Functional Equations With Multiple Gamma Factors and the Average Order of Arithmetical Functions

Abstract

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On the structure of the Selberg class, VI: non-linear twists

“…We remark that Corollary 2 can possibly be proved in a classical way by means of Voronoï-type expansions, although we could not trace this result (in the full generality of the class S ) in ChandrasekharanNarasimhan [3] and related papers. Moreover, the proof of Corollary 2 clearly shows that the exponent in the Ω-estimate is caused by the pole of F (s, α) (with a suitable choice of α) at s = s 0 .…”

“…and by Stirling's formula 3 , say, as required. Now we turn to the J ± -integrals; we consider only the J + -integral since both the treatment of the J − -integral and the resulting bound are the same.…”

“…We first remark that the constants implied in the -symbols below may depend on F and L. Moreover, in view of (2.4) and condition V ≥ c(|t| + L) 3 , the paths in the integrals under consideration have positive distance from the poles of h(w, s). Further, we only prove the bounds for such integrals, since their holomorphy follows by the same argument.…”

“…Since (2.6) can be proved starting with the classical Stirling formula and performing standard manipulations, we omit the proof. Taking z = λ j (|u| − iV ) and a = λ j (1 − s) + µ j or a = 1 − λ j s − µ j we have |arg z| ≤ π/2 and, in view of the assumption V ≥ c(|t| + L) 3 , also |a| ≤ |z| 1/3 . Hence from (2.5), (2.6) and d = 1 we get…”

“…Let (λ, µ) ∈ F, V > 0 be sufficiently large and s belong to the rectangle R defined by −L ≤ σ ≤ R and V ≥ c(|t| + L) 3 …”