arXiv subjects

Wadim Zudilin

Publications and source records attributed to Wadim Zudilin.

At least 109 records · Page 6Linked to original sources

arXiv2011

Densities of short uniform random walks

We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.

math.CA↗

arXiv2011

On the Mahler measure of $1+X+1/X+Y+1/Y$

We prove a conjectured formula relating the Mahler measure of the Laurent polynomial $1+X+X^{-1}+Y+Y^{-1}$, to the $L$-series of a conductor 15 elliptic curve.

math.NT↗

arXiv2010

Tables of Calabi--Yau equations

The main part of this paper is a big table containing what we believe to be a complete list of all fourth order equations of Calabi--Yau type known so far. In the text preceding the tables we explain what a differential equation of Calabi--Yau type is and we briefly discuss how we found these equations. We also describe an electronic version of this list.

math.AG↗

arXiv2010

"Divergent" Ramanujan-type supercongruences

"Divergent" Ramanujan-type series for $1/π$ and $1/π^2$ provide us with new nice examples of supercongruences of the same kind as those related to the convergent cases. In this paper we manage to prove three of the supercongruences by means of the Wilf--Zeilberger algorithmic technique.

math.NT↗

arXiv2010

Experimental Mathematics and Mathematical Physics

One of the most effective techniques of experimental mathematics is to compute mathematical entities such as integrals, series or limits to high precision, then attempt to recognize the resulting numerical values. Recently these techniques have been applied with great success to problems in mathematical physics. Notable among these applications are the identification of some key multi-dimensional integrals that arise in Ising theory, quantum field theory and in magnetic spin theory.

math-ph↗

arXiv2010

A q-rious positivity

The $q$-binomial coefficients $\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$, for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the many combinatorial interpretations of $\qbinom{n}{m}$. In this note we conjecture an arithmetically motivated generalisation of the non-negativity property for products of ratios of $q$-factorials that happen to be polynomials.

math.NT↗

arXiv2010

Dedekind's eta-function and Rogers-Ramanujan identities

We prove a q-series identity that generalises Macdonald's A_{2n}^{(2)} eta-function identity and the Rogers-Ramanujan identities. We conjecture our result to generalise even further to also include the Andrews-Gordon identities.

math.CO↗

arXiv2009

The Erdős--Moser equation $1^k+2^k+...+(m-1)^k=m^k$ revisited using continued fractions

If the equation of the title has an integer solution with $k\ge2$, then $m>10^{9.3\cdot10^6}$. This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark $m>10^{10^7}$. Here we achieve $m>10^{10^9}$ by showing that $2k/(2m-3)$ is a convergent of $\log2$ and making an extensive continued fraction digits calculation of $(\log2)/N$, with $N$ an appropriate integer. This method is very different from that of Moser. Indeed, our result seems to give one of very few instances where a large scale computation of a numerical constant has an application.

math.NT↗

arXiv2008

On the non-quadraticity of values of the q-exponential function and related q-series

We investigate arithmetic properties of values of the entire function $$ F(z)=F_q(z;λ)=\sum_{n=0}^\infty\frac{z^n}{\prod_{j=1}^n(q^j-λ)}, \qquad |q|>1, \quad λ\notin q^{\mathbb Z_{>0}}, $$ that includes as special cases the Tschakaloff function ($λ=0$) and the $q$-exponential function ($λ=1$). In particular, we prove the non-quadraticity of the numbers $F_q(α;λ)$ for integral $q$, rational $λ$ and $α\notin-λq^{\mathbb Z_{>0}}$, $α\ne0$.

math.NT↗

arXiv2008

Ramanujan-type supercongruences

We present several supercongruences that may be viewed as $p$-adic analogues of Ramanujan-type series for $1/π$ and $1/π^2$, and prove three of these examples.

math.NT↗

arXiv2008

On Sp_4 modularity of Picard--Fuchs differential equations for Calabi--Yau threefolds (with an appendix by Vicentiu Pasol)

Motivated by the relationship of classical modular functions and Picard--Fuchs linear differential equations of order 2 and 3, we present an analogous concept for equations of order 4 and 5.

math.NT↗

arXiv2008

Ramanujan-type formulae for $1/π$: A second wind?

In 1914 S. Ramanujan recorded a list of 17 series for $1/π$. We survey the methods of proofs of Ramanujan's formulae and indicate recently discovered generalizations, some of which are not yet proven.

math.NT↗

arXiv2006

Quadratic transformations and Guillera's formulae for $1/π^2$

We prove two new series of Ramanujan type for $1/π^2$.

math.CA↗

arXiv2005

Approximations to -, di- and tri- logarithms

We propose hypergeometric constructions of simultaneous approximations to polylogarithms. These approximations suit for computing the values of polylogarithms and satisfy 4-term Apery-like (polynomial) recursions.

math.CA↗

arXiv2005

An elementary proof of the irrationality of Tschakaloff series

We present a new proof of the irrationality of values of the series $T_q(z)=\sum_{n=0}^\infty z^nq^{-n(n-1)/2}$ in both qualitative and quantitative forms. The proof is based on a hypergeometric construction of rational approximations to $T_q(z)$.

math.NT↗

arXiv2004

Differential equations, mirror maps and zeta values

The aim of this work is an analytic investigation of differential equations producing mirror maps as well as giving new examples of mirror maps; one of these examples is related to (rational approximations to) $ζ(4)$. We also indicate certain observations that might become a subject of further research.

math.NT↗

arXiv2004

An essay on irrationality measures of pi and other logarithms

We present a brief survey of the methods used in deducing upper estimates for irrationality measures of the logarithm values. We particularly expose the best known estimates for $\log2$ (due to E. Rukhadze), $π$ (due to M. Hata) and $\log3$ (due to G. Rhin).

math.NT↗

arXiv2003

Well-poised generation of Apéry-like recursions

The idea to use classical hypergeometric series and, in particular, well-poised hypergeometric series in diophantine problems of the values of the polylogarithms has led to several novelties in number theory and neighbouring areas of mathematics. Here we present a systematic approach to derive second-order polynomial recursions for approximations to some values of the Lerch zeta function, depending on the fixed (but not necessarily real) parameter $α$ satisfying the condition $\Re(α)<1$. Substituting $α=0$ into the resulting recurrence equations produces the famous recursions for rational approximations to $ζ(2)$, $ζ(3)$ due to Apéry, as well as the known recursion for rational approximations to $ζ(4)$. Multiple integral representations for solutions of the constructed recurrences are also given.

math.NT↗