On the mean square of the Riemann zeta-function in short intervals

It is proved that, for $T^ε\le G = G(T) \le {1\over2}\sqrt{T}$, $\int_T^{2T}\Bigl(I_1(t+G)-I_1(t)\Bigr)^2 dt = TG\sum_{j=0}^3a_j\log^j \Bigl({\sqrt{T}\over G}\Bigr) + O_ε(T^{1+ε}+ T^{1/2+ε}G^2)$ with some explicitly computable constants $a_j (a_3>0)$ where, for a fixed natural number $k$, $I_k(t,G) = {1\over\sqrtπ}\int_{-\infty}^\infty |ζ(1/2+it+iu)|^{2k} {\rm e}^{-(u/G)^2} du.$ The generalizations to the mean square of $I_1(t+U,G) - I_1(t,G)$ over $[T, T+H]$ and the estimation of the mean square of $I_2(t+U,G)-I_2(t,G)$ are also discussed.

No reviews yet.

No comments yet.