The modified Mellin transform ${\cal Z}_k(s) = \int_1^\infty |ζ({1\over2}+ix|^{2k}x^{-s}{\rm d} x$ ($k\ge1$ is a fixed integer, $s = σ+ it$) is used to obtain estimates for $$ \sum_{r=1}^R\int_{t_r-G}^{t_r+G}|ζ(1/2+it)|^{2k}{\rm d} t\quad(T < t_1 < >... < t_R < 2T), $$ where $t_{r+1} - t_r \ge G (r =1,..., R-1),
T^ε\le G \le T^{1-ε$. These results can be used to derive bounds for the moments of $}|ζ(1/2+it)|$.
The alternating multiple harmonic sums are partial sums of the infinite series defining the Euler sums which are the alternating version of the multiple zeta value series. In this paper, we present some systematic structural results of the van Hamme type congruences of these sums, collected as finite Euler sums. Moreover, we relate this to the structure of the Euler sums, which generalizes the corresponding result of the multiple zeta values. We also provide a few conjectures with extensive numerical support.
This is part II of our examination of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations.
The aim of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, q-analogue Genocchi type l-function and two-variable q-Genocchi type l-function. Furthermore, we construct new genereting functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations related to q-Hardy-Berndt type sums and q-Genocchi zeta function as well.
A recent paper of Furdui and Valean proves some results about sums of products of "tails" of the series for the Riemann zeta function. We show how such results can be proved with weaker hypotheses using multiple zeta values, and also show how they can be generalized to products of three or more such tails.
The sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values bear a striking resemblance. We explain the resemblance in a rather straightforward manner using an identity that involves the Schur multiple zeta values. We also obtain the sum formula for polynomial multiple zeta(-star) values in terms of generating functions, simultaneously generalizing the sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values.
A typical formula of multiple zeta values is the sum formula which expresses a Riemann zeta value as a sum of all multiple zeta values of fixed weight and depth. Recently weighted sum formulas, which are weighted analogues of the sum formula, have been studied by many people. In this paper, we give two formulas of weighted sums with two parameters of multiple zeta values. As applications of the formulas, we find some linear combinations of multiple zeta values which can be expressed as polynomials of usual zeta values with coeffcients in the rational polynomial ring generated by the two parame…
We consider sums of the form $\sum ϕ(γ)$, where $ϕ$ is a given function, and $γ$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such sums can be accelerated by a simple device, and give examples involving both convergent and divergent infinite sums.
We investigate the distribution of the zeros of partial sums of the Riemann zeta-function, sum_{n\leq X}n^{-s}, estimating the number of zeros up to height T, the number of zeros to the right of a given vertical line, and other aspects of their horizontal distribution.
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.
We define and apply a method to study the non-vanishing of $p$-adic cyclotomic multiple zeta values. We prove the non-vanishing of certain cyclotomic multiple harmonic sums, and, via a formula proved in another paper, which expresses a cyclotomic multiple harmonic sums as an infinite sum of products of $p$-adic cyclotomic multiple zeta values, this implies the non-vanishing of certain $p$-adic cyclotomic multiple zeta values.
Ohno's relation states that a certain sum, which we call an Ohno type sum, of multiple zeta values remains unchanged if we replace the base index by its dual index. In view of Oyama's theorem concerning Ohno type sums of finite and symmetric multiple zeta values, Kaneko looked at Ohno type sums of finite and symmetric multiple zeta-star values and made a conjecture on the generating function for a specific index of depth three. In this paper, we confirm this conjecture and further give a formula for arbitrary indices of depth three.
An identity involving symmetric sums of regularized multiple zeta-star values of harmonic type was proved by Hoffman. In this paper, we prove an identity of shuffle type. We use Bell polynomials appearing in the study of set partitions to prove the identity.
We establish a new decoupling inequality for curves in the spirit of [B-D1], [B-D2] which implies a new mean value theorem for certain exponential sums crucial to the Bombieri-Iwaniec method as developed further in [H]. In particular, this leads to an improved bound $|ζ(\frac 12+it)|\ll t^{53/342+\varepsilon}$ for the zeta function on the critical line
Let $γ$ denote imaginary parts of complex zeros of the Riemann zeta-function $ζ(s)$. Certain sums over the $γ$'s are evaluated, by using the function $G(s) = \sum_{γ>0}γ^{-s}$ and other techniques. Some integrals involving the function $S(T) = (1/π)\argζ(1/2+iT)$ are also considered.
