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**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 p_{1}p_{2}\equiv a (mod q) can be solved in primes p_{1} \leq q and p_{2} \leq q. All considerations are subject to the quasi-Riemann hypothesis H(\theta_{q},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 > \theta_{q}, |Im s| < x^{1-\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 p_{1} and p_{2}.

It is proved that

(i) if H(\theta_{q},q) is true then p_{1}p_{2}\equiv a (mod q), p_{1} \leq q, p_{2} \leq q can be solved for all but cq^{2\thetaq-1} log^{5}q residue classes a\not\equiv 0 modulo q;

(ii) if H(\theta_{q},q) is true with \theta_{q} < 1-(3+\epsilon)\frac{log log q}{log q} then p_{1}p_{2}p_{3}\equiv a (mod q), p_{1} \leq q, p_{2} \leq q, p_{3} \leq q can be solved;

(iii) if the generalized Riemann hypothesis is true then p_{1}p_{2}\equiv a (mod q), p_{1} \leq cq log^{4}q, p_{2} \leq cq log^{4}q 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

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