Julio Andrade
An asymptotic formula for the sum $\sum L(1,χ)$ is established for a family of hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_q$ as $g\rightarrow\infty$ making use of the analogue of the approximate functional equation for such $L$--functions. As a corollary, we obtain a formula for the average of the class number of the associated rings $\mathbb{F}_{q}[T,sqrt{D}]$.
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Assuming the Riemann hypothesis, we obtain a formula for the mean value of the $k$-derivative of $ζ'/ζ$, depending on the pair correlation of zeros of the Riemann zeta-function. This formula allows us to obtain new equivalences to Montgomery's pair correlation conjecture. This extends a result of Goldston, Gonek, and Montgomery where the mean value of $ζ'/ζ$ was considered.
We prove an easy but interesting result about the linear independence of multiple zeta values of different weights.
In this note, we shall discuss a generalization of Thakur's multiple zeta values and allied objects, in the framework of function fields of positive characteristic and more precisely, of periods in Tate algebras.
In this note, by using several ideas of other researchers, we derive several relations among multiple zeta-star values from the hypergeometric identities of C. Krattenthaler and T. Rivoal.
We prove that the algebra of p-adic multi-zeta values are contained in another algebra which is defined explicitly in terms of series.
We prove three theorems on finite real multiple zeta values: the symmetric formula, the sum formula and the height-one duality theorem. These are analogues of their counterparts on finite multiple zeta values.
I obtain new evaluations of special values of multiple polylogarithms by using a limiting case of a basic hypergeometric identity of G. E. Andrews.
Donnay, Garcia and Rouse classified nonsingular quadratic forms $Q$ with integral coefficients and prime numbers $p$ such that the set of quotients of values of $Q$ attained for integer arguments is dense in the field of $p$-adic numbers. The aim of this note is to give another proof of this classification.
Using a new construction of rational linear forms in odd zeta values and the saddle point method, we prove the existence of at least two irrational numbers amongst the 33 odd zeta values $ζ$(5), $ζ$(7),. .. , $ζ$(69).
The Ohno relation is a well-known relation among multiple zeta values. Hirose, Onozuka, Sato, and the author investigated the sum related to the Ohno relation and presented two types of new relations and five conjectural formulas. This paper proves one of these formulas.
A note on values of $ {}_{4}F_{3} $ hypergeometric functions
本文通过建立扩展的高斯求和恒等式，计算了一族 $_4F_3$ 超几何函数的取值，从而推广了 Ferretti 等人的已有结果。
In [Ok] Okounkov studies a specific $q$-analogue of multiple zeta values and makes some conjectures on their algebraic structure. In this note we compare Okounkovs $q$-analogues to the generating function for multiple divisor sums defined in [BK1]. We also state a conjecture on their dimensions that complements Okounkovs conjectural formula and present some numerical evidences for it.
Multiple zeta values (MZVs) with certain repeated arguments or certain sums of cyclically generated MZVs are evaluated as rational multiple of powers of $π^2$. In this paper, we give a short and simple proof of the remarkable evaluations of MZVs established by D. Borman and D. M. Bradley.
We investigate the large values of the derivatives of the Riemann zeta function $ζ(s)$ on the 1-line. We give a larger lower bound for $\max_{t\in[T,2T]}|ζ^{(\ell)}(1+{\rm i} t)|$, which improves the previous result established by Yang.
