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On zeros of Dirichlet’s \(L\)-series. (English) Zbl 0275.10024

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.

MSC:

11L40 Estimates on character sums
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11N13 Primes in congruence classes
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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
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