Yuri Matiyasevich

Steklov Institute of Mathematics at St.Petersburg

We study a class of approximations to the Riemann zeta function introduced earlier by the second author on the basis of Euler product. This allows us to justify Euler Product Sieve for generation of prime numbers. Also we show that Bounded Riemann Hypothesis (stated in a paper by the fourth author) is equivalent to conjunction the Riemann Hypothesi...

It is well-known that the Riemann zeta function does not satisfy any exact polynomial differential equation. Here we present numerical evidence for the existence of approximate algebraic dependencies between the values of the alternating zeta function and its initial derivatives calculated at the same point or at several points in general position....

We consider the following two problems. We are given the values of several initial derivatives of the Riemann zeta function calculated at some (unknown to us) point a. • How could we calculate an approximate value of the function itself at the same point a without prior finding this number? • How could we find an approximate value of a itself? We s...

We consider the following two problems. We are given the values of several initial derivatives of the Riemann zeta function calculated at some (unknown to us) point a. How could we calculate an approximate value of the function itself at the same point a without prior finding this number? How could we find an approximate value of a itself? We...

The Riemann zeta function is an important number-theoretical tool for studying prime numbers. The first part of the paper is a short survey of some known results about this function. The emphasis is given to the possibility to formulate the celebrated Riemann Hypothesis as a statement from class \(\mathrm {\Pi }^0_1\) in the arithmetical hierarchy....

A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into peril. Now may be the time to reconcile theory and practice, i.e. precision and intuition, through the advent...

In the first part (\doi{10.13140/RG.2.2.29328.43528}) and in the second part (\doi{10.13140/RG.2.2.20434.22720}) the author presented numerical examples of calculation of approximate values of the zeros of the zeta-function, the alternating zeta function, and Davenport--Heilbronn function by by tools of linear algebra. This paper presents numerical...

In Part I (doi:10.13140/RG.2.2.29328.43528) the author demonstrated on a numerical example how a zero of the alternating zeta function η(s) = (1 − 2 × 2^ −s)ζ(s) can be approximately calculated by tools of the linear algebra. Unfortunately , this zero was a zero of the factor 1 − 2 × 2^ −s rather than being a zero of the zeta function. In this p...

Abstract: We consider a particular way to calculate approximations to the alternating zeta function via the determinants of certain matrices, and numerically examine characteristic polynomials, eigenvalues and eigenvectors of these matrices. It turns out that the ratios of entries of such eigenvectors lay inside very thin annuluses.

Let η(s)=∑n=1∞(−1)n+1n−s be the alternating zeta function. For a real number τ we define certain complex numbers bM,m(τ) and consider finite Dirichlet series υM(τ,s)=∑m=1MbM,m(τ)m−s and ηN(τ,s)=∑M=1NυM(τ,s). Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far. First, nume...

In this paper we show that there is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{52})=0$ (with integer coefficients) over the Gaussian ring $\mathbb Z[i]$ is solvable.

In this paper, we establish some congruences involving the Apéry numbers βn=∑k=0nnk2n+kk. For example, we show that ∑k=0n−1(11k2+13k+4)βk≡0(mod2n2) for any positive integer n, and ∑k=0p−1(11k2+13k+4)βk≡4p2+4p7Bp−5(modp8) for any prime p>3, where Bp−5 is the (p−5)th Bernoulli number. We also present certain relations between congruence properties of...

This paper demonstrates on numerical examples several plausible ways of calculating values of the Riemann zeta function via the Riemann–Siegel theta function. These new methods are mainly of theoretical interest for the study of the zeta function, they are not claimed to be more efficient than other already known techniques.

Under nearby properties of the Riemann's zeta function we mean properties of approximations to this function, or, more generally, properties of functions which are similar to the zeta function in certain respects. Of these properties the most interesting are those that cannot be formulated in terms of the zeta function alone. In the paper we consi...

The Riemann Hypothesis is reformulated as the statement that particular explicitly presented register machine with 29 registers and 130 instructions never halts.

Riemann’s zeta function (defined by a certain Dirichlet series) satisfies an identity known as the functional equation. H. Hamburger established that the function is identified by the equation inside a wide class of functions defined by Dirichlet series. Riemann’s zeta function is a member of a large family of functions with similar properties, in...

In this paper we establish some congruences involving the Ap\'ery numbers $\beta_{n}=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}$ $(n=0,1,2,\ldots)$. For example, we show that $$\sum_{k=0}^{n-1}(11k^2+13k+4)\beta_k\equiv0\pmod{2n^2}$$ for any positive integer $n$, and $$\sum_{k=0}^{p-1}(11k^2+13k+4)\beta_k\equiv 4p^2+4p^7B_{p-5}\pmod{p^8}$$ for any...

This paper demonstrate on numerical examples several plausible ways of calculating values of the Riemann zeta function via the Riemann-Siegel theta function. These new methods are mainly of theoretical interest for the study of the zeta function, they are not claimed to be more efficient than other already known techniques.

