The corresponding hamiltonian admits a selfadjoint extension that is tuned to the phase of the zeta function, on the critical line, in order to obtain the riemann zeros as bound states. Cogrowth of groups and a matrix of redheffer humphries, stephen p. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the riemann zeta function can be made. Riemann zeta spectrum adjacency matrix mathematica experiment with random 53regular graph. An infinite matrix whose nth leading principal minor is equal to mn for all n. The name is also used for some closely related analogues, such as the riemann hypothesis for curves over finite fields. We begin by defining the redheffer matrix of order n. Old and new arithmetic and analytic equivalences of the. Riemann hypotheses for global lfunctions the riemann hypothesis. A, is of numbertheoretic interest because the riemann hypothesis is true if and only if.
This suggests that matrix analysis methods may come to play a. If w is a nontrivial, cyclically reduced word in f, then there are exactly 2 n 2 choices of b a such that bwb 1 is freely reduced as written and, more generally, there are exactly 2n 22n 1 p1. Under this explanation, the famous riemann hypothesis is equivalent to levison theorem of scattering phaseshifts. Support for this spectral approach did not emerge until the. The zeta functions of riemann, selberg and ruelle are brie. Kudryavtseva 1 filip saidak peter zvengrowski abstract an exposition is given, partly historical and partly mathematical, of the riemann zeta function s and the associated riemann hypothesis. Quantum chaos, random matrix theory, and the riemann function. Symmetric matrices related to the mertens function, linear. Quantum physicists attack the riemann hypothesis quanta. Are there similar equivalent statement for grh generalized riemann hypothesis. They also then study the eigenvalues of this matrix. Dyson wrote a paper in 1975 which related random matrices and inverse scattering problem. We describe the permutations that contribute to its determinant and evaluate its permanent in terms of integer factorizations. Snaith school of mathematics, university of bristol, bristol bs8 1tw, uk december 8, 2009 abstract in the past dozen years random matrix theory has become a useful tool for conjec.
The redheffer matrix of a partially ordered set semantic scholar. The researchers spelled out several arguments for why the eigenvalues of their matrix are probably real, and why, in that case, the riemann hypothesis is probably correct, but they came short of proving it. Now define a, to be the leading n x n principal submatrix of a. In mathematics, a redheffer matrix, often denoted as studied by redheffer 1977, is a square 0,1 matrix whose entries a ij are 1 if i divides j or if j 1.
Experimental observations on the uncomputability of the riemann hypothesis. The determinants of the redheffer matrices are immediately tied to the riemann hypothesis rh. The riemann hypothesis is equivalent to this bound for the mobius function. The riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip. He then proceeds to study the eigenvalues of this matrix to glean insights into the hypothesis from this viewpoint. We use the integral representation 1 of riemann zeta function rz in the critical strip. Riemann s conjecture was that the real part of the nonobvious zeros is exactly 12. Riemann, dedekind, selberg, and ihara zetas audrey terras crm montreal 2009 more details can be. Several applications use the generalized riemann hypothesis for dirichlet lseries or zeta functions of number fields rather than just the riemann hypothesis. Apr 04, 2017 the researchers spelled out several arguments for why the eigenvalues of their matrix are probably real, and why, in that case, the riemann hypothesis is probably correct, but they came short of proving it. It is useful in some contexts to express dirichlet convolution, or convolved divisors sums, in terms of matrix products involving the transpose of the redheffer matrix. Hilbert and p olya put forward the idea that the zeros of the riemann zeta function may have a spectral origin.
Mertens equimodular matrices of redheffer type request pdf. The ihara zeta function of a nite graph is our main topic. Cogrowth of groups and a matrix of redheffer we will also need 105 lemma 2. In this paper, we explore a family of congruences over n. An infinite matrix whose nth leading principal minor is equal to m n for all n. Is it possible to draw connections between the factorization matrix def. Does this allow to come up with an alternative formulation of the riemann hypothesis eq. An infinite matrix whose n th leading principal minor is equal to m n for all n. The riemann hypothesis in characteristic p in historical. Selberg, on the zeros of riemanns zetafunction, skr. Collection of equivalent forms of riemann hypothesis. The riemann zeros as spectrum and the riemann hypothesis. According to 7, 8 the riemann hypothesis is equivalent to.
