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Yuri Matiyasevich

Publications and source records attributed to Yuri Matiyasevich.

18 recordsLinked to original sources
arXiv2026

Hypothetical connection of the theta functions of Dirichlet characters with the real cyclotomic fields

We consider a possible approach to the Lindelöf hypothesis for Dirichlet $L$-functions. It is based on a special form of the functional equation for the corresponding theta functions. To estimate $L_χ(0.5+it)$ we need to solve certain systems of linear equations. The entries to the corresponding matrices are formed by the summands to the series for theta functions and their derivatives. Numerical data suggest that the inverse matrices have a deep structure and allow us to state a number of conjectures. In particular, it seems that for a character modulo $q$ certain entries to the inverse matrices tend to finite limits when the sizes of the matrices run over arithmetical progressions with step $2q$. Moreover, these limits belong to the real cyclotomic field $\mathbb{Q}(\cos(π/q))$ (up to a scaling factor of $\sqrt{q}$).

math.NT↗
arXiv2026

Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function

We prove four explicit continued fraction representations for the Lerch transcendent $Φ(z, s, M+1)$, where $\Re M>0$ and $(z,s)\in\{(-1,1),(-1,2),(1,2),(1,3)\}$. All of the continued fractions have unit partial numerators, while partial denominators depend on the parameter $M$. The proofs combine equivalent transformations of continued fractions, generalized hypergeometric functions, three-term recurrences, and asymptotic analysis of minimal solutions. For $z=1$, the corresponding representations give continued fractions for the values of the Hurwitz zeta function $ζ(2, M+1)$ and $ζ(3, M+1)$.

math.NT↗
arXiv2026

An approach to the Lindelöf Hypothesis for Dirichlet $L$-functions

The suggested approach is based on a known representation of Dirichlet $L$-functions via the incomplete gamma functions. Some properties of the Taylor coefficients of the lower incomplete gamma function at infinity seem to be new. Specifically, these coefficients can be expressed in terms of Touchard polynomials. Furthermore, these same coefficients can be used to reformulate the functional equation for Dirichlet $L$-functions. This relationship "explains"' why $\vert L_χ(1/2+i t)\vert $ should be small. To present the new ideas in a nutshell, we start by giving (in Section 1) a "formula proof" of the Lindelöf hypothesis. This is not a genuine proof, as we are not concerned with the convergence of our series nor do we justify changing the order of summation. In Section 2, we suggest some hypothetical ways of transforming the "proof" from Section 1 into a rigorous mathematical proof. Sections 3-5 contain some technical details and bibliographical references.

math.NT↗
arXiv2025

Undecidability on Diophantine equations over $\mathbb Z[i]$ with $20$ unknowns

It is known that Hilbert's Tenth Problem over the Gaussian ring $\mathbb Z[i]=\{a+bi:\ a,b\in\mathbb Z\}$ is undecidable. In this paper we obtain the following further result: There is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{20})=0$ (with integer coefficients) is solvable over $\mathbb Z[i]$. This improves the previous record involving $52$ variables.

math.NT↗
arXiv2025

Diophantine Equations over $\mathbb Z$: Universal Bounds and Parallel Formalization

This paper explores multiple closely related themes: bounding the complexity of Diophantine equations over the integers and developing mathematical proofs in parallel with formal theorem provers. Hilbert's Tenth Problem (H10) asks about the decidability of Diophantine equations and has been answered negatively by Davis, Putnam, Robinson and Matiyasevich. It is natural to ask for which subclasses of Diophantine equations H10 remains undecidable. Such subclasses can be defined in terms of universal pairs: bounds on the number of variables $ν$ and degree $δ$ such that all Diophantine equations can be rewritten in at most this complexity. Our work develops explicit universal pairs $(ν, δ)$ for integer unknowns, achieving new bounds that cannot be obtained by naive translations from known results over $\mathbb N$. In parallel, we have conducted a formal verification of our results using the proof assistant Isabelle. While formal proof verification has traditionally been applied a posteriori to known results, this project integrates formalization into the discovery and development process. In a final section, we describe key insights gained from this unusual approach and its implications for mathematical practice. Our work contributes both to the study of Diophantine equations and to the broader question of how mathematics is conducted in the 21st century.

