In this paper, we present a different proof of the well known recurrence formula for the Riemann zeta function at positive even integers, the integral representations of the Riemann zeta function at positive integers and at fractional points by means of a probabilistic approach.
The aim of this paper is to give some new identities and relations related to
the some families of special numbers such as the Bernoulli numbers, the
Euler numbers, the Stirling numbers of the first and second kinds, the
central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?)
which are given Simsek [31]. Our method is related to the functional
equations of the generating functions and the fermionic and bosonic p-adic
Volkenborn integral on Zp. Finally, we give remarks and comments on our
results.
Abstract
Let L be a number field. For a given prime p, we define integers
α
p
L
$ \alpha_{p}^{L} $
and
β
p
L
$ \beta_{p}^{L} $
with some interesting arithmetic properties. For instance,
β
p
L
$ \beta_{p}^{L} $
is equal to 1 whenever p does not ramify in L and
α
p
L
$ \alpha_{p}^{L} $
is divisible by p whenever p is wildly ramified in L. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of L. Moreover, if the residue class mod p of
α
p
L
$ \alpha_{p}^{L} $
is not zero for all p then such residues determine the genus of the integral trace.
A Probabilistic Proof for Representations of the Riemann Zeta Function
