Yoshinori Yamasaki

Q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function

Abstract A q-analogue ζ q (s) of the Riemann zeta function ζ(s) was studied in [Kaneko M., Kurokawa N. and Wakayama M.: A variation of Eulers approach to values of the Riemann zeta function. Kyushu J. Math. 57 (2003), 175–192] via a certain q-series of two variables. We introduce in a similar way a q-analogue of the Dirichlet L-functions and make a detailed study of them, including some issues concerning the classical limit of ζ q (s) left open in [Kaneko M., Kurokawa N. and Wakayama M.: A variation of Eulers approach to values of the Riemann zeta function. Kyushu J. Math. 57 (2003), 175–192]. We also examine a “crystal” limit (i.e. q ↓ 0) behavior of ζ q (s). The q-trajectories of the trivial and essential zeros of ζ(s) are investigated numerically when q moves in (0, 1]. Moreover, conjectures for the crystal limit behavior of zeros of ζ q (s), which predict an interesting distribution of “trivial zeros” and an analogue of the Riemann hypothesis for a crystal zeta function, are given. 2000 Mathematics Subject Classification: 11M06.

Integral representations of q-analogues of the Hurwitz zeta function

Abstract.Two integral representations of q-analogues of the Hurwitz zeta function are established. Each integral representation allows us to obtain an analytic continuation including also a full description of poles and special values at non-positive integers of the q-analogue of the Hurwitz zeta function, and to study the classical limit of this q-analogue. All the discussion developed here is entirely different from the previous work in [5].

Evaluations of multiple Dirichlet L-values via symmetric functions

Abstract We explicitly evaluate a special type of multiple Dirichlet L-values at positive integers in two different ways: One approach involves using of symmetric functions, while the other involves using of a generating function of the values. Equating these two expressions, we derive several summation formulae involving the Bernoulli and Euler numbers. Moreover, values at non-positive integers, called central limit values, are also studied.

A variation of multiple

A variation of multiple L-values, which arises from the description of the special values of the spectral zeta function of the non-commutative harmonic oscillator, is introduced. In some special cases, we show that its generating function can be written in terms of the gamma functions. This result enables us to obtain explicit evaluations of them.

L

In this paper, we determine the bound of the valency of Cayley graphs of Frobenius groups with respect to normal Cayley subsets which guarantees to be Ramanujan. We see that if the ratio between the orders of the Frobenius kernel and complement is not so small, then this bound coincides with the trivial one coming from the trivial estimate of the largest non-trivial eigenvalue of the graphs. Moreover, in the cases of the dihedral groups of order twice odd primes, which are special cases of the Frobenius groups, we determine the same bound for the Cayley graphs of the groups with respect to not only normal but also all Cayley subsets. As is the case of abelian groups which we have treated in the previous papers, such a bound is equal to the trivial one in the above sense or, as exceptional cases, exceeds one from it. We then clarify that the latter occurs if and only if the corresponding prime is represented by a quadratic polynomial in a finite family.

-values arising from the spectral zeta function of the non-commutative harmonic oscillator

In this paper, we introduce

Ramanujan Cayley graphs of Frobenius groups

q

On

-analogues of the Barnes multiple zeta functions. We show that these functions can be extended meromorphically to the whole plane, and moreover, tend to the Barnes multiple zeta functions when

q

q\uparrow 1

-Analogues of the Barnes Multiple Zeta Functions

for all complex numbers.