It is well known that the Riemann Zeta function ζ ( p ) = ∑∞n=1 1/np can be represented in closed form for p an even integer. We will define a shifted quadratic Zeta series as ∑∞ n=1 1/ ( 4n2−α2)p . In this paper, we will determine closed-form representations of shifted quadratic Zeta series from a recursion point of view using the Riemann Zeta function. We will also determine closed-form repre...

The history of problems of evaluation of series associated with the Riemann Zeta function can be traced back to Christian Goldbach (1690–1764) and Leonhard Euler (1707–1783). Many di¤erent techniques to evaluate various series involving the Zeta and related functions have since then been developed. The authors show how elegantly certain families of series involving the Zeta function can be eval...

Recently, Storm [10] defined the Ihara-Selberg zeta function of a hypergraph, and gave two determinant expressions of it by the Perron-Frobenius operator of a digraph and a deformation of the usual Laplacian of a graph. We present a new determinant expression for the Ihara-Selberg zeta function of a hypergraph, and give a linear algebraic proof of Storm’s Theorem. Furthermore, we generalize the...

Recently, Storm [8] defined the Ihara-Selberg zeta function of a hypergraph, and gave two determinant expressions of it. We define the Bartholdi zeta function of a hypergraph, and present a determinant expression of it. Furthermore, we give a determinant expression for the Bartholdi zeta function of semiregular bipartite graph. As a corollary, we obtain a decomposition formula for the Bartholdi...

Sharp bounds are obtained for expressions involving Zeta and related functions at a distance of one apart. Since Euler discovered in 1736 a closed form expression for the Zeta function at the even integers, a comparable expression for the odd integers has not been forthcoming. The current article derives sharp bounds for the Zeta, Lambda and Eta functions at a distance of one apart. The methods...

Currently at least 11 protein kinase C (PKC) isoforms have been identified and may play different roles in cell signaling pathways leading to changes in cardiac contractility, the hypertrophic response, and tolerance to myocardial ischemia. The purpose of the present study was to test the hypothesis that responses of individual PKC isoforms to distinct pathological stimuli were differentially r...

Abstract. Multiple q-zeta values are a 1-parameter generalization (in fact, a q-analog) of the multiple harmonic sums commonly referred to as multiple zeta values. These latter are obtained from the multiple q-zeta values in the limit as q → 1. Here, we discuss the sum formula for multiple q-zeta values, and provide a self-contained proof. As a consequence, we also derive a q-analog of Euler’s ...

The purpose of the present article is to survey some mean value results obtained recently in zeta-function theory. We do not mention other important aspects of the theory of zeta-functions, such as the distribution of zeros, value-distribution, and applications to number theory. Some of them are probably treated in the articles of Professor Apostol and Professor Ramachandra in the present volum...