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(1.1) D(Q; x,h)-, max] ,(x+h; q,/)--,(x q, 1)-h--h[ qQ (,q)=l. q(q) .’ where (x;q, l)is the usual ebyev function for the arithmetic progression l mod q and Q, h are appropriate functions of x. Both results of Bombieri and Jutila have been obtained by reducing the problem to the estimation of the total density of zeros of ’many’ L-functions. Bu Gallagher [2] has found a way to prove Bombieri’s result without using the density theorem. In [4] an opinion is expressed that it seems dicult to prove a non-trivial estimation of (1.1) on the similar line. The purpose o the present paper is to offer such a proof. Our main tool is the following beautiful inequality of Gallagher [3]: If E nan +,

On a mean value theorem for the remainder term in the prime number theorem for short arithmetic progressions