtained results using the Mellin transform of functions with its variable substituted by an operator.
文摘In the following pages I will try to give a solution to this very known unsolved problem of theory of numbers. The solution is given here with an important analysis of the proof of formula (4.18), with the introduction of special intervals between square of prime numbers that I call silver intervals . And I make introduction of another also new mathematic phenomenon of logical proposition “In mathematics nothing happens without reason” for which I use the ancient Greek term “catholic information”. From the theorem of prime numbers we know that the expected multitude of prime numbers in an interval is given by formula ?considering that interval as a continuous distribution of real numbers that represents an elementary natural numbers interval. From that we find that in the elementary interval around of a natural number ν we easily get by dx=1 the probability that has the ν to be a prime number. From the last formula one can see that the second part of formula (4.18) is absolutely in agreement with the above theorem of prime numbers. But the benefit of the (4.18) is that this formula enables correct calculations in set N on finding the multitude of twin prime numbers, in contrary of the above logarithmic relation which is an approximation and must tend to be correct as ν tends to infinity. Using the relationship (4.18) we calculate here the multitude of twins in N, concluding that this multitude tends to infinite. But for the validity of the computation, the distribution of the primes in a random silver interval is examined, proving on the basis of catholic information that the density of primes in the same random silver interval is statistically constant. Below, in introduction, we will define this concept of “catholic information” stems of “information theory” [1] and it is defined to use only general forms in set N, because these represent the set N and not finite parts of it. This concept must be correlated to Riemann Hypothesis.
文摘This article B is almost autonomous because it can be read independently from the first published article A [1] using only a few parts of the article A. Be-low are given instructions so to need the reader study only on few places of the article A. Also, in the part A of Introduction, here, you will find simple and useful definitions and the strategy we are going to follow as well useful new theorems (also and in Section 5, which have been produced in this solution). So the published solution of twin’s problem can now be easily understood. The inequalities (4.17), (4.18) of Article A are proved here in Section 4 by a new clear method, without the possible ambiguity of the text between the relations (4.14), (4.16) of the Article A. Also we complete the proof for the twin’s distri-bution which we use. At the end here are presented the Conclusions, the No-menclatures and the numerical control of the proof, which is probably useful as well in coding methods. For a general and convincing picture is sufficient, a study from the beginning of this article B until the end of the part A of the In-troduction here as well a general glance on the Section 5 and on the Conclu-sions below.
文摘The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in the complex domain by pictures. It can be seen how the zeros in finite approximations approach to the genuine zeros in the transition to higher-order approximation and in case of the Gaussian (bell) function that they go with great uniformity to infinity in the complex plane. A limiting transition from the modified Bessel functions to a Gaussian function is discussed and represented in pictures. In an Appendix a new building stone to a full proof of the Riemann hypothesis using the Second mean-value theorem is presented.
文摘In this research we are going to define two new concepts: a) “The Potential of Events” (EP) and b) “The Catholic Information” (CI). The term CI derives from the ancient Greek language and declares all the Catholic (general) Logical Propositions (<img src="Edit_5f13a4a5-abc6-4bc5-9e4c-4ff981627b2a.png" width="33" height="21" alt="" />) which will true for every element of a set A. We will study the Riemann Hypothesis in two stages: a) By using the EP we will prove that the distribution of events e (even) and o (odd) of Square Free Numbers (SFN) on the axis Ax(N) of naturals is Heads-Tails (H-T) type. b) By using the CI we will explain the way that the distribution of prime numbers can be correlated with the non-trivial zeros of the function <em>ζ</em>(<em>s</em>) of Riemann. The Introduction and the Chapter 2 are necessary for understanding the solution. In the Chapter 3 we will present a simple method of forecasting in many very useful applications (e.g. financial, technological, medical, social, etc) developing a generalization of this new, proven here, theory which we finally apply to the solution of RH. The following Introduction as well the Results with the Discussion at the end shed light about the possibility of the proof of all the above. The article consists of 9 chapters that are numbered by 1, 2, …, 9.
文摘Short Retraction Notice? The paper does not meet the standards of 'Applied Mathematics'.? This article has been retracted to straighten the academic record. In making this decision the Editorial Board follows COPE's Retraction Guidelines. The aim is to promote the circulation of scientific research by offering an ideal research publication platform with due consideration of internationally accepted standards on publication ethics. The Editorial Board would like to extend its sincere apologies for any inconvenience this retraction may have caused.? Editor guiding this retraction: Editorial Board of AM.? Please see the article page for more details. The full retraction notice in PDF is preceding the original paper which is marked 'RETRACTED'.
文摘A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppose ξ(s) = ξ1(a,b) + iξ2(a,b) = 0 but ζ(s) = ζ1(a,b) + iζ2(a,b) ≠ 0 with s = a + ib at first. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is to suppose that ζ1(a,b) = 0 and ζ2(a,b) = 0. However, by using the compassion method of infinite series, it is proved that ζ1(a,b) ≠ 0 and ζ2(a,b) ≠ 0. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.
文摘In this paper, we introduce the application of random matrices in mathe- matical physics including Riemann-Hilbert problem, nuclear physics, big data, image processing, compressed sensing and so on. We start with the Riemann- Hilbert problem and state the relation between the probability distribution of nontrivial zeros and the eigenvalues of the random matrices. Through the random matrices theory, we derive the distribution of Neutron width and prob- ability density between energy levels. In addition, the application of random matrices in quantum chromo dynamics and two dimensional Einstein gravity equations is also present in this paper.
文摘The Riemann hypothesis is a well-known mathematical problem that has been in suspense for 162 years. Its difficulty lies in the fact that it is involved in an infinite integral which includes infinite series with complex variables. To detour this is in vain, since all the messages are hid in it. To unscramble them, there is a totally new idea, that is, the “periodicity”! By investigating the numerical approximate values of zero points, an explicit distribution law on the critical line was found. To accord with this, a periodic form for the real part of Xi function was constructed and rigidly proved. The Riemann hypothesis can be divided into three progressive propositions. The first proposition (the number of zero points in the critical strip satisfies a certain estimation) had been proved in 1905. The second proposition (the number of zero points on the critical line satisfies the same estimation as in the critical strip) is ever in suspense. It can be solved perfectly with the newly found “periodicity”. The third proposition (all the nontrivial zero points are on the critical line), that is, the Riemann hypothesis, is also true. The proof is a combination of the symmetry, monotonicity, periodicity of the Xi function and the extremum principle of the harmonic functions. It is the moment to draw full stop for this suspending problem.
文摘We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other information which has been waiting for a rigorous proof. Such a task is partially fulfilled in this paper, where we succeeded among other things, to prove a theorem about general Dirichlet series having as corollary the Speiser’s theorem. We have also proved that those functions do not possess multiple zeros of order higher than 2 and the double zeros have very particular locations. Moreover, their derivatives have only simple zeros. With these results at hand, we revisited GRH for a simplified proof.
文摘The Riemann hypothesis is part of Hilbert’s eighth problem in David Hilbert’s list of 23 unsolved problems. It is also one of the Clay Mathematics Institute’s Millennium Prize Problems. Some mathematicians consider it the most important unresolved problem in pure mathematics. Many mathematicians made a lot of efforts;they don’t have to prove the Riemann hypothesis. In this paper, I use the analytic methods to deny the Riemann Hypothesis;if there’s something wrong, please criticize and correct me.
