: We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exists a subgroup H of the reduced residue system, modulo q, which contains the squares, such that `(x; q; a) ¸ 2x=OE(q) for each a 62 H and `(x; q; a) = o(x=OE(q)), otherwise. From here, we deduce that if the second case holds at all, then it holds only for the multiples of some fixed integer q 0 ? 1. Actually, even if the Prime Number Theorem for arithmetic progressions, modulo q, does hold, these methods allow us to deduce the behaviour of a possible `Siegel zero' from Selberg's formula. We also propose a new method for determining explicit upper and lower bounds on `(x; q; a), which uses only elementary number theoretic computations. 1. Introduction. Define `(x) = P px log p, where p only denot...

ektivn, dv konkrtn funkci f : N ! N takovou, e mnoina f1; 2; : : : ; f(k)g pro kad k obsahuje aritmetickou posloupnost dlky k sloenou z prvosel. Tao v [49] uvd, e lze vzt f(k) = 2 2 2 2 2 100k Green a Tao modi kac dkazu vty 1.1 dokzali jej zeslen, vtu 1.2: Je-li P mnoina vech prvosel a podmnoina Q P spluje lim sup n!1 Q(n) P (n) = c > 0 (Q(n) je poet prvk q v Q, q n, podobn P (n)), mus Q obsahovat libovoln dlouh aritmetick posloupnosti. Je znmo, e pro Q 1 = fp 2 P : p = 4n+1g mme c = 1=2 a kad p 2 Q 1 je souet dvou tverc (p = a +b pro dv pirozen sla a; b, viz st 3). Vta 1.2 tedy dv (napklad) dosud neznm fakt, e existuj libovoln dlouh aritmetick posloupnosti tvoen souty dvou tverc. (Nap. 37 = 1 , 61 = 5 , 85 = 9 + 2 , 109 = 10 + 3 je takov posloupnost.) 2 Dkaz Greenovy a Taovy vty o prvoslech Pirozen sla f1; 2; : : :g ozname N a mnoinu f1; 2; : : : ; Ng, pro N 2 N, jako [N ]. Symboly Z, Q, R a C oznauj mnoiny celch, racionlnch, relnch a komplex