A fast algorithm to compute the Ramanujan-Deninger gamma function and some number-theoretic applications

Numerical verification of Littlewood's bounds for $\vert L(1,χ)\vert$

“…, q − 2}. This idea was first formulated by Rader [17] and it was used in [2,11,12,13] to speed-up the computation of similar quantities via the use of Fast Fourier Transform dedicated software libraries.…”

“…We follow the argument used in Languasco-Righi [13] to study the Ramanujan-Deninger Gamma function Γ 1 (x). We immediately remark that the series in (12) absolutely converges for x ∈ (0, 2); this fact and the well-known relation…”

Small values of $| L^\prime/L(1,χ) |$

“…Theorem 1.1 gives the first known non-trivial upper bound for m q . Furthermore, using the algorithm developed in Languasco-Righi [13] together with the results of Languasco [12], we were able to compute the values of m q for q ≤ 10 7 and obtain the following computational result. Theorem 1.2.…”

“…In this way we can compute m q = min(m odd q , m even q ), 3 ≤ q ≤ 10 7 , q prime, using the values of log Γ and S obtained in [12] for q ≤ 10 6 and with a new set of computation for 10 6 < q ≤ 10 7 . We recall that computing the needed values of S(a/q) is the most time consuming step of the whole procedure; for a detailed description on how to obtain such values, we refer to [12] and to a new, much faster, algorithm developed by Languasco and Righi in [13]. In fact it is the latter method that made it possible to obtain the new set of results for 10 6 < q ≤ 10 7 .…”

“…. , q − 1 obtained in [12] and, for 10 6 < q ≤ 10 7 , computed with the algorithm described in [13], we obtained the values of m q for every odd prime q ≤ 10 7 using the long double precision (80 bits) of the C programming language. For q up to 10 6 this just required about one day of time on the Dell Optiplex machine previously mentioned since the data on S(a/q) were already available from [12].…”