## The mean square of the error term for the fourth power moment of the zeta-function.(English)Zbl 0805.11060

Let $\int^ T_ 0 \left | \zeta \Bigl( {1 \over 2} + it \Bigr) \right |^ 4dt = Tf (\log T) + E_ 2(T),$ where $$f$$ is an appropriate quartic polynomial. It is shown here that $\int^ T_ 0 E_ 2(t)^ 2dt \ll T^ 2 (\log T)^ C$ for some constant $$C$$. This remarkable result implies the estimates $$E_ 2 (T) \ll T^{2/3} (\log T)^ C$$, and hence $$\zeta ({1 \over 2} + it) \ll t^{1/6} (\log t)^ C$$, as well as the bound $\int^ T_ 0 \left | \zeta \Bigl( {1 \over 2} + it \Bigr) \right |^{12} dt \ll T^ 2 (\log T)^ C,$ with differing values of $$C$$. Further theorems describe the mean value of the error terms for $$\sum^ N_{n = 1} d(n) d(n+k)$$ and $$\sum^{N- 1}_{n=1} d(n)d(N-n)$$. In particular, the latter has an asymptotic formula with an error term which is $$O(N^{{1 \over 2} + \varepsilon})$$ in mean.
The proofs use the spectral theory of the non-Euclidean Laplacian.

### MSC:

 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 11N37 Asymptotic results on arithmetic functions
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