Let φ denote Euler’s phi function. For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n 6 x such that q ∤ φ(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the Hardy-Littlewood conjecture about counts of prime k-tuples and a conjecture o...

We nd a condition for weights on the edges of a graph which insures that the Ihara zeta function has a 3-term determinant formula. Then we investigate the locations of poles of abelian graph coverings and compare the results with random covers. We discover that the zeta function of the random cover satis es an approximate Riemann hypothesis while that of the abelian cover does not.

Let φ denote Euler’s phi function. For a fixed odd prime q we give an asymptotic series expansion in the sense of Poincaré for the number Eq(x) of n ≤ x such that q ∤ φ(n). Thereby we improve on a recent theorem by B.K. Spearman and K.S. Williams [Ark. Mat. 44 (2006), 166–181]. Furthermore we resolve, under the Generalized Riemann Hypothesis, which of two approximations to Eq(x) is asymptotical...

This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function ζ0(s) which generalizes the Artin-Mazur zeta function was given by Lind for Z2-action φ. The n-th order zeta function ζn of φ on Zn×∞, n ≥ 1, is studied first. The trace operator Tn which is the transition matrix for x-periodic patterns of period n with height 2 is rotationally s...

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 Riemann zeta process is a stochastic process {Z(σ), σ > 1} with independent increments and marginal distributions whose characteristic functions are proportional to the Riemann zeta function along vertical lines < s = σ . We establish functional limit theorems for the zeta process and other related processes as arguments σ approach the pole at s = 1 of the zeta function (from above).

This paper investigates a generalization the classical Hurwitz zeta function. It is shown that many of the properties exhibited by this special function extends to class of functions called hypergeometric Hurwitz zeta functions, including their analytic continuation to the complex plane and a pre-functional equation satisfied by them. As an application, a formula for moments of hypergeometric H...

For a d-dimensional real hyperbolic manifold with cusps, we obtain more refined error terms in the prime geodesic theorem (PGT) using the Ruelle zeta function instead of the Selberg zeta function. To do this, we prove that the Ruelle zeta function over this type manifold is a meromorphic function of order d over C.