A Mean Value Theorem for Dirichlet Series and a General Divisor Problem

On mean values of some zeta-functions in the critical strip

“…Theorem 1 will provide good results for values of σ close to 1. Results of similar type for the general case and the case of the Rankin-Selberg series can be found in [12] and [13]. However in the proof of Theorem 1 we shall avoid using the Cauchy-Schwarz inequality and therefore obtain a sharper value of the exponent than we would obtain by following the ideas of [12] and [13].…”

“…Similarly, by using the first derivative test (see [4, Lemma 2.1]), we obtain 12) where the interchange of the order of integration is justified by absolute convergence. Therefore from (3.10)-(3.12) it follows that…”

On general divisor problems involving Hecke eigenvalues

“…For example, Kanemitsu, Sankaranarayanan and Tanigawa [13] studied a general divisor problem, and as examples showed that for k 2…”

“…which can be regarded as the general divisor problems considered by Kanemitsu, Sankaranarayanan and Tanigawa [13]. In fact, when k = 1, these are two classical problems, which have received attention of many authors.…”

On the value-distribution of Epstein zeta-functions

“…We leave out the proof since this mean-square estimate follows immediately from a far more general result due to Kanemitsu et al [17].…”