On the singular series for primes in arithmetic progressions*

Abstract

Using $$ \mathfrak{G} $$ (d, k) to denote the singular series for primes in arithmetic progressions and using the word “tail” to denote the difference of $$ \mathfrak{G} $$ (d, k) and its partial sums, we establish some asymptotic formulas for weighted sums of the square of the tail and so give more information on the singular series $$ \mathfrak{G} $$ (d, k).