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The distribution of the zeros of \(L\)-functions and the least prime in some arithmetic progression. (English) Zbl 1014.11055

If \((a,q)=1\) then there is a prime \(p\equiv a\pmod q\) with \(p\ll q^{5.5}\), as was shown by the reviewer [Proc. Lond. Math. Soc. (3) 64, 265-338 (1992; Zbl 0739.11033)]. This is currently the best known form of Linnik’s Theorem. The argument uses zero-free regions for Dirichlet \(L\)-functions \(L(s,\chi)\) to modulus \(q\), which take the form \[ \sigma\geq 1- \frac{c}{\log q}, \qquad |t|\leq 1, \tag \(*\) \] for suitable constants \(c>0\).
The present paper examines the special case in which the modulus \(q\) is prime. For characters to prime modulus (and in general for cube-free modulus) Burgess’ bounds for \(L\)-functions are stronger than in the general case. Consequently one can establish wider zero-free regions for such moduli, and an improved Linnik theorem. For \(q\) prime and \(c= 0.372\) it is shown that the region \((*)\) contains at most an exceptional real zero. Moreover with \(c= 0.808\) it is shown that there is at most one pair \(\chi_1\), \(\overline{\chi}_1\) of characters for which \(L(s,\chi)\) can vanish in \((*)\); and if \(c= 1.1428\) there are at most two such pairs.
As a result, if \(q\) is prime then there is a prime \(p\equiv a\pmod q\) with \(p\ll q^{4.5}\), whenever \((a,q)=1\).
These results basically arise by inserting the improved Burgess bound into the reviewer’s method. However there are a number of complications of detail. As a result it is not clear whether the results here would remain true if one merely supposed \(q\) to be square-free.

MSC:

11N13 Primes in congruence classes
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses

Citations:

Zbl 0739.11033
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References:

[1] Motohashi, Y., A note on the least prime in an arithmetic progression with a prime difference, Acta Arith., 17, 283-283 (1970) · Zbl 0215.35302
[2] Heath-Brown, D. R., Zero-free regions for DirirhletL-functions, and the least prime in an arithmetic progression, Proc. Iondon Math. Soc., 64, 3, 265-265 (1992) · Zbl 0739.11033 · doi:10.1112/plms/s3-64.2.265
[3] Heath-Brown, D. R., Siegel zeros and the least prime in an arithmetic progession, Quart. J. Math. Oxford, 41, 2, 405-405 (1990) · Zbl 0715.11049 · doi:10.1093/qmath/41.4.405
[4] Chengdong, Pan; Chengbiao, Pan, The Foundation of Analytic Number Theory (in Chinese) (1997), New York: Science Press, New York · Zbl 0703.11045
[5] Wang, W., On the least prime in an arithmetic progression, Acta Math. Sinica (N.S.), 7, 3, 279-279 (1991) · Zbl 0742.11044
[6] Ming-chit, Liu; Tianze, Wang, A numerical bound for small prime solutions of some ternary linear equations, Acta Arith., 86, 4, 343-343 (1998) · Zbl 0918.11053
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