A segment of Euler product associated to a certain Dirichlet series (Q6613266)

The main goal in this paper is to express a Dirichlet series associated to a generalized divisor function \(\sigma_\alpha(n) :=\sum_{d|n} d^\alpha\) in terms of a segment of an Euler product in the critical strip where the Dirichlet series and its Euler product are not convergent. More precisely, a precise formula is given for \[\zeta(s) \zeta(s-\alpha)\prod_{p\in \mathbb{P}}\left(1-\frac{1}{p^s}\right)\left(1-\frac{1}{p^{s-\alpha}}\right), \] where \(\mathbb{P}\) is a finite set of primes, \(\alpha\in \mathbb{C}^*\) and \(\Re(s)>\max\lbrace0, \Re(\alpha)\rbrace\) with \(s\neq 1, 1+\alpha\).\N\NAs consequence, the authors derive a special case of their main theorem as \(\alpha \to 0\), the identity for \(\zeta^2(s)\) in terms of a segment of an Euler product and a series involving the modified Bessel function of the second kind of order zero. More particularly, if we let \(\mathbb{P}\) be the collection of all primes, then we recover the well-known Euler identity, namely, we get for \(\Re(s) > 1\) \[\zeta^2(s)\prod_p \left( 1-\frac{1}{p^s} \right)^2=1 . \]