On the singular series for primes in arithmetic progressions*

A singular series average and the zeros of the Riemann zeta-function

“…The problem that is encountered in doing this for S 1 (x) is that the Cesaro mean is not smooth enough for absolute convergence in the contour integrals one encounters. Inspired by recent work on a related problem [GZHN17], [GL17] concerning the square of the singular series, we can sidestep this problem by using smoother weights. We found while writing this paper that this idea has already been applied to other arithmetic functions, and some results of the same type as ours have been obtained by similar methods for these functions.…”

A singular series average and the zeros of the Riemann zeta-function

“…The problem that is encountered in doing this for S 1 (x) is that the Cesàro mean is not smooth enough for absolute convergence in the contour integrals one encounters. Inspired by recent work on a related problem [GZHN17], [GL17] concerning the square of the singular series, we can sidestep this problem by using smoother weights. We found while writing this paper that this idea has already been applied to other arithmetic functions, and some results of the same type as ours have been obtained by similar methods for these functions.…”