The two-variable zeta function and the Riemann hypothesis for function fields

We present Bombieri's proof of the Riemann hypothesis for the zeta function of a curve over a finite field. We first briefly describe this zeta function and discuss the two-variable zeta function of Pellikaan. Then we give Naumann's proof that the numerator of this function is irreducible.     

Asymptotic expansions of the Hurwitz-Lerch zeta function

The Hurwitz-Lerch zeta function Φ(z,s,a) is considered for large and small values of a∈C, and for large values of z∈C, with |Arg(a)|<π, z∉[1,∞) and s∈C. This function is originally defined as a power series in z, convergent for |z|<1, s∈C and 1−a∉N. An integral representation is obtained for Φ(z,s,a) which define the analytical continuation of the Hurwitz-Lerch zeta function to the cut complex z-plane C?[1,∞). From this integral we derive three complete asymptotic expansions for either large or small a and large z. These expansions are accompanied by error bounds at any order of the approximation. Numerical experiments show that these bounds are very accurate for real values of the asymptotic variables.     

Localization of the first zero of the Dedekind zeta function

Using Weil's explicit formula, we propose a method to compute low zeros of the Dedekind zeta function. As an application of this method, we compute the first zero of the Dedekind zeta function associated to totally complex fields of degree less than or equal to 30 having the smallest known discriminant.

     

Remarks on the universality of the periodic zeta function

We study the universality of a Dirichlet series with periodic coefficients. This property is proved in the case of multiplicative coefficients, and in the general case we establish universality in a certain set of analytic functions related to a probability distribution.     

Riemann zeta函数的收敛区域

给出了Riemann zeta函数收敛区域的几种证明.     

Continuity of the hyperbolic zeta function of lattices

The continuity of the hyperbolic zeta function of lattices _H (¦),=+it, >1, as a function on the metric space of lattices is proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 522–526, April, 1998.The authors wish to express their gratitude to Professor N. M. Korobkov and Professor V. I. Nechaev for fruitful discussions.     

The cotype zeta function of Zd

We give an asymptotic formula for the number of sublattices $\Lambda \subseteq {\mathbb{Z}}^d$ of index at most $X$ for which ${\mathbb{Z}}^d/\Lambda$ has rank at most $m$, answering a question of Nguyen and Shparlinski. We compare this result to work of Stanley and Wang on Smith normal forms of random integral matrices and discuss connections to the Cohen–Lenstra heuristics. Our arguments are based on Petrogradsky’s formulas for the cotype zeta function of ${\mathbb{Z}}^d$, a multivariable generalization of the subgroup growth zeta function of ${\mathbb{Z}}^d$.     

On values of the Riemann zeta function at positive integers

Lithuanian Mathematical Journal - We give new proofs of some known results on the values of the Riemann zeta function at positive integers and obtain some new theorems related to these values....     

Higher moment of coefficients of Dedekind zeta function

Let be a non-normal cubic extension over Q.We study the higher moment of the coefficients aK3(n)of Dedckind zeta function over sum of two squares∑n21+n22≤xa1K3(n21+n22),where 2≤l≤8 and n1,n2,l∈Z.     

Evaluation of the Dedekind zeta functions of some non-normal totally real cubic fields at negative odd integers

Let {K _m } _{m ≥ 4} be the family of non-normal totally real cubic number fields defined by the irreducible cubic polynomial f _m (x) = x ³ − mx ² − (m + 1)x − 1, where m is an integer with m ≥ 4. In this paper, we will apply Siegel’s formula for the values of the zeta function of a totally real algebraic number field at negative odd integers to K _m , and compute the values of the Dedekind zeta function of K _m . This work was supported by grant No.R01-2006-000-11176-0 from the Basic Research Program of KOSEF.     

An integral representation of the Mordell-Tornheim double zeta function and its values at non-positive integers

A surface integral representation of the Mordell-Tornheim double zeta function is given, which is a direct analogue of a well-known integral representation of the Riemann zeta function of Hankel’s type. As an application, we investigate its values and residues at integers, where generalizations of a generating function of Bernoulli numbers naturally appear.      

Many values of the Riemann zeta function at odd integers are irrational

In this note, we announce the following result: at least $2^{(1?\epsilon )\frac{\log ?s}{\log ?\log ?s}}$ values of the Riemann zeta function at odd integers between 3 and s are irrational, where ε is any positive real number and s is large enough in terms of ε. This improves on the lower bound $\frac{1?\epsilon }{1+\log ?2}\log ?s$ that follows from the Ball–Rivoal theorem. We give the main ideas of the proof, which is based on an elimination process between several linear forms in odd zeta values with related coefficients.     

Representations for the derivative at zero and finite parts of the Barnes zeta function

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes gamma function, including the particular case also known as multiple gamma function.     

New inequalities for the Hurwitz zeta function

We establish various new inequalities for the Hurwitz zeta function. Our results generalize some known results for the polygamma functions to the Hurwitz zeta function.     

The joint value-distribution of the Riemann zeta function and Hurwitz zeta functions

In this paper, we investigate the joint value-distribution for the Riemann zeta function and Hurwitz zeta function attached with a transcendental real parameter. Especially, we establish the joint universality theorem for these two zeta functions. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 39–57, January–March, 2007.     

Another discrete Fourier transform pairs associated with the Lipschitz-Lerch zeta function

It is demonstrated that the alternating Lipschitz-Lerch zeta function and the alternating Hurwitz zeta function constitute a discrete Fourier transform pair. This discrete transform pair makes it possible to deduce, as special cases and consequences, many (mainly new) transformation relations involving the values at rational arguments of alternating variants of various zeta functions, such as the Lerch and Hurwitz zeta functions and Legendre chi function.     

The average behavior of the coefficients of Dedekind zeta function over square numbers

In this paper, we are interested in the average behavior of the coefficients of Dedekind zeta function over square numbers. In Galois fields of degree d which is odd, when l?1 is an integer, we have

     

Remark on a double-inequality for the Riemann zeta function

Let ζ be the Riemann zeta function and δ(x)=1/(2^x-1). For all x>0 we have

(1-δ(x))ζ(x)+αδ(x)<ζ(x+1)<(1-δ(x))ζ(x)+βδ(x),