Perturbed Dirichlet Series and the Difference Operator

The most well-known perturbed Dirichlet series is the Hurwitz zeta-function. Its analytic continuation via the binomial expansion has been studied extensively, beginning with Wilton’s work. In this paper, we shall provide, above all things, two striking instances of the binomial expansion. One is elucidation of Mikolás an integral formula for the Hurwitz zeta-function valid in the critical strip to the effect that it is a manifestation of the picking-up principle of the values at the poles of the gamma function of the binomial expansion. The other is a new proof of Hasse’s formula by the binomial expansion. Also, we show the effectiveness of the difference operator in dealing with a series of the form ∑n=0∞(n+a1)−s1(n+a2)−s2(n+a3)−s3⋯,Resj>2,j=1,2,⋯ where 0<aj≤1 or aj∈H (in the upper half-plane). Furthermore, elucidation of the above results is made in the light of the Hardy–Hecke transform.