On the singular series for primes in arithmetic progressions*

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).