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...

A weak version of the Ihara formula is proved for zeta functions attached to quotients of the Bruhat-Tits building of PGL3. This formula expresses the zeta function in terms of Hecke-Operators. It is the first step towards an arithmetical interpretation of the combinatorially defined zeta function.

We show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta function which implies a lower bound of the class numbers of imaginary quadratic fields.

In this paper, partial Epstein zeta functions on binary linear codes, which are related with Hamming weight enumerators of binary linear codes, are newly defined. Then functional equations for those zeta functions on codes are presented. In particular, it is clarified that simple functional equations hold for partial Epstein zeta functions on binary linear self-dual codes.

Engagement of the T cell antigen receptor (TCR) results in activation of several tyrosine kinases leading to tyrosine phosphorylation of protein substrates and activation of multiple biochemical pathways. TCR-mediated activation of the src-family kinases, Lck and Fyn, results in tyrosine phosphorylation of the TCR zeta and CD3 chains. The site of phosphorylation in these chains is the tyrosine-...

Previous studies have suggested that 1) atypical protein kinase C (PKC) isoforms are required for insulin stimulation of glucose transport, and 2) 3-phosphoinositide-dependent protein kinase-1 (PDK-1) is required for activation of atypical PKCs. Presently, we evaluated the role of PDK-1, both in the activation of PKC-zeta, and the translocation of epitope-tagged glucose transporter 4 (GLUT4) to...

In this note, we describe various theoretical results, numerical computations, and speculations concerning the analytic properties of the Shintani zeta functions associated to the space of binary cubic forms. We describe how these zeta functions almost fit into the general analytic theory of zeta and L-functions, and we discuss the relationship between this analytic theory and counting problems...

We study a polynomial interpolation of finite multiple zeta and zeta-star values with variable $t$, which is an analogue interpolated introduced by Yamamoto. introduce several relations among them and, in particular, prove the cyclic sum formula, Bowman--Bradley type weighted formula. The harmonic relation, shuffle duality derivation relation are also presented.