On the nontrivial zeros of modified Epstein zeta functions

We study the zeros of modified Epstein zeta functions having functional equations. The result is that for any $\delta >0$, all but finitely many nontrivial zeros of such a function in $\{s\in \mathbb{C}:|s?\frac{1}{2}|<\delta \}$ are simple and on the critical line. As an immediate consequence of this theorem, all but finitely many nontrivial zeros of many modified Epstein zeta functions are simple and on the critical line. To cite this article: H. Ki, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Hecke-Siegel's pull-back formula for the Epstein zeta function with a harmonic polynomial

In this paper, we discuss the generalization of the Hecke's integration formula for the Epstein zeta functions. We treat the Epstein zeta function as an Eisenstein series come from a degenerate principal series. For the Epstein zeta function of degree two, Siegel considered the Hecke's formula as the constant term of a certain Fourier expansion of the Epstein zeta function and obtained the other Fourier coefficients as the Dedekind zeta functions with Grössencharacters of a real quadratic field. We generalize this Siegel's Fourier expansion to more general Eisenstein series with harmonic polynomials. Then we obtain the Dedekind zeta functions with Grössencharacters for arbitrary number fields.     

On the value distribution and moments of the Epstein zeta function to the right of the critical strip

We study the Epstein zeta function En(L,s) for and a random lattice L of large dimension n. For any fixed we determine the value distribution and moments of En(⋅,cn) (suitably normalized) as n→∞. We further discuss the random function c?En(⋅,cn) for c∈[A,B] with and determine its limit distribution as n→∞.     

On the prime zeta function

The basic properties of the prime zeta function are discussed in some detail. A certain Dirichlet series closely connected with the function is introduced and investigated. Its dependence on the structure of the natural numbers with respect to their factorization is particularly stressed.     

Ramanujan series for Epstein zeta functions

In the spirit of Ramanujan, we derive exponentially fast convergent series for Epstein zeta functions \(E^{\varGamma _0(N)}(z,s)\) on the Hecke congruence groups \( \varGamma _0(N),N\in \mathbb {Z}_{>0}\), where z is an arbitrary point in the upper half-plane \( \mathfrak {H}\) and \(s\in \mathbb {Z}_{>1}\). These Ramanujan series can be reformulated as integrations of modular forms, in the framework of Eichler integrals. Particular cases of these Eichler integrals recover part of the recent results reported by Wan and Zucker (arXiv:1410.7081v1, 2014).     

On the Rédei zeta function

Let L be a locally finite lattice. An order function ν on L is a function defined on pairs of elements x, y (with x ≤ y) in L such that ν(x, y) = ν(x, z) ν(z, y). The Rédei zeta function of L is given by ?(s; L) = Σ_x∈Lμ(Ô, x) ν(Ô, x)^?s. It generalizes the following functions: the chromatic polynomial of a graph, the characteristic polynomial of a lattice, the inverse of the Dedekind zeta function of a number field, the inverse of the Weil zeta function for a variety over a finite field, Philip Hall's φ-function for a group and Rédei's zeta function for an abelian group. Moreover, the paradigmatic problem in all these areas can be stated in terms of the location of the zeroes of the Rédei zeta function.     

On q-analogues of Riemann's zeta function

In the paper, we introduce q-deformations of the Riemann zeta function, extend them to the whole complex plane, establish a q-counterpart of the TlogT-formula for the number of zeros, and discuss theoretically and (mainly) numerically a q-variant of the Riemann hypothesis. The construction is closely related to the recent difference generalization of the Harish-Chandra theory of zonal spherical functions. The q-zeta functions do not satisfy the functional equation but have some analytic advantages.     

On spectral theory of the Riemann zeta function

Science China Mathematics - Every nontrivial zero of the Riemann zeta function is associated as eigenvalue with an eigenfunction of the fundamental differential operator on a Hilbert-Pólya...     

On the zeta function of a family of quintics

In this article, we give a proof of the link between the zeta function of two families of hypergeometric curves and the zeta function of a family of quintics that was observed numerically by Candelas, de la Ossa, and Rodriguez Villegas. The method we use is based on formulas of Koblitz and various Gauss sums identities; it does not give any geometric information on the link.     

On some series representations of the Hurwitz zeta function

A variety of infinite series representations for the Hurwitz zeta function are obtained. Particular cases recover known results, while others are new. Specialization of the series representations apply to the Riemann zeta function, leading to additional results. The method is briefly extended to the Lerch zeta function. Most of the series representations exhibit fast convergence, making them attractive for the computation of special functions and fundamental constants.     

On the mean square value of Hurwitz zeta function

R Balasubramanian has shown that $$\mathop \smallint \limits_1^{\rm T} |\zeta (\tfrac{1}{2} + it)|^2 dt = T\log \tfrac{T}{{2\pi }} + (2\gamma - 1)T + O(T^{\theta + \in } )$$ with θ = 1/3. In this paper we develop a hybrid analogue for the mean square value of the Hurwitz zeta function ζ (s, a) and show that (i) new asymptotic terms arise in the expression for ζ (s, a) which are not present in the above expression for the ordinary zeta function and (ii) the corresponding error term is given by $$O(T^{5/12} log^2 T) + O\left( {\frac{{logT}}{{\left\| {2a} \right\|}}} \right)$$ for 0 <a < 1.     

On an equivariant analogue of the monodromy zeta function

We offer an equivariant analogue of the monodromy zeta function of a germ invariant with respect to an action of a finite group G as an element of the Grothendieck ring of finite (?×G)-sets. We state equivariant analogues of the Sebastiani-Thom theorem and of the A’Campo formula.     

On the p-adic meromorphy of the function field height zeta function

In this brief note, we will investigate the number of points of bounded height in a projective variety defined over a function field, where the function field comes from a projective variety of dimension greater than or equal to 2. A first step in this investigation is to understand the p-adic analytic properties of the height zeta function. In particular, we will show that for a large class of projective varieties this function is p-adic meromorphic.     

On some new properties of the gamma function and the Riemann zeta function

In this paper, we have exhibited, by utilizing value distribution theory, some new properties of the Gamma function Γ(z) and the Riemann zeta function ζ(z). Specifically, we have proved that both of the two functions are prime and the Riemann zeta function, like Γ(z), does not satisfy any algebraic differential equation with coefficients in ??₀. Moreover, the two functions do not satisfy any functional equation of the form P(Γ, ζ, z) ≡ 0, where P(x, y, z) is a nonconstant polynomial in x, y and z.