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Quadratic residues and the distribution of prime numbers

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Abstract

D. Shanks [11] has given a heuristical argument for the fact that there are “more” primes in the non-quadratic residue classes modq than in the quadratic ones. In this paper we confirmShanks' conjecture in all casesq<25 in the following sense. Ifl 1 is a quadratic residue,l 2 a non-residue modq, ε(n, q, l 1,l 2) takes the values +1 or −1 according ton≢l 1 orl 2 modq, then

$$\mathop {\lim }\limits_{x \to \infty } \sum\limits_p {\varepsilon (p,q,l_1 ,l_2 )} \log pp^{ - \alpha } \exp ( - (\log p)^2 /x) = - \infty$$

for 0≤α<1/2. In the general case the same holds, if all zeros ϱ=β+yγ of allL(s, χ modq),q fix, satisfy the inequality β2−γ2<1/4.

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Authors and Affiliations

  1. FB 6 Mathematik/Philosophie, Universität Osnabrück, D-4500, Osnabrück, Germany

    H. -J. Bentz

  2. Mathematical Institute of the, Hungarian Academy of Sciences, H-1053, Budapest, Hungary

    J. Pintz

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  1. H. -J. Bentz
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  2. J. Pintz
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Bentz, H.J., Pintz, J. Quadratic residues and the distribution of prime numbers. Monatshefte für Mathematik 90, 91–100 (1980). https://doi.org/10.1007/BF01303260

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