53 pages Latex, one figure; Version 3 revised introduction, new section 9 added, various small changes in text, revised abstract. Version 4 adds to appendix Riemann-Roch theorem for number fields, formula for two-variable zeta function for general number fields, and corrects misprints. Version 5 corrects sign in definition of xi_Q(w,s), corrects signs throughout paper, changing some theorem statements. Proof of Theorem 7.4 revised
This paper studies a zeta function of two complex variables (w, s) attached to an algebraic number field K, introduced by van der Geer and Schoof, which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov divisors. We mainly consider the case of the rational field Q, where for w = 1 one recovers the Riemann zeta function with the factors at infinity added. The analogue of the Riemann xi-function analytically continues to an entire function of two complex variables, and satisfies a functional equation holding w fixed and sending s to w-s. For real w the "critical line" is therefore Re(s) = w/2. For fixed nonnegative real w the zeros are confined to a vertical strip in s and have the same asymptotics as zeta zeros.
For negative real w the function is positive real on the critical line, so has no zeros there. This phenomonon is associated to a positive convolution semigroup of infinitely divisible probability distributions, and the Khintchine canonical measure of this family is explicitly determined.
In [P] R. Pellikaan introduced a two variable zeta-function for a curve over a finite field and proved that it is a rational function. Here we show that its denominator is absolutely irreducible. This is motivated by work of J. Lagarias and E. Rains on an analogous two variable zeta-funtion for number fields.
In this paper we prove a regularized product expansion for the two-variable zeta functions of number fields introduced by van der Geer and Schoof. The proof is based on a general criterion for zeta-regularizability due to Illies. For number fields of non-zero unit rank our method involves a result of independent interest about the asymptotic behaviour of certain oscillatory integrals in the geometry of numbers. We also explain the cohomological motivation for the paper.
We consider twisted zeta series of several variables associated to polynomials of several variables. Thanks to a totally new method (exchange lemma) we calculate the values at vectors formed of negative integers.After transformation of the formula we obtained, we make p-adic interpolation. ----- Nous considerons des series zetas multivariables associees a des polynomes de plusieurs variables. Grace a une methode totalement nouvelle (dite du lemme d'echange) nous calculons les valeurs aux T-uplets d'entiers negatifs. Apres transformation des formules ainsi obtenues nous faisons l'interpolation …
For the multiple zeta function zeta2(s1,s2) of two variables,we obtain its integral representation(involving product of Hurwitz zeta functions) over the interval [1,infinity),with respect to second variable of Hurwitz zeta function and also obtain a good approximate to it as a function of s1 and s2,for s1>=1 and s2>1.In particular this approximate is explicitly computable,when s1,s2 differ by an even integer and is good,when s2 is large.We treat zeta3(s1,s2,s3) likewise.
We give a new and very concise proof of the existence of a holomorphic continuation for a large class of twisted multivariable zeta functions. To do this, we use a simple method of "decalage" that avoids using an integral representation of the zeta function. This allows us to derive explicit recurrence {\it relations} between the values at $T-$tuples of negative integers. This also extends some earlier results of several authors where the underlying polynomials were products of linear forms.
Starting from a recent result expressing the Lerch zeta function as a fractional derivative, we consider further fractional derivatives of the Lerch zeta function with respect to different variables. We establish a partial differential equation, involving an infinite series of fractional derivatives, which is satisfied by the Lerch zeta function.
The higher rank Lefschetz formula for p-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a p-adic group on its Bruhat-Tits building. By specializing to certain lines one gets one-variable zeta functions, which then can be related to geometrically defined zeta functions
For Hurwitz zeta function, we obtain power series expression in second variable for its higher order derivatives (with respect to first variable) at non-positive integer arguments and consequently obtain rapidly decreasing series expression for Riemann zeta fuction at positive odd integer arguments. Further, we obtain corresponding results for Dirichlet L-series. We also a unified proof of various classical identities involving Riemann zeta values.
