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

Special value formula for the spectral zeta function of the non-commutative harmonic oscillator

This series is absolutely convergent in the region Rs > 1, and defines a holomorphic function in s there. We call this function ζQ(s) the spectral zeta function for the non-commutative harmonic oscillator Q, which is introduced and studied by Ichinose and Wakayama [1]. The zeta function ζQ(s) is analytically continued to the whole complex plane as a single-valued meromorphic function which is h...

Sphere Packing, Lattices, and Epstein Zeta Function

On the harmonic index and harmonic polynomial of Caterpillars with diameter four

The harmonic index H(G) , of a graph G is defined as the sum of weights 2/(deg(u)+deg(v)) of all edges in E(G), where deg (u) denotes the degree of a vertex u in V(G). In this paper we define the harmonic polynomial of G. We present explicit formula for the values of harmonic polynomial for several families of specific graphs and we find the lower and upper bound for harmonic index in Caterpill...

On the Universality of the Epstein Zeta Function

We study universality properties of the Epstein zeta function En(L, s) for lattices L of large dimension n and suitable regions of complex numbers s. Our main result is that, as n → ∞, En(L, s) is universal in the right half of the critical strip as L varies over all n-dimensional lattices L. The proof uses an approximation result for Dirichlet polynomials together with a recent result on the d...

On the Minima and Convexity of Epstein Zeta Function

to the complex plane. We show that for fixed s 6= n/2, the function Zn(s; a1, . . . , an), as a function of (a1, . . . , an) ∈ (R) with fixed Qn i=1 ai, has a unique minimum at the point a1 = . . . = an. When Pn i=1 ci is fixed, the function (c1, . . . , cn) 7→ Zn (s; e c1 , . . . , en ) can be shown to be a convex function of any (n−1) of the variables {c1, . . . , cn}. These results are then ...