Proof of some conjectures on the mean-value of Titchmarsh series - III

Abstract

With some applications in view, the following problem is solved in some special case which is not too special. Let F(s) = ∑_n=1^∞ a_n λn^-s be a generalized Dirichlet series with 1 = λ₁ < λ₂ < ..,λ_n ≤Dn, and λ_n+1 - λ_n ≥ D ^-1 λ_n+1^-α where α>0 and D(≥ 1) are constants. Then subject to analytic continuation and some growth conditions, a lower bound is obtained for (1/H) ∫_O^H|F(it)|²dt These results will be applied in other papers to appear later.