The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number. The existence of such an operator implies the celebrated riemann hypothesis that all complex zeros lie on the critical line res 12. We all know that riemann hypothesis rh has many equivalent statements. The book is gentle on the reader with definitions repeated, proofs split into logical sections, and graphical descriptions of the relations between.
Riemanns 1859 manuscript clay mathematics institute. Many statements equivalent to the riemann hypothesis have been found, though so far none of them have led to much progress in proving or disproving it. Equivalents of the riemann hypothesis by kevin broughan. Riemann hypothesis 2 however, the series on the right converges not just when s is greater than one, but more generally whenever s has positive real part. The first matrix theoretical approach to the riemann hypothesis was done by redheffer in 1977. The important relationship between riemann hypothesis and random matrices was found by freeman j. Jun 20, 2019 several applications use the generalized riemann hypothesis for dirichlet lseries or zeta functions of number fields rather than just the riemann hypothesis. In mathematics, the riemann hypothesis, proposed by bernhard riemann, is a conjecture about the distribution of the zeros of the riemann zetafunction which states that all nontrivial zeros of the riemann zeta function have real part 12. We generalize the redheffer matrix to finite posets that have a 0 element and find the analogous results in the more general situation. This paper seeks to explore whether the riemann hypothesis falls into a class of putatively unprovable mathematical conjectures, which arise as a result of unpredictable irregularity. Redheffer described an n\times n matrix of 0s and 1s the size of whose determinant is connected to the riemann hypothesis. Define the redheffer matrix to be the 01 matrix with entries and 0 otherwise.
This twovolume text presents the main known equivalents to rh using analytic and computational methods. Riemann hypothesis german sierraquantization of nonstandard hamiltonians and the riemann zeros rami ahmad elnabulsion the riemann hypothesis, complex scalings and logarithmic time reversal. Whether it will be difficult or easy to fill in the missing steps, at this point we cannot speculate, said brody. The hamiltonian of a quantum mechanical system has an a liated spectrum. Quantum chaos, random matrix theory, and the riemann. In this paper we show how some properties of riemann zeta function lead to the proof of the prime number theorem, the prime ideal theorem, and chebotarev density theorem. Equivalent forms of the riemann hypothesis the lumber room. Riemann checked the first few zeros of the zeta function by hand. Introduction if the riemann hypothesis rh is true then the nontrivial zeros of the riemann zeta function. From the results of numerical experiments, we formulate a conjecture about the growth of the quadratic norm of these matrices, which implies the riemann hypothesis.
Then, the riemann hypothesis is true if and only if the lefthand side, which is the sum of the divisors of, is also denoted. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number theorist and wrote just one paper on number theory in. Jul 06, 2010 then, the riemann hypothesis is true if and only if the lefthand side, which is the sum of the divisors of, is also denoted. Riemann hypothesis depending on an integer triangular matrix. Cogrowth of groups an d a ma trix of redheffer we will also need 105 lemma 2. As such, results on the growth of the mertens function are useful to the general study of prime numbers and the zeta function. The determinant of the redheffer divisibility matrix redheffer. In combination with riemanns and other estimates for number theoretic functions based on the zeta function, the riemann hypothesis would. The riemann hypothesis rh is perhaps the most important outstanding problem in mathematics. Booker 1208 notices of the ams volume 53, number 10 a lan turings final research paper1 11 described a numerical method for verifying the riemann hypothesis and its implementation on the manchester mark i, one of the earliest general purpose digital computers.
The determinant of the order n redheffer matrix is equal to m nso the riemann hypothesis upotesi also be stated as a condition on the growth of these determinants. Investigations of the redheffer matrix have been extended to group theory by. Riemann hypothesis and quantum mechanics article pdf available in journal of physics a mathematical and theoretical 4414 december 2010 with 378 reads how we measure reads. The determinant of the order n redheffer matrix is equal to mn, so the riemann hypothesis can also be stated as a condition on the growth of these determinants. Riemann conjectured his hypothesis in a remarkable short paper presented to the berlin academy in 1859, an english translation of which is an appendix to edwards book 4. One might vaguely hope that exploration in this relatively novel direction can shed insight on the riemann hypothesis.
