短区间内黎曼$\zeta$函数均方值
数论
2010-01-23 v3
摘要
证明了对于 T ϵ ≤ G = G ( T ) ≤ 1 2 T T^\epsilon\le G = G(T) \le {1\over2}\sqrt{T} T ϵ ≤ G = G ( T ) ≤ 2 1 T ∫ T 2 T ( I 1 ( t + G ) − I 1 ( t ) ) 2 d t = T G ∑ j = 0 3 a j log j ( T G ) + O ϵ ( T 1 + ϵ + T 1 / 2 + ϵ G 2 ) \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_\epsilon(T^{1+\epsilon}+ T^{1/2+\epsilon}G^2) ∫ T 2 T ( I 1 ( t + G ) − I 1 ( t ) ) 2 d t = T G j = 0 ∑ 3 a j log j ( G T ) + O ϵ ( T 1 + ϵ + T 1/2 + ϵ G 2 ) a j a_j a j a 3 > 0 a_3>0 a 3 > 0 k k k I k ( t , G ) = 1 π ∫ − ∞ ∞ ∣ ζ ( 1 / 2 + i t + i u ) ∣ 2 k e − ( u / G ) 2 d u . I_k(t,G) = {1\over\sqrt{\pi}}\int_{-\infty}^\infty |\zeta(1/2+it+iu)|^{2k} {\rm e}^{-(u/G)^2} du. I k ( t , G ) = π 1 ∫ − ∞ ∞ ∣ ζ ( 1/2 + i t + i u ) ∣ 2 k e − ( u / G ) 2 d u . I 1 ( t + U , G ) − I 1 ( t , G ) I_1(t+U,G) - I_1(t,G) I 1 ( t + U , G ) − I 1 ( t , G ) [ T , T + H ] [T, T+H] [ T , T + H ] I 2 ( t + U , G ) − I 2 ( t , G ) I_2(t+U,G)-I_2(t,G) I 2 ( t + U , G ) − I 2 ( t , G )
引用
@article{arxiv.0803.0132,
title = {On the mean square of the Riemann zeta-function in short intervals},
author = {Aleksandar Ivić},
journal= {arXiv preprint arXiv:0803.0132},
year = {2010}
}
备注
19 pages