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.