A mean value estimate for real character sums

where Σ∗ indicates summation over primitive characters only. These last two bounds follow respectively from Theorem 6.2 of Montgomery [6] and from the large sieve in the form due to Gallagher [2], for example. In each case one may interpret the upper bound as being composed of two terms, the first of which reflects the long term average, and the second of which reflects the contribution of a single point where the Dirichlet polynomial is large. Thus for example one has ∫ T