We prove some identities, which involve the non-trivial zeros of the Riemann zeta function. From them we derive some convergent asymptotic expansions related to the work by Cramér, and also new representations for some arithmetical functions in terms of the non-trivial zeros.
In this paper, we give identities involving cyclic sums of regularized multiple zeta values of depth less than $5$. As a corollary, we present two kinds of extensions of Hoffman's theorem for symmetric sums of multiple zeta values for this case.
We construct an analytic approach to evaluate odd Euler sums, multiple zeta value $ζ(3,2,\ldots,2)$ and multiple $t$-value $t\left(3,2,\ldots,2\right)$. Moreover, we also conjecture a closed expression for multiple $t$-value $t\left(2,\ldots,2,1\right)$.
In this paper, we shall show that certain signed cyclic sums of Mordell-Tornheim L-values are rational linear combinations of products of multiple L-values of lower depths (i.e., reducible). This simultaneously generalizes some results of Subbarao and Sitaramachandrarao, and Matsumoto et al. As a direct corollary, we can prove that for any integer k>2 and positive integer n, the Mordell-Tornheim sums zeta_\MT(\{n\}_k) is reducible.
In this paper, we establish some expressions of series involving harmonic numbers and Stirling numbers of the first kind in terms of multiple zeta values, and present some new relationships between multiple zeta values and multiple zeta star values. The relationships obtained allow us to find some nice closed form representations of nonlinear Euler sums through Riemann zeta values and linear sums. Furthermore, we show that the combined sums \[H\left( {a,b;m,p} \right) := \sum\limits_{a + b = m - 1} {ζ\left( {{\left\{ p \right\}_a},p + 1,{\left\{ p \right\}_b}} \right)}\quad (m\in \N,p>1) \] an…
Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation their rationality is surprising, and even simple examples are delicate to compute. For instance, we give a detailed description of the partial zeta function of an affine curve where the number of unit poles varies, a property different from classical zeta functions. On the other hand, they do retain some properties similar to the classical case. To this end, …
In this paper, we study various twisted A-harmonic sums, named following the seminal log-algebraicity papers of G. Anderson. These objects are partial sums of new types of special zeta values introduced by the first author and linked to certain rank one Drinfeld modules over Tate algebras in positive characteristic by Anglès, Tavares Ribeiro and the first author. We prove, by using techniques introduced by the second author, that various infinite families of such sums may be interpolated by polynomials, and we deduce, among several other results, properties of analogues of finite zeta values b…
Ohno's relation is a well-known family of relations among multiple zeta values, which can naturally be regarded as a type of duality for a certain power series which we call an Ohno sum. In this paper, we investigate $\mathbb{Q}$-linear relations among Ohno sums which are not contained in Ohno's relation. We prove two new families of such relations, and pose several further conjectural families of such relations.
In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begins the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of length up to $T^3$ with divisor functions as coefficients.
In this paper, we formally introduce the notion of Ap{é}ry-like sums and we show that every multiple zeta values can be expressed as a $\bf Z$-linear combination of them. We even describe a canonical way to do so. This allows us to put in a new theoretical context several identities scattered in the literature, as well as to discover many new interesting ones. We give in this paper new integral formulas for multiple zeta values and Apéry-like sums. They enable us to give a short direct proof of Zagier's formulas for $ζ(2,\ldots,2,3,2,\ldots,2)$ as well as of similar ones in the context of Ap{é…
A discussion involving the evaluation of the sum $\sum_{0<γ\le T} |ζ(1/2+iγ)|^2$ is presented, where $γ$ denotes imaginary parts of complex zeros of the Riemann zeta-function $ζ(s)$. Three theorems involving certain integrals related to this sum are proved, and the sum is unconditionally shown to be $\ll T\log^2T\log\log T$.
In this series of papers we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper completes the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of arbitrary length with divisor functions as coefficients.
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.
We prove that any Mordell-Tornheim sum with positive integer arguments can be expressed as a rational linear combination of multiple zeta values of the same weight and depth. By a result of Tsumura, it follows that any Mordell-Tornheim sum with weight and depth of opposite parity can be expressed as a rational linear combination of products of multiple zeta values of lower depth.
Bowman and Bradley proved an explicit formula for the sum of multiple zeta values whose indices are the sequence (3,1,3,1,...,3,1) with a number of 2's inserted. Kondo, Saito and Tanaka considered the similar sum of multiple zeta-star values and showed that this value is a rational multiple of a power of π. In this paper, we give an explicit formula for the rational part. In addition, we interpret the result as an identity in the harmonic algebra.
We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.
In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper is concerned with the precise input of the conjectural formula for the classical shifted convolution problem for divisor sums so as to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of length up to $T^2$ with divisor functions as coefficients.
