Certain relationships among polygamma functions, Riemann zeta function and generalized zeta function

q-Riemann zeta function

We consider the modified q-analogue of Riemann zeta function which is defined by ζq(s)= ∑∞ n=1(qn(s−1)/[n]s), 0< q < 1, s ∈ C. In this paper, we give q-Bernoulli numbers which can be viewed as interpolation of the above q-analogue of Riemann zeta function at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Also, we will treat some...

Lagrangians with Riemann Zeta Function

We consider construction of some Lagrangians which contain the Riemann zeta function. The starting point in their construction is p-adic string theory. These Lagrangians describe some nonlocal and nonpolynomial scalar field models, where nonlocality is controlled by the operator valued Riemann zeta function. The main motivation for this research is intention to find an effective Lagrangian for ...

Mollifying the Riemann Zeta-function

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

A q-analogue ζq(s) of the Riemann zeta function ζ(s) was studied in [Kaneko et al. 03] 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 ζq(s) left open in [Kaneko et al. 03]. We also examine a “crystal” limit (i.e. q ↓ 0) behavior of ζq(s). T...

Spectral zeta functions of graphs and the Riemann zeta function in the critical strip

We initiate the study of spectral zeta functions ⇣X for finite and infinite graphs X, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann’s zeta funct...