The Riemann Hypothesis has many equivalent reformulations. Some of them are arithmetical, that is, thewy are statements about properties of integers or natural numbers. Among them the reformulations with the simplest logical structure are those from the class Π 0 1 from the arithmetical hierachy, that is, having the form "for every x 1 ,. .. , x m...

The Riemann Hypothesis is reformulated as the statement that a particular explicitly presented register machine with 24 registers and 128 instructions never halts. _____________________________________________________________________ The original preprint contains errors. The corrected version is published in Theoretical Computer Science, volume 8...

According to a converse theorem of Hamburger type, Ramanujan's tau numbers are completely determined by the functional equation for Ramanujan's tau L-function. The paper presents a computational method for "extracting" the numbers from the equation. Published in INTEGERS (ISSN 1553-1732), vol. 18A, 2018 (http://math.colgate.edu/~integers/vol18a.ht...

The paper present a naturally arising function vanishing at the non-trivial zeroes of Riemann's zeta function and having one more zero lying on the critical line.

Riemann's zeta-function (defined by a certain Dirichlet series) satisfies an identity known as the functional equation. H. Hamburger established that the function is identified by the equation inside a wide class of functions defined by Dirichlet series. Riemann's zeta-function is a member of a large family of functions with similar properties, i...

The paper presents a technique for the automatic calculation of Belyi functions for trees with weighted edges. Bibliography: 20 titles.

The paper demonstrates by numerical examples a nontraditional way to get high precision values of Riemann’s zeta function inside the critical strip by using the functional equation and the factors from the Euler product corresponding to a very small number of primes. For example, the three initial primes produce more than 50 correct decimal digits...

The termination problem for semi-Thue systems asks whether all derivations for a given word in a given semi-Thue system are finite, i.e., all derivations terminate after finite number of steps. This problem is known to be undecidable, there is a standard reduction of the halting problem of the Turing machines into termination problem; moreover, one...

The paper describes computer experiments for calculating zeros and values of Riemann's zeta function and of its first derivative inside the critical strip and to the left of it with the help of finite Dirichlet series the coefficients of which are defined via initial nontrivial zeros of the zeta function.

The paper presents the history of the negative solution of Hilbert’s tenth problem , the role played in it by Martin Davis, consequent modifications of the original proof of DPRM-theorem, its improvements and applications, and a new (2010) conjecture of Martin Davis .

The Four Color Conjecture, which in 1977 became the Four Color Theorem of Kenneth Appel and Wolfgang Haken, is famous for the number of its reformulations. Three of them found by the author at different time are discussed in this paper.

We present a parallel algorithm for calculating determinants of matrices in arbitrary precision arithmetic on computer clusters. This algorithm limits data movements between the nodes and computes not only the determinant but also all the minors corresponding to a particular row or column at a little extra cost, and also the determinants and minors...

The positivity of the sum from the title is the first condition in the well-known criterium for the validity of the Riemann Hypothesis suggested by X.-J. Li. In the paper this value is represented as an infinite sum with positive summands.

The finite Dirichlet series from the title are defined by the condition that they vanish at as many initial zeroes of the zeta function as possible. It turned out that such series can produce extremely good approximations to the values of Riemann's zeta function inside the critical strip. In addition, the coefficients of these series have remarkabl...

We present a parallel algorithm for calculating very large determinants with arbitrary precision on computer clusters. This algorithm minimises data movements between the nodes and computes not only the determinant but also all minors corresponding to a particular row or column at a little extra cost, and also the determinants and minors of all sub...

Nikolai Aleksandrovich Shanin (obituary) Abstract: Nikolai Aleksandrovich Shanin, a remarkable mathematician, pedagogue, and person, passed away on 17 September 2011. He was born on 25 May 1919 in Pskov in a doctor’s family. In 1935 he entered the Mathematics-Mechanic s Faculty of Leningrad State University, and in 1939 he began Ph.D. studies ther...

This survey presents various theorems (obtained mainly by specialists in mathematical logic and computability theory) stating the impossibility of algorithms for solving certain Diophantine problems. Often the technique developed for obtaining such “negative” results also allows one to prove many “positive” theorems on the possibility of formulatin...

The talk is devoted to methods for calculating approximate values of the non-trivial zeros of Riemann's zeta functions, its values and values of its first derivative inside and outside the critical strip. These methods have been recently discovered by the author in the course of intensive numerical calculations, and so far there is no theoretical e...

In mathematics sometimes methods from one area can be fruitfully applied for getting results in another area, occasionally looking very remote from the other area. A well-known example is given by analytic geometry that enables us, besides proving "elementary" geometrical theorems, to establish otherwise untractable results like unsolvability of th...

It is shown that the absolute values of Riemann's zeta function and two related functions strictly decrease when the imaginary part of the argument is fixed to any number with absolute value at least 8 and the real part of the argument is negative and increases up to 0; extending this monotonicity to the increase of the real part up to 1/2 is shown...

The celebrated theorem established by Martin Davis, Hilary Putnam, and Julia Robinson in 1961 states that every effectively enumerable set of natural numbers has an exponential Diophantine representation. This theorem was improved by the author in two ways: to the existence of a Diophantine representation, to the existence of a so-called single-fol...

