Iwaniec, H. On zeros of Dirichlet’s \(L\)-series. (English) Zbl 0275.10024 Invent. Math. 23, 97-104 (1974). A. G. Postnikov [Izv. Akad. Nauk SSSR, Ser. Mat. 19, 11–16 (1955; Zbl 0068.04003)] has shown how sums of characters to a prime-power modulus can be estimated using known estimates for Weyl sums. This discovery implied important results on \(L\)-functions and distribution of primes. Further developments have been given by S. M. Rozin [Izv. Akad. Nauk SSSR, Ser. Mat. 23, 503–508 (1959; Zbl 0089.02803)] and by P. X. Gallagher [Invent. Math. 16, 191–201 (1972; Zbl 0246.10030)]. The author generalizes Gallagher’s variant of Postnikov’s method to a general modulus. We quote an application to zero-free regions. Let \(q\geq 3\), \(d\) the product of different prime factors of \(q\), \(\ell=\log q(| t|+3)\), \(\vartheta^{-1}=4\cdot 10^4\left(\log d+(\ell\log 2\ell)^{3/4}\right)\). Then the region \(\text{Re}\,s>1-\vartheta\) does not contain any zero of any \(L\)-function \(\pmod q\), except for the possible Siegel-zero. Reviewer: Matti Jutila (Turku) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 8 ReviewsCited in 25 Documents MSC: 11L40 Estimates on character sums 11M06 \(\zeta (s)\) and \(L(s, \chi)\) 11N13 Primes in congruence classes Keywords:cestimate of character sums; general modulus; applications to zero-free regions; L-functions mod q; least prime in arithmetic progression Citations:Zbl 0068.04003; Zbl 0089.02803; Zbl 0246.10030 PDFBibTeX XMLCite \textit{H. Iwaniec}, Invent. Math. 23, 97--104 (1974; Zbl 0275.10024) Full Text: DOI EuDML References: [1] Forti, M., Viola, C.: Density estimates for thezeros ofL-functions. Acta Arith., XXIII 379-391 (1973) · Zbl 0268.10028 [2] Gallagher, P.X.: Primes in progressions to prime-power modulus. Inventiones math.16, 191-201 (1972) · Zbl 0246.10030 · doi:10.1007/BF01425492 [3] Prachar, K.: Primzahlverteilung, Berlin-Göttingen-Heidelberg: Springer 1957 [4] Vinogradov, I.M.: General theorems concerning the upper estimation of absolute value of trigonometrical sum (In Russian). Izv. Akad. Nauk SSSR Ser. Mat.15, 109-130 (1951) · Zbl 0042.04205 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.