The distribution of the zeros of \(L\)-functions and the least prime in some arithmetic progression
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.
scientific article; zbMATH DE number 1782029