The paper presents yet another way to reformulate the Four Colour Conjecture as a statement concerning conditional probabilities of certain events involving planar graphs.

We propose a new class of formulas, which, just as the Euler-Maclaurin formula, can be used to calculate the approximate value of an infinite sum by approximating the corresponding integral. Key wordsEuler-Maclaurin formula-Bernoulli number-quadrature formula-Stieltjes constant-Riemann zeta function-Simpson’s formula

We prove that for given morphisms g,h:{a 1 ,a 2 ,⋯,a n }→B * , it is decidable whether or not there exists a word w in the regular language a 1 * a 2 * ⋯a n * such that g(w)=h(w). In other words, we prove that the Post Correspondence Problem is decidable if the solutions are restricted to be from this special language. This yields a nice example of...

The representation is essentially the same as that given by J.P.Nagle in J. Comb. Theory (B), 1971, 10:1, 42--59. The distinction is in the definition of the weighting function via the number of flows. This new definition allows one to deduce a number of corollaries, in particular, the following. A) The chromatic polynomial of a connected planar gr...

A new method of coding Diophantine equations is introduced. This method allows (i) checking that a coded sequence of natural numbers is a solution of a coded equation without decoding; (ii) defining by a purely existential formula, the code of an equation equivalent to a system of indefinitely many copies of an equation represented by its code.The...

Given two trees (a target T and a pattern P) and a natural number w, window embedded subtree problems consist in deciding whether P occurs as an embedded subtree of T and/or finding the number of size (at most) w windows of T which contain pattern P as an embedded subtree. P is an embedded subtree of T if P can be obtained by deleting some nodes fr...

It is proved that all recursively enumerable sets of natural numbers can be represented by arithmetic formulas (of two kinds) with only 3 quantifiers.

L.M.Vitaver [1962] and G.I.Minty [1962] suggested criteria for vertex colorability of a graph in at most a given number of colors; these criteria are stated in terms of the orientation of the edges. One additional criterion of this kind is given here.

This is a survey of a century-long history of interplay between Hilbert’s tenth problem (about solvability of Diophantine equations) and different notions and ideas from Computability Theory. The present paper is an extended version of [83].

In this paper we give a detailed account of some results obtained by a group of specialists in mathematical logic in connection with an investigation of Hilbert's 10th problem. This problem was formulated in his well-known lecture [1], in the following way. "10. THE PROBLEM OF THE SOLUBILITY OF DIOPHANTINE EQUATIONS. Given a Diophantine equation in...

The Riemann Hypothesis can be reformulated as statements about the eigenvalues of certain matrices whose entries are defined in terms of the Taylor coefficients of the zeta function. These eigenvalues exhibit interesting visual patterns allowing one to state a number of conjectures. The Hankel matrices introduced here are obtained, by rearranging o...

The Riemann Hypothesis is reformulated as statements about eigenvalues of some matrices entries of which are defined via Taylor coefficient of the zeta function. These eigenvalues demonstrate interesting visual patterns allowing one to state a number of conjectures.

Given two strings (a text t of length n and a pattern p) and a natural number w, window subsequence problems consist in deciding whether p occurs as a subsequence of t and/or nding,the number of size (at most) w windows of text t which contain pattern p as a subsequence, i.e. the letters of pattern p occur in the text window, in the same order as i...

Given q+1q+1 strings (a text t of length n and q patterns m1,…,mqm1,…,mq) and a natural number w, the multiple serial episode matching problem consists in finding the number of size w windows of text t which contain patterns m1,…,mqm1,…,mq as subsequences, i.e., for each mimi, if mi=p1,…,pkmi=p1,…,pk, the letters p1,…,pkp1,…,pk occur in the window,...

The notion of a word, considered as an element of a free monoid, has been long ago generalized to the notion of a trace, an element of a partially commutative monod. Traces turned out to be useful tool for studying concurrency. Every decision problem for words can be generalized to corresponding problem for traces. The main content of the paper is...

This is a survey of a century long history of interplay between Hilbert’s tenth problem (about solvability of Diophantine equations) and different notions and ideas from the Computability Theory.

We show that the accessibility problem, the common descendant problem, the termination problem and the uniform termination problem are undecidable for 3-rules semi-Thue systems. As a corollary we obtain the undecidability of the Post correspondence problem for 7 rules.

The results of elimination of quantifiers over natural numbers and some implifications of these results on the power of computer algebra systems are discussed. A mathematician managed to eliminate the assumptiion about long arithmatical progressions of prime numbers, replacing it by the trivial existence of arbitrary long arithmatical progressions...

With every triangulation of sphere we associate in a natural way a probabilistic space and define several random events. The Four Color Conjecture turns out to be equivalent to di#erent statements about positive correlation among some pairs of these events.

With an arbitrary graph G having n vertices and m edges, and with an arbitrary natural number p, we associate in a natural way a polynomial R(x1,...,xn) with integer coefficients such that the number of colorings of the vertices of the graph G in p colors is equal to pm-nR(0,...,0). Also with an arbitrary maximal planar graph G, we associate severa...