Spectral Theory of the Riemann Zeta-Function: Chapter 6: Appendix

Apéry's theorem and problems for the values of Riemann's zeta function and their $q$-analogues

This monograph is intended to be considered as my habilitation (D.Sc.) thesis; because of that and as everything has already appeared in English, it is performed exclusively in Russian. The monograph comprises a detailed introduction and seven chapters that represent part of my work influenced by Apéry's proof from 1978 of the irrationality of $ζ(2)$ and $ζ(3)$, the values of Riemann's zeta function. Chapter 1 is about "at least one of the four numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ and $ζ(11)$ is irrational" (based in part on arXiv:math.NT/0206176). Chapter 2 explains a connection between the general…

有效欧拉和

Effective Euler Sums

本文提出并构造了深度一和深度二下与经典欧拉和对应的有效欧拉和，并在附录中给出了深度二偶数权重的候选，证明了与金子-扎吉尔定义一致。

The Riemann hypothesis for Weng's zeta function of $Sp(4)$ over $\mathbb{Q}$

As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple poles. The rank one zeta function is the Dedekind zeta function. For the rank two case, the Riemann hypothesis is proved for a general number field. Recently, he defined more general new zeta function associated to a pair of reductive group and its maximal parabolic subgroup. As well as high rank zeta functions, the new zeta function satisfies standard prope…

Odd zeta motive and linear forms in odd zeta values

We study a family of mixed Tate motives over $\mathbb{Z}$ whose periods are linear forms in the zeta values $ζ(n)$. They naturally include the Beukers-Rhin-Viola integrals for $ζ(2)$ and the Ball-Rivoal linear forms in odd zeta values. We give a general integral formula for the coefficients of the linear forms and a geometric interpretation of the vanishing of the coefficients of a given parity. The main underlying result is a geometric construction of a minimal ind-object in the category of mixed Tate motives over $\mathbb{Z}$ which contains all the non-trivial extensions between simple objec…

Lectures on height zeta functions: At the confluence of algebraic geometry, algebraic number theory, and analysis

This is a survey on the theory of height zeta functions, written on the occasion of a French-Japanese winter school, held in Miura (Kanagawa, Japan) in Jan. 2008. It does not presuppose much knowledge in algebraic geometry. The last chapter of the survey explains recent results obtained in collaboration with Yuri Tschinkel concerning asymptotics of volumes of height balls in analytic geometry over local fields, or in adelic spaces.

曲线上的Zeta函数

Zeta functions over curves

本文综述了David Goss发展的zeta函数理论，并与另一种基于有限域上曲线的zeta函数构造方法进行比较，介绍了Ferraro的有理性定理及其在相对zeta函数整函数性质中的应用。

Edge reconstruction of the Ihara zeta function

We show that if a graph $G$ has average degree $\bar d \geq 4$, then the Ihara zeta function of $G$ is edge-reconstructible. We prove some general spectral properties of the edge adjacency operator $T$: it is symmetric for an indefinite form and has a "large" semi-simple part (but it can fail to be semi-simple in general). We prove that this implies that if $\bar d>4$, one can reconstruct the number of non-backtracking (closed or not) walks through a given edge, the Perron-Frobenius eigenvector of $T$ (modulo a natural symmetry), as well as the closed walks that pass through a given edge in bo…

关于zeta函数的Ser表示与Hasse表示的三则注记

Three notes on Ser's and Hasse's representations for the zeta-functions

本文研究zeta函数的Ser与Hasse级数表示，证明二者等价，推导了涉及多种特殊数与多项式的zeta函数、Hurwitz zeta函数及Dirichlet级数的新级数表示，还得到Stieltjes常数等的新展开式。

Euler 和数的加权与受限求和公式

Weighted and Restricted Sum Formulas of Euler Sums

本文利用 Euler 和数的正则化双洗牌关系，统一证明了多重 zeta 值的求和公式及其变体，并在附录中以三个表格总结了包括新公式在内的多种结果。

On the Shintani zeta function for the space of pairs of binary Hermitian forms

Throughout this paper, k is a number field. We fix a quadratic extension k_1=k(a_0) of k, where a_0=sqrt(B_0) for a certain B_0 in k^\times/(k^\times)^2. In this paper, we consider the zeta function defined for the space of pairs of binary Hermitian forms. This is the prehomogeneous vector space we discussed in section 2 [1] and is a non-split form of the D_4 case in [4]. The purpose of this paper is to determine the principal part of the adjusted zeta function. Our main result is Theorem (8.15). Our case resembles the space of pairs of binary quadratic forms which we discussed in Chapter 5 [5…

Dynamics on abelian varieties in positive characteristic

We study periodic points for endomorphisms $σ$ of abelian varieties $A$ over algebraically closed fields of positive characteristic $p$. We show that the dynamical zeta function $ζ_σ$ of $σ$ is either rational or transcendental, the first case happening precisely when $σ^n-1$ is a separable isogeny for all $n$. We call this condition very inseparability and show it is equivalent to the action of $σ$ on the local $p$-torsion group scheme being nilpotent. The "false" zeta function $D_σ$, in which the number of fixed points of $σ^n$ is replaced by the degree of $σ^n-1$, is always a rational funct…

Ramanujan Summation and the Exponential Generating Function $ \sum_{k=0}^{\infty}\frac{z^{k}}{k!}ζ^{\prime}(-k)$

In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from w…

Scattering, determinants, hyperfunctions in relation to Gamma(1-s)/Gamma(s)

The method of realizing certain self-reciprocal transforms as (absolute) scattering, previously presented in summarized form in the case of the Fourier cosine and sine transforms, is here applied to the self-reciprocal transform f(y)-> H(f)(x) = \int_0^\infty J_0(2\sqrt{xy})f(y) dy, which is isometrically equivalent to the Hankel transform of order zero and is related to the functional equations of the Dedekind zeta functions of imaginary quadratic fields. This also allows to re-prove and to extend theorems of de Branges and V. Rovnyak regarding square integrable functions which are self-or-sk…