The mean-value theorem in the theory of prime numbers (Q1385964)

Let \(\psi(y,\chi)\) denote the Chebyshev function and \(T(x; Q)= \sum_{q\leq Q}{q\over \rho(q)} \sum_\chi^*\max_{y\leq x}|\psi (y,\chi) | \), where \(\sum^*\) denotes summation over all primitive characters modulo \(q\). The author proves the following statement. Theorem: Let \(x\geq 2\) and \(Q\geq 1\). Then \[ T(x;Q)\leq x{\mathcal L}^3+ x^{4/5}Q{\mathcal L}^{34}+ x^{1/2}Q^2{\mathcal L}^c, \] where \({\mathcal L}= \log(xQ)\), \(c=34\), if \(Q\leq x^{1/3}(\log x)^{-5/6}\) and \(c=3.5\) if \(Q>x^{1/3}(\log x)^{-5/6}\). This estimate is more exact for \(x^{1/5} <Q< x^{1/3}\) than one of \textit{R. C. Vaughan} [J. Lond. Math. Soc. (2) 10, 153-162 (1975; Zbl 0314.10028)].