Barban-Davenport-Halberstam average sum and exceptional zero of L-functions

摘要

We prove a formula for the Barban-Davenport-Halberstam average sum [GRAPHICS] where x is sufficiently large, Lambda(n) is the von Mangoldt function, and xe(-c(logx)1/2)<= Q <= x, c > 0 being an absolute constant. The formula, which involves the exceptional zero of L-functions, comes from the intention of investigating the asymptotic behaviour of S(Q, x) via the circle method and the zero-density method for Q in the range (a) (presently unknown without assuming GRH). The formula not only implies a weaker version of the known asymptotic formula for S(Q, x) due to Montgomery and Hooley whenever x(log x)(-A)<= Q <= x, for any constant A > 0, but also improves a lower bound for S(Q, x) obtained by Hooley recently for Q satisfying (alpha).