Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 625.10035
Autor: Erdös, Paul; Odlyzko, Andrew M.; Sárközy, A.
Title: On the residues of products of prime numbers. (In English)
Source: Period. Math. Hung. 18, 229-239 (1987).
Review: This paper contains a modest attack to the problem proposed by P.Erdös that for any sufficiently large prime q and any residue class a\not\equiv 0 modulo q the congruence p1p2\equiv a (mod q) can be solved in primes p1 \leq q and p2 \leq q. All considerations are subject to the quasi-Riemann hypothesis H(\thetaq,x), i.e., it is supposed that for all characters \chi modulo q the L(s,\chi) do not vanish in the domain Re s > \thetaq, |Im s| < x1-\thetaq.
The generalized Riemann hypothesis is H(½,oo) but this is not enough to imply the above conjecture. There are three possible ways to weaken it, which can be satisfied (i) with almost all residue classes mod q, (ii) with the product of three primes instead of two, and (iii) with a little bit larger primes p1 and p2.
It is proved that
(i) if H(\thetaq,q) is true then p1p2\equiv a (mod q), p1 \leq q, p2 \leq q can be solved for all but cq2\thetaq-1 log5q residue classes a\not\equiv 0 modulo q;
(ii) if H(\thetaq,q) is true with \thetaq < 1-(3+\epsilon)\frac{log log q}{log q} then p1p2p3\equiv a (mod q), p1 \leq q, p2 \leq q, p3 \leq q can be solved;
(iii) if the generalized Riemann hypothesis is true then p1p2\equiv a (mod q), p1 \leq cq log4q, p2 \leq cq log4q can be solved.
Reviewer: A.Balog
Classif.: * 11N13 Primes in progressions
11N05 Distribution of primes
Keywords: distribution of primes in residue classes; product of two primes
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag