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Wadim Zudilin

Publications and source records attributed to Wadim Zudilin.

At least 19 recordsLinked to original sources
arXiv2026

Modular regulators and multiple Eisenstein values

We introduce a new methodology for length reduction of multiple modular values as developed by Brown; it involves an interpolation of multiple Eisenstein values and differentiation with respect to their continuous elliptic parameters. We apply our method to computing explicitly the Goncharov regulator integral associated to $K_4$ classes on modular curves in terms of $L$-values of modular forms. We use this expression to connect it with the Beilinson regulator integral. Our general approach reveals new interesting arithmetic phenomena and prospects for the general $K$-groups of modular curves.

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arXiv2026

Continued Fractions and Irrationality Measures for Chowla--Selberg Gamma Quotients

We give 39 rapidly convergent continued fractions for Chowla--Selberg gamma quotients, and deduce good irrationality measures for 20 of them, including for $\operatorname{CS}(-3)=(Γ(1/3)/Γ(2/3))^3$, for $a^{1/4}\operatorname{CS}(-4)=a^{1/4}(Γ(1/4)/Γ(3/4))^2$ with $a=12$ and $a=1/5$, and for $\operatorname{CS}(-7)=Γ(1/7)Γ(2/7)Γ(4/7)/(Γ(3/7)Γ(5/7)Γ(6/7))$. These appear to be the first proved and reasonable irrationality measures for gamma quotients.

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arXiv2026

A note on the irrationality of $ζ_2(5)$

In a spirit of Apéry's proof of the irrationality of $ζ(3)$, we construct a sequence $p_n/q_n$ of rational approximations to the $2$-adic zeta value $ζ_2(5)$ which satisfy $0 < |ζ_2(5)-p_n/q_n|_2 < \max\{|p_n|,|q_n|\}^{-1-δ}$ for an explicit constant $δ>0$. This leads to a new proof of the irrationality of $ζ_2(5)$, the result established recently by Calegari, Dimitrov and Tang using a different method. Furthermore, our approximations allow us to obtain an upper bound for the irrationality measure of this $2$-adic quantity; namely, we show that $μ(ζ_2(5)) \le (16\log2)/(8\log2-5) = 20.342\dots$.

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arXiv2026

An integrality phenomenon

We prove a general statement about the integrality of the sequences generated by a recursion of the following form: $nu_n$ equals a linear combination of $u_{n-1},u_{n-2},\dots,u_0$ with polynomial coefficients in $n$ of special form. This includes a conjectural integrality of the sequence related to the Hörmander-Bernhardsson extremal function, for which we further give a direct proof as well.

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arXiv2026

On Schultz's generalization of Borweins' cubic identity

In 1991, the Borweins established a cubic analogue of Jacobi's identity for theta functions, which is used by B.C. Berndt, S. Bhargava, and F.G. Garvan in the development of Ramanujan's cubic theory of elliptic functions. In 2013, D. Schultz discovered an identity for theta series in three variables which generalizes the Borweins' identity. In this article, we revisit Schultz's identity and present two distinct approaches to its derivation. Our investigation not only provides new proofs but also yields several new Schultz-type identities.

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arXiv2026

Poor man's transcendence for Frobenius traces of elliptic curves

Let $E$ be an elliptic curve without complex multiplication defined over $\mathbb Q$. Viewing the sequence of its Frobenius traces $(a_p(E))_p$ indexed by primes $p$ as an element in the "poor man's adèle ring", we prove its transcendence over $\mathbb Q$.

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arXiv2026

On cellular rational approximations to $ζ(5)$

We analyse a certain family of cellular integrals, which are period integrals on the moduli space $\mathcal{M}_{0,8}$ of curves of genus zero with eight marked points, and give rise to simultaneous rational approximations to $ζ(3)$ and $ζ(5)$. By exploiting the action of a large symmetry group on these integrals, we construct an infinite $effective$ sequence of rational approximations $p/q$ to $ζ(5)$ satisfying \[ 0<\bigg|ζ(5)-\frac pq\bigg|<\frac1{q^{0.86}}. \]

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arXiv2025

$q$-rious unimodality

We generalise our still-wide-open $q$-rious positivity conjecture from 2011 to a $q$-rious unimodality conjecture.

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arXiv2025

Variations on a theme of Apéry

Apéry's remarkable discovery of rapidly converging continued fractions with small coefficients for $ζ(2)$ and $ζ(3)$ has led to a flurry of important activity in an incredible variety of different directions. Our purpose is to show that modifications of Apéry's continued fractions can give interesting results including new rapidly convergent continued fractions for certain interesting constants.

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arXiv2025

Galois Groups of Apéry-like Series Modulo Primes

We compute the Galois groups of the reductions modulo the prime numbers $p$ of the generating series of Apéry numbers, Domb numbers and Almkvist--Zudilin numbers. We observe in particular that their behavior is governed by congruence conditions on p.

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arXiv2025

Linear independence measures for Chowla--Selberg periods

We use simultaneous Padé approximations to $_3F_2$ hypergeometric functions to estimate from below linear forms in $1$, $π\sqrt d$, $Ω_D/π$ and $π/Ω_D$ with integral coefficients, for certain choices of positive integer $d$ and negative integer $D$, where $Ω_D$ is (the square of) a Chowla--Selberg period attached to the imaginary quadratic field $Q(\sqrt{D})$.

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arXiv2025

An evolution of matrix-valued orthogonal polynomials

We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating functions, distribution of zeros for individual entries of the matrices and new type of differential-difference structure. We further speculate about other potentials of the connection formulas found. Part of our proofs makes use of creative telescoping in a matrix setting$-$the strategy which is not yet developed algorithmically.

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arXiv2025

Irrationality and transcendence questions in the "poor man's adèle ring"

We discuss arithmetic questions related to the "poor man's adèle ring" $\mathcal A$ whose elements are encoded by sequences $(t_p)_p$ indexed by prime numbers, with each $t_p$ viewed as a residue in $\mathbb Z/p\mathbb Z$. Our main theorem is about the $\mathcal A$-transcendence of the element $(F_p(q))_p$, where $F_n(q)$ (Schur's $q$-Fibonacci numbers) are the $(1,1)$-entries of $2\times2$-matrices $$ \bigg(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}\bigg) \bigg(\begin{matrix} 1 & 1 \\ q & 0 \end{matrix}\bigg) \bigg(\begin{matrix} 1 & 1 \\ q^2 & 0 \end{matrix}\bigg) \cdots \bigg(\begin{matrix} 1 & 1 \\ q^{n-2} & 0 \end{matrix}\bigg) $$ and $q>1$ is an integer. This result was previously known for $q>1$ square free under the GRH.

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arXiv2025

A partial-sum deformation for a family of orthogonal polynomials

There are several questions one may ask about polynomials $q_m(x)=q_m(x;t)=\sum_{n=0}^mt^mp_n(x)$ attached to a family of orthogonal polynomials $\{p_n(x)\}_{n\ge0}$. In this note we draw attention to the naturalness of this partial-sum deformation and related beautiful structures. In particular, we investigate the location and distribution of zeros of $q_m(x;t)$ in the case of varying real parameter $t$.

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arXiv2024

First memoir on the asymptotics of certain infinite products

The product sides of the Rogers--Ramanujan identities and alike often appear to be "transparently modular" (functions). The old work by Rogers (1894) and recent work by Rosengren make use (somewhat implicitly) of this fact for proving the identities with the help of underlying modular equations$-$the main challenge is verifying the latter for the sum sides. Here we speculate on the potentials of using the asymptotics of such $q$-identities or their finite versions for proving them.

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