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The prime number race and zeros of \(L\)-functions off the critical line. (English) Zbl 1010.11051

Let \(\pi_{q,a}(x)\) count the primes \(p\leq x\) with \(p\equiv a\pmod q\). For \(r>1\) distinct residues \(a_1,a_2,\ldots,a_r\) with \((a_i,q)=1\), there are \(r!\) possible oderings of \(\pi_{q,a}(x)\) by size, and it is generally believed that each such ordering is valid for various \(x\to\infty\). When \(q=4\) and \(q=3\), one can only have \(r=2\), and such a result on the ordering was first proved by Littlewood. Later S. Knapowski and P. Turán [Acta Arith. 12, 85-96 (1966; Zbl 0144.28304)] dealt with many more cases with \(r=2\), some under the assumption of the Extended Riemann Hypothesis, because the distribution of the counting functions \(\pi_{q,a}(x)\) is closely related to the distribution of the zeros of the Dirichlet \(L\)-functions \(L(s,\chi)\) for the characters \(\chi\) modulo \(q\). There is also the phenomenon called Chebyshev’s bias (see, for example, M. Rubinstein and P. Sarnak [Exp. Math. 3, 173-197 (1994; Zbl 0823.11050)]) that \(\pi_{q,a_1}(x)<\pi_{q,a_2}(x)\) tends to be valid more often when \(a_1\) and \(a_2\) are quadratic residues and non-residues, respectively.
When \(r>2\) there is not much that has been established without assumptions such as the the Extended Riemann Hypothesis. Taking \(r=3\), so that there are six orderings for the three counting functions \(\pi_{q,a_i}(x)\), the authors establish a certain ‘negative’ result, which is not easy to state precisely without introducing notations and concepts relevant to their method, which depends heavily on a formula for \(\pi_{q,a}(x)\) expressed in terms of the zeros of \(L(s,\chi)\). Roughly speaking the authors show that if certain reasonable assumptions on the zeros of \(L(s,\chi)\) are not valid then one may manipulate the dependence of \(\pi_{q,a}(x)\) on two specific zeros to deduce that one of the six orderings of \(\pi_{q,a_i}(x)\) will not be valid for large \(x\). Perhaps it should be added that their result does not mean that the falsity of Extended Riemann Hypothesis implies that one of the six orderings is not valid for large \(x\).

MSC:

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