We study the algebra MD of generating function for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in Q arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra MD is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in MD. The (quasi-)modular forms for the full modular group Sl_2(Z) constitute a sub-algebra of MD this also yields linear relations in MD. Generating functions of mul…
We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously this approach has proved unsuccessful in computing moments beyond the eighth, even heuristically. A careful analysis of the second and fourth moments illustrates the nature of the problem and enables us to identify the terms that are missed in the standard application of these methods.
Recently, the present authors jointly with Tauraso found a family of binomial identities for multiple harmonic sums (MHS) on strings $(\{2\}^a,c,\{2\}^b)$ that appeared to be useful for proving new congruences for MHS as well as new relations for multiple zeta values. Very recently, Zhao generalized this set of MHS identities to strings with repetitions of the above patterns and, as an application, proved the two-one formula for multiple zeta star values conjectured by Ohno and Zudilin. In this paper, we extend our approach to $q$-binomial identities and prove $q$-analogues of two-one formulas…
We prove an asymptotic for the sum of $ζ^{(n)} (ρ)X^ρ$ where $ζ^{(n)} (s)$ denotes the $n$th derivative of the Riemann zeta function, $X$ is a positive real and $ρ$ denotes a non-trivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height $T$, as $T \rightarrow \infty$. We also specify what the asymptotic formula becomes when $X$ is a positive integer, highlighting the differences in the asymptotic expansions as $X$ changes its arithmetic nature.
We define a parametric variant of generalized Euler sums and construct contour integration to give some explicit evaluations of these parametric Euler sums. In particular, we establish several explicit formulas of (Hurwitz) zeta functions, linear and quadratic parametric Euler sums. Furthermore, we also give an explicit evaluation of alternating double zeta values $\ze(\overline{2j},2m+1)$ in terms of a combination of alternating Riemann zeta values by using the parametric Euler sums.
In this article, we study the zeros of the partial sums of the Dedekind zeta function of a cyclotomic field $K$ defined by the truncated Dirichlet series \[ ζ_{K, X} (s) = \sum_{\|\mathfrak{a}\| \leq X} \frac{1}{\|\mathfrak{a}\|^{s}}, \] where the sum is to be taken over nonzero integral ideals $\mathfrak{a}$ of $K$ and $\|\mathfrak{a}\|$ denotes the absolute norm of $\mathfrak{a}$. Specifically, we establish the zero-free regions for $ζ_{K, X} (s)$ and estimate the number of zeros of $ζ_{K, X} (s)$ up to height $T$.
In this paper we present many new families of identities for multiple harmonic sums using binomial coefficients. Some of these generalize a few recent results of Hessami Pilehrood et al. As applications we prove several conjectures involving multiple zeta star values (MZSV): the Two-one formula conjectured by Ohno and Zudilin, and a few conjectures of Imatomi et al. involving 2-3-2-1 type MZSV, where 2 means some finite string of 2's
We establish a connection between the ratios conjecture for the Riemann zeta-function and a conjecture concerning correlations of convolutions of Möbius and divisor functions. Specifically, we prove that the ratios conjecture and an arithmetic correlations conjecture imply the same result. This provides new support for the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe. Our main theorem generalises a recent calculation pertaining to the special case of two-over-two ratios.
In this paper, we investigate the sums of mutliple zeta(-star) values of height one: $Z_{\pm}(n)=\sum_{a+b=n} (\pm 1)^bζ(\{1\}^a,b+2)$, $Z_{\pm}^{\star}(n)=\sum_{a+b=n} (\pm 1)^bζ^{\star}(\{1\}^a,b+2)$. In particular, we prove that the weighted sum $\sum_{\substack{0\leq m\leq p\\ m: {\rm even}}} \sum_{\mid\boldsymbolα\mid=p+3} 2^{α_{m+1}\ +1}ζ(α_0,α_1,\ldots,α_m,α_{m+1}+1) $ can be evaluated through the convolution of $Z_{-}(m)$ and $Z_{+}(n)$ with $m+n=p$.
Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. Since then no other moments have been asymptotically evaluated. In this article we study the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula with power savings error term for the sixth moment of th…
In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers. We discuss the analytic representations of Euler sums through values of polylogarithm function and Riemann zeta function. Moreover, we provide explicit evaluations for almost all Euler sums with weight 5, which can be expressed in terms of zeta values and polylogarithms. Furthermore, we give explicit formula for several classes of Euler-related sums in terms of zeta values and harmonic numbers, and several examples are given. The given representations are new.
