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Wadim Zudilin

Publications and source records attributed to Wadim Zudilin.

At least 55 records · Page 3Linked to original sources
arXiv2020

Dwork-type supercongruences through a creative $q$-microscope

We develop an analytical method to prove congruences of the type $$ \sum_{k=0}^{(p^r-1)/d}A_kz^k \equiv ω(z)\sum_{k=0}^{(p^{r-1}-1)/d}A_kz^{pk} \pmod{p^{mr}\mathbb Z_p[[z]]} \quad \text{for}\; r=1,2,\dots, $$ for primes $p>2$ and fixed integers $m,d\ge1$, where $f(z)=\sum_{k=0}^\infty A_kz^k$ is an "arithmetic" hypergeometric series. Such congruences for $m=d=1$ were introduced by Dwork in 1969 as a tool for $p$-adic analytical continuation of $f(z)$. Our proofs of several Dwork-type congruences corresponding to $m\ge2$ (in other words, supercongruences) are based on constructing and proving their suitable $q$-analogues, which in turn have their own right for existence and potential for a $q$-deformation of modular forms and of cohomology groups of algebraic varieties. Our method follows the principles of creative microscoping introduced by us to tackle $r=1$ instances of such congruences; it is the first method capable of establishing the supercongruences of this type for general $r$.

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arXiv2020

A case study for $ζ(4)$

Using symbolic summation tools in the setting of difference rings, we prove a two-parametric identity that relates rational approximations to $ζ(4)$.

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arXiv2020

The Irrationality Measure of Pi is at most 7.103205334137...

We use a variant of Salikhov's ingenious proof that the irrationality measure of $π$ is at most $7.606308\dots$ to prove that, in fact, it is at most $7.103205334137\dots$. Accompanying Maple package: While this article has a fully rigorous human-made and human-readable proof of the claim in the title, it was discovered thanks to the Maple package available from http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/pimeas.html

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arXiv2019

Automatic Discovery of Irrationality Proofs and Irrationality Measures

We illustrate the power of Experimental Mathematics and Symbolic Computation to suggest irrationality proofs of natural constants, and the determination of their irrationality measures. Sometimes such proofs can be fully automated, but sometimes there is still need for a human touch.

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arXiv2019

The method of creative microscoping

We outline basic principles of a new method that gives a conceptual reasoning for and, at the same time, proofs of (super)congruences for truncated sums of arithmetic hypergeometric evaluations.

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arXiv2019

A common $q$-analogue of two supercongruences

We give a $q$-congruence whose specializations $q=-1$ and $q=1$ correspond to supercongruences (B.2) and (H.2) on Van Hamme's 1997 list: $$ \sum_{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\equiv p(-1)^{(p-1)/2}\pmod{p^3} \quad\text{and}\quad \sum_{k=0}^{(p-1)/2}A_k\equiv a(p)\pmod{p^2}, $$ where $p>2$ is prime, $$ A_k=\prod_{j=0}^{k-1}\biggl(\frac{1/2+j}{1+j}\biggr)^3=\frac1{2^{6k}}{\binom{2k}k}^3 \quad\text{for}\ k=0,1,2,\dots, $$ and $a(p)$ is the $p$-th coefficient of (the weight 3 modular form) $q\prod_{j=1}^\infty(1-q^{4j})^6$. We complement our result with a general common $q$-congruence for related hypergeometric sums.

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arXiv2019

$q$-rious and $q$-riouser

Dick Askey is known not just for his beautiful mathematics and his many amazing theorems, but also for posing numerous interesting and important open problems. Dick being Dick, these problems are hardly ever isolated, and often intended to demonstrate the unity of analysis, number theory and combinatorics. We take the reader down the rabbit hole created by one such problem, published as Advanced Problem 6514 by the American Mathematical Monthly in April 1986.

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arXiv2019

Arithmetic of Catalan's constant and its relatives

We prove that at least one of the six numbers $β(2i)$ for $i=1,\dots,6$ is irrational. Here $β(s)=\sum_{k=0}^\infty(-1)^k(2k+1)^{-s}$ denotes Dirichlet's beta function, so that $β(2)$ is Catalan's constant.

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arXiv2019

Hypergeometric rational approximations to $ζ(4)$

We give a new hypergeometric construction of rational approximations to $ζ(4)$, which absorbs the earlier one from 2003 based on Bailey's ${}_9F_8$ hypergeometric integrals. With the novel ingredients we are able to get a better control of arithmetic and produce a record irrationality measure for $ζ(4)$.

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arXiv2019

Congruences for $q$-binomial coefficients

We discuss $q$-analogues of the classical congruence $\binom{ap}{bp}\equiv\binom{a}{b}\pmod{p^3}$, valid for primes $p>3$, as well as its generalisations. In particular, we prove related congruences for ($q$-analogues of) integral factorial ratios.

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arXiv2019

On a $q$-deformation of modular forms

There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work arXiv:1803.01830, such partial sums are related to the radial asymptotics of infinite $q$-hypergeometric sums at roots of unity. Here we combine the two features to construct a hypergeometric $q$-deformation of two CM modular forms of weight 3 and discuss the corresponding $q$-congruences.

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arXiv2019

A $q$-microscope for supercongruences

By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a $q$-analogue of Ramanujan's formula $$ \sum_{n=0}^\infty\frac{\binom{4n}{2n}{\binom{2n}{n}}^2}{2^{8n}3^{2n}}\,(8n+1) =\frac{2\sqrt{3}}π, $$ of the two supercongruences $$ S(p-1)\equiv p\biggl(\frac{-3}p\biggr)\pmod{p^3} \quad\text{and}\quad S\Bigl(\frac{p-1}2\Bigr) \equiv p\biggl(\frac{-3}p\biggr)\pmod{p^3}, $$ valid for all primes $p>3$, where $S(N)$ denotes the truncation of the infinite sum at the $N$-th place and $(\frac{-3}{\cdot})$ stands for the quadratic character modulo $3$.

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arXiv2018

A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds

We examine instances of modularity of (rigid) Calabi-Yau manifolds whose periods are expressed in terms of hypergeometric functions. The $p$-th coefficients $a(p)$ of the corresponding modular form can be often read off, at least conjecturally, from the truncated partial sums of the underlying hypergeometric series modulo a power of $p$ and from Weil's general bounds $|a(p)|\le2p^{(m-1)/2}$, where $m$ is the weight of the form. Furthermore, the critical $L$-values of the modular form are predicted to be $\mathbb Q$-proportional to the values of a related basis of solutions to the hypergeometric differential equation.

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arXiv2018

Hypergeometry inspired by irrationality questions

We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $π^2$, their connection with already known ones, and underlying "permutation group" structures. Our principal arithmetic achievement is a new partial irrationality result for the values of Riemann's zeta function at odd integers.

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arXiv2018

Short walk adventures

We review recent development of short uniform random walks, with a focus on its connection to (zeta) Mahler measures and modular parametrisation of the density functions. Furthermore, we extend available "probabilistic" techniques to cover a variation of random walks and reduce some three-variable Mahler measures, which are conjectured to evaluate in terms of $L$-values of modular forms, to hypergeometric form.

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