The riemann hypothesis has various weaker consequences as well. For any m 0 let 91m denote the transpose of redheffers matrix of size m. It is the opinion of many mathematicians that the riemann hypothesis is probably the most important open problem in pure mathematics 3. In mathematics, a redheffer matrix, often denoted as studied by redheffer 1977, is a square 0,1 matrix whose entries aij are 1 if i divides j or if j 1. The determinant of the order n redheffer matrix is equal to m nso the riemann hypothesis can also be stated as a condition on the growth of these determinants. Dyson 2009 suggested trying to prove the riemann hypothesis by classifying. Riemann hypothesis, in number theory, hypothesis by german mathematician bernhard riemann concerning the location of solutions to the riemann zeta function, which is connected to the prime number theorem and has important implications for the distribution of prime numbers. Thus, this alternative series extends the zeta function from res 1 to the larger domain res 0, excluding the zeros of see dirichlet eta function. The riemann hypothesis for hilbert spaces of entire functions 2 is a condition on stieltjes spaces of entire functions which explains the observed shift in zeros and which implies the riemann conjecture if it can be applied to the euler zeta function. This bold idea is supported by several results and analogies involving number theory, random matrix theory and quantum chaos 917. Keating university of bristol university walk, clifton bristol bs8 1tw, uk. Using techniques similar to those of riemann, it is shown how to locate and count nontrivial zeros of s. A meeting on the riemann hypothesis, and on the theory of the zetafunction and other lfunctions. We then introduce some results related to riemann hypothesis, and artins conjecture as a corollary of generalized riemann hypothesis.
Riemann hypothesis is not easy to state in terms a nonmathematician can easily grasp. Riemann hypothesis for function elds, or curves, of characteristic pstarting with artins thesis in the year 1921, covering hasses work in the 1930s on elliptic elds and more, until weils nal proof in 1948. It is impossible to separate a cube into two cubes, or a biquadrate. The riemann hypothesis, the statement that the nontrivial zeros of the riemann zeta function have real part 1 2, is usually viewed as one of the most 1. Fermats last theorem was stated in 1637 and solved in 1994. Moxley iii, solving the riemann hypothesis with greens function and a gelfand triplet june 2018 abstract.
The dyadic monoid is a certain subset of the modular group sl2,z, which is a subgroup of the fuchsian group sl2,r, in turn a subgroup of the kleinian group sl2,c, all of which are intertwined with the riemann zeta and the structure of the set of rational numbers. The matrix a, is of numbertheoretic interest because the riemann hypothesis is true if and only if det a, onife for every positive e, and a proof that ldeta,i o n would constitute a new proof of the primenumber theorem. The riemann hypothesis is named after the fact that it is a hypothesis, which, as we all know, is the largest of the three sides of a right triangle. We use riordan matrices to find a large class of mertens equimodular. The redheffer matrix of a partially ordered set nasaads. Now we can use the redheffer matrices to translate the riemann hypothesis into the language of graph theory. New insight into proving maths milliondollar problem. Eigenvalues of the redheffer matrix and their relation to. This is the m x m which has j entry 1 if either i 1 or jli. Cogrowth of groups and a matrix of redheffer cogrowth of groups and a matrix of redheffer humphries, stephen p. Eigenvalues of the redheffer matrix and their relation to the.
Ive posted a paper on arxiv that outlines a linear algebra approach to number theory specifically, i have the following questions. And riemann hypothesis1 automorphic functions and fermats last theorem1 chunxuan jiang p. The fourcolor problem was stated in 1852 and solved in 1976. In other words, the riemann hypothesis is in some sense equivalent to saying that. We consider two determinant formulas for the ihara zeta, the riemann hypothesis, and connections with random matrix theory and quantum chaos. See jeffrey lagarias, an elementary problem equivalent to the riemann hypothesis pdf, arxiv. The model suggests a proof of the riemann hypothesis in the limit where the potentials vanish. Aug 04, 2019 the determinant of the order n redheffer matrix is equal to m nso the riemann hypothesis can also be stated as a condition on the growth of these determinants.
A 1927 result of polya is apparently that rh is equivalent to all of these polynomials being hyperbolic. Similarly, redheffer matrices can be used to relate the. That is, they all lie on a specific vertical line in the complex plane. We also provide an alternative proof of redheffers theorem based upon a lu. Yeah, im jealous the riemann hypothesis is named after the fact that it is a hypothesis, which, as we all know, is the largest of the three sides of a right triangle. Mertens equimodular matrices of redheffer type sciencedirect. This relationship is determined using a certain matrix of redheffer. Since the invertibility of the redheffer matrices are complicated by the initial column of ones in the matrix, it is often convenient to express.
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