On $p$-adic Hurwitz-type Euler zeta functions

Details On $p$-adic Hurwitz-type Euler zeta functions On $p$-adic Hurwitz-type Euler zeta functions

2010-10-12

arxiv.org/abs/1010.2269v2

Mathematics

Number Theory

22 Pages, Corrected typos

Scientific paper

Henri Cohen and Eduardo Friedman constructed the $p$-adic analogue for Hurwitz zeta functions, and Raabe-type formulas for the $p$-adic gamma and zeta functions from Volkenborn integrals satisfying the modified difference equation. In this paper, we define the $p$-adic Hurwitz-type Euler zeta functions. Our main tool is the fermionic $p$-adic integral on $\mathbb Z_p$. We find that many interesting properties for the $p$-adic Hurwitz zeta functions are also hold for the $p$-adic Hurwitz-type Euler 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.

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