We present drawings on the complex plane of the lines Im(zeta(s))=0 and Re(zeta(s))=0. This allow to illustrate many properties of the zeta function of Riemann. This is an expository paper. It does not pretend to prove any new result about the zeta function. But it gives a new perspective to many known results. Also it may prove useful to show to students of Complex Variables or Analytic Number Theory. (This was my initial motivation to make the drawings).
We define polynomials of one variable t whose values at t=0 and 1 are the multiple zeta values and the multiple zeta-star values, respectively. We give an application to the two-one conjecture of Ohno-Zudilin, and also prove the cyclic sum formula for these polynomials.
We show that for a non-positive value of the first variable,Hurwitz zeta function becomes a polynomial in the second variable. We show this, using 'integration approach', instead of 'power series approach', which we had resorted to, in our earlier paper with the same title. This, in particular, explains why Riemann zeta function at positive even integer arguments, can be evaluated and why it cannot be evaluated explicitly at positive odd integer arguments.
In recent work, Fyodorov and Keating conjectured the maximum size of $|ζ(1/2+it)|$ in a typical interval of length O(1) on the critical line. They did this by modelling the zeta function by the characteristic polynomial of a random matrix; relating the random matrix problem to another problem from statistical mechanics; and applying a heuristic analysis of that problem. In this note we recover a conjecture like that of Fyodorov and Keating, but using a different model for $|ζ(1/2+it)|$ in terms of a random Euler product. In this case the probabilistic model reduces to studying the supremum of …
In this paper we show that the iterated integrals on products of one variable multiple polylogarithms from 0 to 1 are actually multiple zeta values if they are convergent. In the divergent case, we define regularized iterated integrals from 0 to 1. By the same method, we show that the regularized iterated integrals are also multiple zeta values. As an application, we give new series representations for multiple zeta values and calculate some interesting examples of iterated integrals.
For the higher order derivative(with respect to the first variable) of Hurwitz zeta function,we discuss as a function of the second variable,the location and the nature of its singularities and obtain the formulae for its derivative and primitive and also discuss its Riemann integrability on the unit interval and on the interval[1,infinity).We also evaluate certain integrals on the unit interval involving these functions.
In this paper we develop the analytic theory of a multiple zeta function in d independent complex variables defined over a global function field. This is the function field analog of the Euler-Zagier multiple zeta function of depth d.
A natural problem in the context of the coupon collector's problem is the behavior of the maximum of independent geometrically distributed random variables (with distinct parameters). This question has been addressed by Brennan et al. (British J. of Math. & CS. 8 (2015), 330-336). Here we provide explicit asymptotic expressions for the moments of that maximum, as well as of the maximum of exponential random variables with corresponding parameters. We also deal with the probability of each of the variables being the maximal one. The calculations lead to expressions involving Hurwitz's zeta func…
This is the first of four papers that study algebraic and analytic structures associated to the Lerch zeta function. This paper studies "zeta integrals" associated to the Lerch zeta function using test functions, and obtains functional equations for them. Special cases include a pair of symmetrized four-term functional equations for combinations of Lerch zeta functions, found by A. Weil, for real parameters (a,c) with 0< a, c< 1. It extends these functions to real a, and c, and studies limiting cases of these functions where at least one of a and c takes the values 0 or 1. A main feature is th…
We first review our previous works of Arakawa and the authors on two, closely related single-variable zeta functions. Their special values at positive and negative integer arguments are respectively multiple zeta values and poly-Bernoulli numbers. We then introduce, as a generalization of Sasaki's work, level 2 analogue of one of the two zeta functions and prove results analogous to those by Arakawa and the first named author.
We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding…
By restricting the variables running over various (possibly different) subfields, we introduce the notion of a partial zeta function. We prove that the partial zeta function is rational in an interesting case, generalizing Dwork's well known rationality theorem. In general, the partial zeta function is probably not rational. But a theorem of Faltings says that the partial zeta function is always nearly rational.
