Barban-Davenport-Halberstam average sum and exceptional zero of \(L\)-functions
Set \[ S(Q, x) = \sum_{q\leq Q}\sum_{(a, q)=1}\Big(\frac{x}{\varphi(q)}-\underset{n\leq x}{\sum_{n\equiv a\pmod q}}\Big)^2. \] The Barban-Davenport-Halberstam-theorem states that \(S(Q, x)\ll Qx\log x\) holds true for \(x\log^{-A} x\leq Q\leq x\). Montgomery proved \(S(Q, x)\sim Qx\log x\) for \(Q\) in the same range. In the present work the author gives an asymptotic formula for \(S(Q,x)\) valid in the range \(xe^{-c\sqrt{\log x}}\leq Q\leq x\), which, unavoidably, involves Siegel's zero. If no Siegel zero exists, this formula becomes \(S(Q,x)=Qx\log x + O\big(Qx\log^{1/2} x(\log\log x)^5\big)\). For the proof one has to evaluate the sum \[ \sum_{r\leq x-1}\tau_Q(r)\sum_{1\leq n\leq x-r}\Lambda(n)\Lambda(n+r), \] where \(\tau_Q(r)\) denotes the number of divisors \(d\) of \(r\) which are less than \(Q\). In this article the author applies the circle method to do so. The minor arc estimate is straightforward, on the major arcs the inner sum is expressed via the explicit formula. Since the contribution of an exceptional zero has to be treated like a second main term, this yields rather long expressions. The contribution of non-exceptional zeros is bounded using zero density estimates, the contribution of the main term and the exceptional zero yield rather complicated sums which have to be evaluated explicitly. These computations form the technical main part of the present article.
zero-density estimates for Dirichlet \(L\)-functions
scientific article; zbMATH DE number 5378841