An additive theory of the zeros of the Riemann zeta function (Q921039)

The author presents several results on the additive theory of the zeros of the Riemann zeta-function. For example, it is announced that \[ \sum_{0<\gamma,\gamma '\leq T;\gamma +\gamma '\leq T}1=\frac{1}{8\pi^ 2}T^ 2\log^ 2T-\frac{1}{8\pi^ 2}(3+2 \log (2\pi))T^ 2\log T+\frac{1}{16\pi^ 2}(7+6 \log (2\pi)+2 \log^ 2(2\pi)-2\zeta (2))T^ 2+O(T\frac{\log^ 2T}{\log \log T}) \] and \(\sum_{0<\gamma,\gamma '\leq T;\gamma +\gamma '\leq T}x^{\rho +\rho '}=\frac{1}{8\pi^ 2}T\Lambda^ 2(x)+O(T \log^ 2T)\). Here, as usual, \(\rho =\beta +i\gamma\) denotes complex zeros of \(\zeta\) (s), \(\Lambda (x)=\log p\) if \(x=p^ k\) (p prime) and zero otherwise. Under the Riemann hypothesis the above formulas are sharpened. Detailed proofs will appear elsewhere.