We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure on its residues. The special case of two variables can be written as a partial Mellin transform of a real analytic Eisenstein series, which enables us to relate its values at pairs of positive even points to periods of (simple extensions of symmetric powers of the cohomology of) the CM elliptic curve corresponding to the Gaussian integers. In general, the t…
The cyclic relation obtained in a study by Hirose, Murakami, and the first-named author, is a wide class of relations, which includes the well-known cyclic sum formula for multiple zeta and zeta-star values, and the derivation relation for multiple zeta values. In this paper, we present its generalization to complex variables. Our proof includes a new proof of the cyclic relation.
Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring O_{P,X} at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if O_{P,X} is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincare series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the gene…
As a function of second variable, we identify the Fourier series of Hurwitz zeta function and its derivatives on the unit interval. Consequently, we obtain results based on the formula for Fourier coefficients and also on Parseval's theorem. We do likewise in the case of Lerch's zeta function and its derivatives.
The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet $L$ series, to Eisenstein series, and to general Mellin transforms. The Hurwitz number…
In this paper we present a polynomial time algorithm to compute the local zeta function Z(s,f) attached to a polynomial f(x) in Z[x] (in one variable, with splitting field Q) and a prime p. The algorithm reduces in polynomial time the computation of Z(s,f) to the computation of a factorization of f(x) over Q. This reduction is accomplished by constructing a weighted tree from the p-adic expansion of the roots of f(x) modulo a certain power of p, and then associating a generating function to this tree. The generating function constructed in this way coincides with the local zeta function of f(x…
We give an instant evaluation of multiple Zeta function at non-positive integers by elementary methods and discuss the Fourier theory (on unit interval) of the product of Bernoulli polynomials.We also show that the polynomial expression for Hurwitz Zeta function(at non-positive integral values of the first variable) and the polynomial expression for Bernoulli polynomials are equivalent.
We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length $(\log T)^θ$, where $θ$ is either fixed or tends to zero at a suitable rate. It is shown that the deterministic level of the maximum interpolates smoothly between the ones of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition 'from $\frac34$ to $\frac14$' in the second order. This provides a natural context where extreme value statistics of log-correlated variables with time-dependent variance and rate occur. A key ingredient of the proof is …
In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.
We introduce a Selberg type zeta function of two variables which interpolates several higher Selberg zeta functions. The analytic continuation, the functional equation and the determinant expression of this function via the Laplacian on a Riemann surface are obtained.
This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators $\{ T_m: \, m \ge 1\}$ given by $T_m(f)(a, c) = \frac{1}{m} \sum_{k=0}^{m-1} f(\frac{a+k}{m}, mc)$ acting on certain spaces of real-analytic functions, including Lerch zeta functions for various parameter values. It determines the action of various related operators on these function spaces. It characterizes Lerch zeta functions (for fixed $s$ in the following way. It shows that there is for each $s \in {\bf C}$ a two-dimensional vector space spanned…
In this paper, we generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give new integral representations of several zeta functions, an extension of the parity result to the whole domain of convergence, concrete expressions of Tornheim's double zeta function at non-positive integers and some results for the behavior of a certain Witten's zeta function at each integer. As an appendix, we show a functional equation for Euler's double ze…
Kaneko and Tsumura introduced a new kind of multiple zeta functions $η(k_1,\ldots,k_r;s_1,\ldots,s_r)$. This is an analytic function of complex variables $s_1,\ldots,s_r$, while $k_1,\ldots,k_r$ are non-positive integer parameters. In this paper, we first extend this function to an analytic function $η(s'_1,\ldots,s'_r;s_1,\ldots,s_r)$ of $2r$ complex variables. Then we investigate its special values at positive integers. In particular, we prove some linear relations among these $η$-values and the multiple zeta values $ζ(k_1,\ldots,k_r)$ of Euler-Zagier type.
Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some $q$-series identity for proving the zeta function has an Euler product and then, describe the location of zeros. We study further multi-variable and multi-parameter versions of the multiple finite Riemann zeta functions and their infinite counterparts in connection with symmetric polynomials and some arithmetic quantities called powerful numbers.