This article considers the positive integers $N$ for which $ζ_{N}(s) = \sum_{n=1}^{N} n^{-s}$ has zeroes in the half-plane $\Re(s)>1$. Building on earlier results, we show that there are no zeroes for $1\leq N\leq 18$ and for $N=20, 21, 28$. For all other $N$ there are infinitely many zeroes.
Flajolet and Salvy pointed out that every Euler sum is a $\mathbb{Q}$-linear combination of multiple zeta values. However, in the literature, there is no formula completely revealing this relation. In this paper, using permutations and compositions, we establish two explicit formulas for the Euler sums, and show that all the Euler sums are indeed expressible in terms of MZVs. Moreover, we apply this method to the alternating Euler sums, and show that all the alternating Euler sums are reducible to alternating MZVs. Some famous results, such as the Euler theorem, the Borwein--Borwein--Girgensoh…
The alternating zeta function zeta*(s) = 1 - 2^{-s} + 3^{-s} - ... is related to the Riemann zeta function by the identity (1-2^{1-s})zeta(s) = zeta*(s). We deduce the vanishing of zeta*(s) at each nonreal zero of the factor 1-2^{1-s} without using the identity. Instead, we use a formula connecting the partial sums of the series for zeta*(s) to Riemann sums for the integral of x^{-s} from x=1 to x=2. We relate the proof to our earlier paper "The Riemann Hypothesis, simple zeros, and the asymptotic convergence degree of improper Riemann sums," Proc. Amer. Math. Soc. 126 (1998) 1311-1314.
Of what use are the zeros of the Riemann zeta function? We can use sums involving zeta zeros to count the primes up to $x$. Perron's formula leads to sums over zeta zeros that can count the squarefree integers up to $x$, or tally Euler's $ϕ$ function and other arithmetical functions. This is largely a presentation of experimental results.
Multiple zeta values are real numbers defined by an infinite series generalizing values of the Riemann zeta function at positive integers. Finite truncations of this series are called multiple harmonic sums and are known to have interesting arithmetic properties. When the truncation point is one less than a prime $p$, the mod $p$ values of multiple harmonic sums are called finite multiple zeta values. The present work introduces a new class of congruence for multiple harmonic sums, which we call weighted congruences. These congruences can hold modulo arbitrarily large powers of $p$. Unlike res…
We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also introduce generating functions of special values of those lattice sums, and study their properties by virtue of the theory of convex polytopes. Consequently we evaluate special values of those lattice sums, especially certain special values of zeta-functions of root systems and their affine analogues. In some special cases it is possible to treat sums running o…
Let $γ$ denote the imaginary parts of complex zeros $ρ= β+iγ$ of $ζ(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{γ> 0}γ^{-s}$ to the left of the line $\Re s = -1$ is investigated, and its Laurent expansion at the pole $s=1$ is obtained. Estimates for the second moment on the critical line $\int_1^T|G(1/2+it)|^2\,dt$ are revisited. This paper is a continuation of work begun by the second author in 2001.
We prove conjecturally sharp upper bounds for the Dirichlet character moments $\frac{1}{r-1} \sum_{χ\; \text{mod} \; r} |\sum_{n \leq x} χ(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r$, and $0 \leq q \leq 1$ is real. In particular, if both $x$ and $r/x$ tend to infinity with $r$ then $\frac{1}{r-1} \sum_{χ\; \text{mod} \; r} |\sum_{n \leq x} χ(n)| = o(\sqrt{x})$, and so the sums $\sum_{n \leq x} χ(n)$ typically exhibit "better than squareroot cancellation". We prove analogous better than squareroot bounds for the moments $\frac{1}{T} \int_{0}^{T} |\sum_{n \leq x} n^{it}|^{2q} dt$ of…
The Kaneko--Zagier conjecture describes a correspondence between finite multiple zeta values and symmetric multiple zeta values. Its refined version has been established by Jarossay, Rosen and Ono--Seki--Yamamoto. In this paper, we explicate these conjectures through studies of multiple harmonic $q$-sums. We show that the (generalized) finite/symmetric multiple zeta value are obtained by taking an algebraic/analytic limit of multiple harmonic $q$-sums. As applications, new proofs of reversal, duality and cyclic sum formulas for the generalized finite/symmetric multiple zeta values are given.
In this paper we prove a weighted sum formula for multiple harmonic sums modulo primes, thereby proving a weighted sum formula for finite multiple zeta values. Our proof utilizes difference equations for the generating series of multiple harmonic sums. We also conjecture several weighted sum formulas of similar flavor for finite multiple zeta values.
