Su Hu (McGill)

On p-adic Hurwitz-type Euler zeta functions

The p-adic Hurwitz zeta function is the p-adic analogue of the classical Hurwitz zeta function $\zeta(s,x)=\sum_{n=0}^\infty\frac{1}{(n+x)^s}$. It can be defined by using Volkenborn's p-adic integral and it interpolates the Bernoulli polynomials p-adically.

In this talk, using the fermionic p-adic integral, we give a definition for the p-adic analogue of Euler's deformation of the Hurwitz zeta function: $\zeta_{E}(s,x)=2\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+x)^{s}}, the so-called p-adic Hurwitz-type Euler zeta function. We show that it interpolates the Euler polynomials p-adically and that it also shares many fundamental properties with the p-adic Hurwitz zeta functions, including the convergent Laurent series expansion, the distribution formula, the functional equation, the reflection formula, the derivative formula, the p-adic Raabe formula, and so on.

As in the Gross-Stark conjecture, the derivative of p-adic Hurwitz-type Euler zeta functions at s=0 will be connected to a special case of the p-adic analogue of the (S,T)-version of the abelian rank one Stark conjecture.