A conjecture for the sixth power moment of the Riemann zeta-function. (English) Zbl 0920.11060

Conrey, J. B.; Ghosh, A.

Int. Math. Res. Not. 1998, No. 15, 775-780 (1998).

The authors give several heuristic arguments leading to the same conjecture, that \[ \int^T_0\biggl| \zeta\bigl( \tfrac 12+it\bigr) \biggr|^6dt\sim\tfrac{49}{9!} \left\{\prod_p \left(1-\tfrac 1p\right)^4 \left( 1+ \tfrac 4p+ \tfrac{1}{p^2} \right)\right\} T(\log T)^9. \] The unproven assumptions concern the extension of various mean-value estimates [J. B. Conrey and A. Ghosh, Proceedings of the Amalfi conference on analytic number theory, Maiori, 1989, Salerno, 35-59 (1992; Zbl 0792.11033)] beyond their known range of validity.

Reviewer: D.R.Heath-Brown (Oxford)

MSC:

11M06

\(\zeta (s)\) and \(L(s, \chi)\)

Keywords:

Riemann Zeta-function; critical line; sixth power moment; conjecture; mean-value estimates