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

We prove a formula for the Barban–Davenport–Halberstam average sumS(Q,x)=∑q⩽Q∑(a,q)=11⩽a⩽q(∑n≡a(modq)n⩽xΛ(n)−xφ(q))2, where x   is sufficiently large, Λ(n)Λ(n) is the von Mangoldt function, andequation(α)xe−c(logx)12⩽Q⩽x,c>0c>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)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)S(Q,x) due to Montgomery and Hooley wheneverx(logx)−A⩽Q⩽xx(logx)−A⩽Q⩽x for any constant A>0A>0, but also improves a lower bound for S(Q,x)S(Q,x) obtained by Hooley recently for Q satisfying (α).