Sign changes in \(\pi_{q,a}(x) - \pi_{q,b}(x)\) (Q2773293)

An unconditional upper bound for the first sign change of \(\text{li}(x)-\pi(x)\) was first obtained by \textit{S.~Skewes} [Proc. Lond. Math. Soc. (3) 5, 48-70 (1955; Zbl 0068.26802)]. Later \textit{R. S. Lehman} [Acta Arith. 11, 397-410 (1966; Zbl 0151.04101)] developed a method based on an explicit formula for the expression concerned averaged over a Gaussian kernel, and knowledge of zeros of \(\zeta(s)\) in \(|\text{Im} s|\leq 12000\), in order to drastically improve on the enormous `Skewes' number. More recently \textit{C. Bays} and the second author [Math. Comput. 69, 1285-1296 (2000; Zbl 1042.11001)] further reduced this upper bound to \(1.39822{\cdot}10^{316}\).NEWLINENEWLINENEWLINEThe authors generalise Lehman's method and apply computations of the zeros of \(L\)-functions in order to study `Chebyshev's bias', the phenomenon that negative values of \(\Delta_{q,b,1}(x)=\pi_{q,b}(x)-\pi_{q,1}(x)\), where \(\pi_{q,a}(x)\) counts the primes \(p\leq x\) in the arithmetic progression \(nq+a\), may be relatively infrequent if \(b\) is a quadratic nonresidue of~\(q\). The bias seems particularly pronounced when \(q|24\); for example, \textit{C. Bays} and the second author [Math. Comput. 32, 571-576 (1978; Zbl 0388.10003)] showed that the smallest \(x\) with \(\Delta_{3,2,1}(x)<0\) is \(x=608981813029\), and it is reported that exhaustive computational search shows that \(\Delta_{24,b,1}(x)\geq 0\) for \(x\leq 10^{12}\) and \(b=5,7,11,17,19,23\).NEWLINENEWLINENEWLINEThe following results are consequences of the main theorem, which is too lengthy to be quoted here.NEWLINENEWLINENEWLINE(i) For each \(b\in\{3,5,7\}\), there is some \(x<5{\cdot}10^{19}\) such that \(\Delta_{8,b,1}(x)<0\);NEWLINENEWLINENEWLINE(ii) For each \(b\in\{5,7,11\}\), there is some \(x<10^{84}\) such that \(\Delta_{12,b,1}(x)<0\);NEWLINENEWLINENEWLINE(iii) For each \(b\in\{5,7,11,13,17,19,23\}\), there is some \(x<10^{353}\) such that \(\Delta_{24,b,1}(x)<0\).NEWLINENEWLINENEWLINEMoreover, if the zeros of \(L(s,\chi_4)\) lying in the critical strip to height 630000 all have real part equal to \({1\over 2}\), then \(\pi_{4,1}(x)-\pi_{4,3}(x)>\sqrt x/\log x\) for some \(x\) in the vicinity of \(\exp(78683.7)\). Thus, subject to checking such zeros, there is a specific region where \(\pi_{4,1}(x)\) runs ahead of \(\pi_{4,3}(x)\) by as much as it usually runs behind.NEWLINENEWLINENEWLINEAlthough the method cannot be used to make direct comparison for prime counts in more than two arithmetic progressions, it does give results such as \(3\pi_{8,1}(x)>\pi_{8,3}(x)+\pi_{8,5}(x)+\pi_{8,7}(x)\) for some \(x\) near \(\exp(158.64233)\).