Abstract :
We prove a formula for the Barban–Davenport–Halberstam average sumimage where x is sufficiently large, Λ(n) is the von Mangoldt function, andimage 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 (α) (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 wheneverx(logx)−Aless-than-or-equals, slantQless-than-or-equals, slantx for any constant A>0, but also improves a lower bound for S(Q,x) obtained by Hooley recently for Q satisfying (α).