The aim of the present note is to develop a study on the feasibility of a unified theory of mean values of automorphic L-functions, a desideratum in the field. This is an outcome of the investigation commenced with Part XII of this series, where a framework was laid on the basis of the theory of automorphic representations, and a general approach to the mean values was envisaged. Specifically, it is shown here that the inner-product method, which was initiated by A. Good and greatly enhanced by M. Jutila, ought to be brought to perfection so that the mean square of the L-function attached to a…
A note on isomorphism types of complete discretely valued fields of mixed characteristic
本文确定了混合特征完备离散赋值域由其截断赋值环同构类决定所需的最小截断水平。
A note on transcendence of special values of functions related to modularity
本文研究具有代数傅里叶系数的模函数及亚纯（拟）模形式在任意同余子群下特殊值的超越性，统一并推广了 Schneider、Nesterenko 等人的经典定理。
The aim of the present article is to render the spectral theory of mean values of automorphic $L$-functions -- in a unified fashion. This is an outcome of the investigation commenced with the parts XII and XIV, where a framework was laid on the basis of the theory of automorphic representations and a general approach to the mean values was envisaged. We restrict ourselves to the situation offered by the full modular group, solely for the sake of simplicity. Details and extensions are to be published elsewhere. In this new version we added details to all aspects of our argument especially for t…
The derivation relation is a well known relation among multiple zeta values, which was first obtained by Ihara, Kaneko and Zagier. The analogous formula for finite multiple zeta values, which we call the derivation relation for finite multiple zeta values, was conjectured by the third author and proved by the second author. In this paper, we show these two kinds of derivation relations are respectively equivalent to the Ohno type relations for multiple zeta values and finite multiple zeta values. We also reprove these derivation relations in several different ways.
For several evaluations of special values and several relations known only in $\mathcal{A}_n$-multiple zeta values or $\mathcal{S}_n$-multiple zeta values, we prove that they are uniformly valid in $\mathcal{F}_n$-multiple zeta values for both the case where $\mathcal{F}=\mathcal{A}$ and $\mathcal{F}=\mathcal{S}$. In particular, the Bowman-Bradley type theorem and sum formulas for $\mathcal{S}_2$-multiple zeta values are proved.
A note on the mean square of the Riemann zeta-function
本文针对黎曼ζ函数均方值的已知结果，提出了一种新的证明方法。
We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta values, and the Witten zeta function. We discuss a heuristic for finding or dismissing the existence of similar simple sums. We also produce some new sums from recursions involving the Riemann zeta and the Dirichlet beta functions.
A note on zero density results implying large value estimates for Dirichlet polynomials
本文证明黎曼 zeta 函数的零密度估计可蕴含 Dirichlet 多项式的大值估计，填补了反向蕴含关系的空白。
We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa-Kaneko zeta function, complementing and generalizing earlier results. Moreover, we survey properties of certain truncated multiple zeta and zeta star values, pointing out their relation to finite sums of harmonic numbers. We also discuss the duality result of Hoffman, relating binomial sums and truncated multiple zeta star values.
Note on the mean value of the Erdős--Hooley Delta-function
本文针对埃尔德什-胡利Δ函数的均值问题，改进了其近期上界，证明对大x，该函数的均值满足∑_{n≤x}Δ(n)≪x(log₂x)^(5/2)。
We prove the cyclic sum formulas for certain two-parameter multiple series. These are new and non-trivial generalizations of the cyclic sum formulas for multiple zeta values and multiple zeta-star values.
For a number field $K$, let $ζ_{K}(s)$ be the Dedekind zeta function associated to $K$. In this note, we study non-vanishing and transcendence of $ζ_{K}$ as well as its derivative $ζ_{K}'$ at $s= 1/2$. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of $e^γ$ and non-vanishing of $L$-series at $s= 1/2$, J. Number Theory 161 (2016) 444-456].
Let $F$ be a nearly holomorphic vector-valued Siegel modular form of weight $ρ$ with respect to some congruence subgroup of $\mathrm{Sp}_{2n}(\mathbb Q)$. In this note, we prove that the function on $\mathrm{Sp}_{2n}(\mathbb R)$ obtained by lifting $F$ has the moderate growth (or "slowly increasing") property. This is a consequence of the following bound that we prove: $\|ρ(Y^{1/2})F(Z) \| \ll \prod_{i=1}^n (μ_i(Y)^{λ_1/2} + μ_i(Y)^{-λ_1/2})$ where $ λ_1 \ge \ldots \ge λ_n$ is the highest weight of $ρ$ and $μ_i(Y)$ are the eigenvalues of the matrix $Y$.
We give explicit formulae for all of the terms in the asymptotic expansion of the mean fourth power of the Riemann zeta-function on the critical line.
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