There has been an avalanche of recent research on multiple zeta values. We propose dividing identities for multiple zeta values into structural and specific types. Structural identities are valid for any generalized multiple zeta function, and we systematically investigate them through symmetric functions. Specific identities are only valid for a specific zeta function, and we show how these can be used in conjunction with structural identities to find closed form multiple zeta values. This allows us to interpret generalized multiple zeta values as the moments of a random variable, which we ch…
For the Tornheim double zeta function T(s1,s2,s3) of complex variables,we obtain its functional equations,which are new.Using the calculus of r-th order derivative of zeta(s,alpha) as a function of alpha(developed in author[7])as the tool,we obtain with ease an expression for T(s,s,s) for any complex s;and evaluate T(n,n,n) for any integer n>=1;and also evaluate T(n1,0,n2)for integers n1,n2>1 with n1+n2 odd.We obtain for Tornheim zeta function,the counterparts of Euler's formula for zeta(2n)and its analogue for zeta(2n+1),where Zeta(s) is Riemann zeta function.
In this paper, we consstruct a new extended q-Bernoulli numbers and poly nomials. From these numbers, we derive the multiple zeta functions and give some relations between multiple Bernoulli numbers and multiple zeta functions.
The analogue of the Bruhat-Tits building of a p-adic group in F1-geometry is a single apartment. In this setting, the trace formula gives rise to a several variable zeta function analogously to the p-adic case. The analogy carries on to the fact that the restriction to certain lines yield zeta functions which are defined in geometrical terms. Also, the classical formula of Ihara has an analogue in this setting.
This paper studies algebraic and analytic structures associated with the Lerch zeta function, extending the complex variables viewpoint taken in part II. The Lerch transcendent $Φ(s, z, c)$ is obtained from the Lerch zeta function $ζ(s, a, c)$ by the change of variable $z=e^{2 πi a}$. We show that it analytically continues to a maximal domain of holomorphy in three complex variables $(s, z, c)$ as a multivalued function defined over the base manifold ${\bf C} \times P^1({\bf C} \smallsetminus \{0, 1, \infty\}) \times ({\bf C}\smallsetminus {\bf Z})$. and compute the monodromy functions definin…
This is the second of four papers that study algebraic and analytic structures associated with the Lerch zeta function. In this paper we analytically continue it as a function of three complex variables. We that it is well defined as a multivalued function on the manifold M equal to C^3 with the hyperplanes corresponding to integer values of the two variables a and c removed. We show that it becomes single valued on the maximal abelian cover of M. We compute the monodromy functions describing the multivalued nature of this function on M, and determine various of their properties.
Deriving motivic coactions and single-valued maps at genus zero from zeta generators
Institute for Advanced Study（高等研究院）；ETH Zürich（苏黎世联邦理工学院）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）；Uppsala University（乌普萨拉大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学数学研究所）
本文通过 zeta 生成器证明了零 genus 多重 polylogarithms 的动机共作用和单值映射猜想，结合了非交换变量的操作和生成函数。
The purpose of this article is to present closed forms for various types of infinite series involving Fibonacci (Lucas) numbers and the Riemann zeta function at integer arguments.
The purpose this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials and we construct multiple q-zeta function which interpolates multiple q-Euler numbers at negative integers.
We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at non-positive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the $ξ$-function defined by Arakawa and the first-named author. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.
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.
We show the recurrence relations of the Euler-Zagier multiple zeta-function which describes the $r$-fold function with one variable specialized to a non-positive integer as a rational linear combination of $(r-1)$-fold functions, which extends the previous results of Akiyama-Egami-Tanigawa and Matsumoto. As an application, we obtain an explicit method to calculate the special values of the multiple zeta-function at any integer point (the arguments could be neither all-positive nor all-non-positive) as a rational linear summation of the multiple zeta values.
New formulas for approximation of zeta-constants were derived on the basis of a number-theoretic approach constructed for the irrationality proof of certain classical constants. Using these formulas it's possible to approximate certain zeta-constants and their combinations by rational fractions and construct a method for their evaluation.
In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Finally we woll treat some identities of the q-extension of the euler numbers by using fermionic p-adic q-integration on Z_p.
This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic L-series. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of central values of Dirichlet L-series.…
