Lower bounds for sums of Barban-Davenport-Halberstam type. (Supplement)
Let \[ S(Q, x)= \sum_{q\leq Q} \sum _{\substack{ 0< a\leq q\\ (a,q)=1 }} \biggl( \psi (x; q, a)- {x\over {\varphi (q)}} \biggr)^2, \quad S^* (Q, x)= \sum_{q\leq Q} \sum _{\substack{ 0< a\leq q\\ (b, q)=1 }} \biggl( \psi (x; q, a)- {{\psi(x, \chi^0)} \over {\varphi (q)}} \biggr)^2, \] where \(\varphi (q)\) is Euler's function, \(\psi (x; q, a)\) is the Chebyshev function which sums the von Mangoldt function \(\Lambda (n)\) over the arithmetic progression \(n\equiv a\pmod q\), \(n\leq x\), and \(\psi (x, \chi^0)= \sum_{n\leq x} \chi^0 (n) \Lambda (n)\) with \(\chi^0 \pmod q\) being the principal character. In this supplement to [J. Reine Angew. Math. 438, 163-174 (1993; Zbl 0776.11056)], the author proves that if \(c\) and \(x\) are positive numbers which are small and large, respectively, and \(x\exp (-c \log x/ \log \log x)\leq Q\leq x\), then \[ S^* (Q, x)\geq \bigl( {\textstyle {1\over 4}} +O(c) \bigr) Qx\log x. \] On applying the Vinogradov-Korobov version of the prime number theorem the author deduces that if \(Q\) further satisfies \[ x\exp \bigl(- B(\log x)^{3\over 5} (\log \log x)^{-{1\over 5}} \bigr)\leq Q\leq x \] for some positive constant \(B\), then \[ S(Q, x)\geq \bigl( {\textstyle {1\over 8}}+ O(c)\bigr) Qx\log x. \] Although there are asymptotic formulae for \(S(Q, x)\) in the range \(x/ (\log x)^A\leq Q\leq x\), the derivation of such formulae makes use of the Siegel-Walfisz theorem which renders them ineffective, whereas the estimates in this supplement are effective.
sums of Barban-Davenport-Halberstam type
scientific article; zbMATH DE number 784068
scientific article