Abstract
We prove a formula for the Barban–Davenport–Halberstam average sum S ( Q , x ) = ∑ q ⩽ Q ∑ ( a , q ) = 1 1 ⩽ a ⩽ q ( ∑ n ≡ a ( mod q ) n ⩽ x Λ ( n ) − x φ ( q ) ) 2 , where x is sufficiently large, Λ ( n ) is the von Mangoldt function, and ( α) x e − c ( log x ) 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 ( 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 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 ( α) .
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