Accumulation theorems for primes in arithmetic progressions (Q1072588)

Knapowski and Turán have produced two long series of papers on the subject of comparative prime number theory which is concerned about the comparison of certain number theoretic functions summed over two residue classes (mod q). The first series investigated the sign changes of, for example, the function \[ \Delta_ 3(x)=\sum_{p\leq x, p\equiv l_ 1 (q)}\log p\quad -\sum_{p\leq x, p\equiv l_ 2 (q)}\log p. \] The second series investigated the asymptotic behavior as \(x\to \infty\) of weighted sums of a similar type over prime powers with various weight functions \(W_ x(n)\). In particular both investigated ''Chebyshev type problems'' using weight functions \[ W_{k,x}(n)=\exp [-(x-\log n)^ 2/4k],\quad k\leq x,\quad x\to \infty. \] The present authors have obtained improvements in several of the results of Knapowski and Turán of this latter type.