Irregularities in the distribution of primes in arithmetic progressions. I
Assuming that all Dirichlet L-functions (mod q) have no zeros in \(\sigma >\), \(| t| \leq cq^ 2 \log^ 6q\), where \(s=\sigma +it\), lower bounds are obtained for various definite integrals of \(| \Delta_ i(x)| /x\), \(i=2,4\), where \[ \Delta_ 4(x)=\sum_{n\leq x, n\equiv \ell_ 1 (q)}\Lambda (n)-\sum_{n\leq x, n\equiv \ell_ 2 (q)}\Lambda (n) \] for arbitrary \(\ell_ 1,\ell_ 2\) satisfying \((\ell_ 1,q)=(\ell_ 2,q)=1\), \(\ell_ 1\not\equiv \ell_ 2 (q)\) and where \(\Lambda_ 2(x)\) is the same difference with summands \(\Lambda\) (n)/log n. The results improve the previous bounds of Knapowski.
scientific article; zbMATH DE number 3889651
scientific article