An elementary bound on Siegel zeroes

Abstract

We consider Dirichlet L-functions L(s, χ) where χ is a real, non-principal character modulo q. Using Pintz's refinement of Page's theorem, we prove that for q ≥ 3 the function L(s, χ) has at most one real zero β with 1 − 1.011/ log q < β < 1.

Numerical estimates on the Landau-Siegel zero and other related quantities

“…Recalling that Watkins [19] showed that there are no Siegel zeroes for 𝑞 ≤ 3 • 10 8 whenever 𝜒 is odd, and Platt [15] reached the same conclusion for 𝜒 even and 𝑞 ≤ 4 • 10 5 , our Theorem 1 is meaningful only for 4 • 10 5 < 𝑞 ≤ 10 7 , 𝑞 prime, and 𝜒 even. Moreover, we also recall that Morrill and Trudgian [14] proved that for 𝑞 ≥ 3, the function 𝐿 (𝑠, 𝜒 ) has at most one real zero 𝛽 with 1 − 0.933/log 𝑞 < 𝛽 < 1.…”

“…Using the setting explained in the previous subsection, we were able to compute, using the long double precision (80 bits) of the C programming language, the values of 𝐿(1, 𝜒 ) for every odd prime 𝑞 ≤ 10 7 and we provide here the scatter plots of such values normalised using log 𝑞 (in other words, the values of 𝑐 1 (𝑞) = 𝐿 (1, 𝜒 )/log 𝑞, see ( 12)), and the corresponding values for 𝑐 2 (𝑞) as defined in (14), see Figures 1-4. We also inserted the histograms obtained with the same values, see Figures 5 and 6.…”

“…, attained at 𝑞 = 23, respectively 𝑞 = 311 (in both cases 𝜒 is odd). The dependence of 𝑐 2 (𝑞) from 𝑐 1 (𝑞) is stated in (14) below and it involves also a computed value for 𝑆(𝑞) := 𝑞 𝑛=2 (log 𝑛)/𝑛 and an effective version of the Pólya-Vinogradov inequality proved by Lapkova [11] in 2018.…”

An explicit Pólya-Vinogradov inequality via Partial Gaussian sums

“…Using computations from [23,25,26], we can focus on real zeros of nonprincipal real characters with modulus q ≥ 4 · 10 5 . Previous explicit estimates for β 0 have been of the form…”