Some mean value theorems for the Riemann zeta-function and Dirichlet \(L\)-functions (Q2910117)

The authors announce the unconditional result that NEWLINE\[NEWLINE \sum_{0<\gamma\leqslant T}\zeta^{(j)}(\rho)= \frac{(-1)^{j+1}}{j+1}\frac{T}{2\pi} \Bigl(\log\frac{T}{2\pi}\Bigr)^{j+1} + O_j(T\log^jT)\qquad(j\geqslant1), NEWLINE\]NEWLINE and some interesting analogous formulas for Dirichlet \(L\)-functions. Here \(\rho = \beta+i\gamma\) denotes complex zeros of the Riemann zeta-function \(\zeta(s) = \sum_{n=1}^\infty n^{-s}\;(\Re s>1)\). The above formula, when \(j=1\), reproves and sharpens a result of \textit{A. Fujii} (see e.g., his work in [Comment. Math. Univ. St. Pauli 40, No. 2, 125--231 (1991; Zbl 0743.11043)]). The starting point of the proofs is the fact that, by the residue theorem, NEWLINE\[NEWLINE \sum_{c<\gamma'<T}f(\rho') = {1\over2\pi i}\int_{\mathcal D}f(s){g'\over g}(s)\,ds, NEWLINE\]NEWLINE where \(\rho' = \beta' + i\gamma'\) denotes complex zeros of \(g(s)\), and \(\mathcal D\) is a suitable rectangular contour which contains no zeros of \(g(s)\). Detailed proofs of the results are to be found in the M. Sc. theses (Bogazici University, 2009) of the first two authors.