## 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

### Citations:

Zbl 0144.28304; Zbl 0823.11050
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### References:

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