math.NT↗
arXiv2024

Euler Product Sieve

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 Hypothesis + the simplicity of zeros. 16 pages.

math.NT↗
arXiv2022

Mathematical Proof Between Generations

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 of computer proof assistants. For the most time this has been a topic for experts in specialized communities. However, mathematical proofs have become increasingly sophisticated, stretching the boundaries of what is humanly comprehensible, so that leading mathematicians have asked for formal verification of their proofs. At the same time, major theorems in mathematics have recently been computer-verified by people from outside of these communities, even by beginning students. This article investigates the gap between the different definitions of a proof and possibilities to build bridges. It is written as a polemic or a collage by different members of the communities in mathematics and computer science at different stages of their careers, challenging well-known preconceptions and exploring new perspectives.

math.HO↗
arXiv2020

Congruences for Apéry numbers $β_{n}=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}$

In this paper we establish some congruences involving the Apéry numbers $β_{n}=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}$ $(n=0,1,2,\ldots)$. For example, we show that $$\sum_{k=0}^{n-1}(11k^2+13k+4)β_k\equiv0\pmod{2n^2}$$ for any positive integer $n$, and $$\sum_{k=0}^{p-1}(11k^2+13k+4)β_k\equiv 4p^2+4p^7B_{p-5}\pmod{p^8}$$ for any prime $p>3$, where $B_{p-5}$ is the $(p-5)$th Bernoulli number. We also present certain relations between congruence properties of the two kinds of Apery numbers, $β_n$ and $A_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2$.

math.NT↗
arXiv2014

Approximation of Riemann's zeta function by finite Dirichlet series: multiprecision numerical approach

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 remarkable number-theoretical properties discovered in large scale high accuracy numerical experiments. So far no theoretical explanation to the observed phenomena was found.

math.NT↗
arXiv2013

A Parallel Algorithm for Calculation of Large Determinants with High Accuracy for GPUs and MPI clusters

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 submatrices in the top left corner at no extra cost. We implemented the algorithm in arbitrary precision arithmetic, suitable for very ill conditioned matrices, and empirically estimated the loss of precision. The algorithm was applied to studies of Riemann's zeta function.

cs.DC↗
arXiv2012

Horizontal Monotonicity of the Modulus of the Riemann Zeta Function and Related Functions

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 to be equivalent to the Riemann Hypothesis. Another result is a double inequality relating the real parts of the logarithmic derivatives of the three functions under consideration.

math.NT↗
arXiv2007

Hidden Life of Riemann's Zeta Function 1. Arrow, Bow, and Targets

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.

math.NT↗
arXiv2007

Hidden Life of Riemann's Zeta Function 2. Electrons and Trains

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 of columns, from Toeplitz matrices whose eigenvalues were considered in arXiv:0707.1983 . The present paper is a continuation of that publication.

math.NT↗
arXiv2006

Multiple serial episode matching

In a previous paper we generalized the Knuth-Morris-Pratt (KMP) pattern matching algorithm and defined a non-conventional kind of RAM, the MP--RAMs (RAMS equipped with extra operations), and designed an O(n) on-line algorithm for solving the serial episode matching problem on MP--RAMs when there is only one single episode. We here give two extensions of this algorithm to the case when we search for several patterns simultaneously and compare them. More preciseley, given $q+1$ strings (a text $t$ of length $n$ and $q$ patterns $m\_1,...,m\_q$) and a natural number $w$, the {\em multiple serial episode matching problem} consists in finding the number of size $w$ windows of text $t$ which contain patterns $m\_1,...,m\_q$ as subsequences, i.e. for each $m\_i$, if $m\_i=p\_1,..., p\_k$, the letters $p\_1,..., p\_k$ occur in the window, in the same order as in $m\_i$, but not necessarily consecutively (they may be interleaved with other letters).} The main contribution is an algorithm solving this problem on-line in time $O(nq)$.

cs.DS↗