Meng, Zaizhao The distribution of the zeros of \(L\)-functions and the least prime in some arithmetic progression. (English) Zbl 1014.11055 Sci. China, Ser. A 43, No. 9, 937-944 (2000). 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. Reviewer: Roger Heath-Brown (Oxford) Cited in 2 Documents MSC: 11N13 Primes in congruence classes 11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses Keywords:primes; arithmetic progression; prime modulus; Linnik’s constant; Dirichlet \(L\)-function; zero-free region Citations:Zbl 0739.11033 PDFBibTeX XMLCite \textit{Z. Meng}, Sci. China, Ser. A 43, No. 9, 937--944 (2000; Zbl 1014.11055) Full Text: DOI 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 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.