On the mean square of the Riemann zeta-function in short intervals
نویسنده
چکیده
It is proved that, for T ε 6 G = G(T ) 6 2 √ T , ∫︁ 2T T (︁ I1(t+G,G)− I1(t, G) )︁2 dt = TG 3 ∑︁ j=0 aj log (︁√ T G )︁ +Oε(T 1+εG1/2 + T 1/2+εG2) with some explicitly computable constants aj (a3 > 0) where, for fixed k ∈ N, Ik(t, G) = 1 √ π ∫︁ ∞ −∞ |ζ( 1 2 + it+ iu)| 2ke−(u/G) 2 du. The generalizations to the mean square of I1(t+U,G)−I1(t, G) over [T, T+H] and the estimation of the mean square of I2(t + U,G) − I2(t, G) are also discussed.
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