Mean values of the Riemann zeta-function. III. (English) Zbl 0792.11033
Bombieri, E. (ed.) et al., Proceedings of the Amalfi conference on analytic number theory, held at Maiori, Amalfi, Italy, from 25 to 29 September, 1989. Salerno: Universitá di Salerno, 35-59 (1992).
[For Parts I, II, cf. Mathematika 31, 159-161 (1984; Zbl 0528.10026), Acta Arith. 52, 367-371 (1987; Zbl 0688.10036).]
Let \(I_ k= I_ k(T)= \int_ 1^ T | \zeta({1\over 2}+it)|^{2k} dt\). The asymptotic formulas for \(I_ k\) are known for \(k=0,1\) and 2 only. It is also known for some other values of \(k\) that \(I_ k\) has order \(T(\log T)^{k^ 2}\), and there is a conjecture that \(I_ k\sim C_ k\) \(T(\log T)^{k^ 2}\).
In this paper the authors obtain some related results. If \[ A_ k(s,p)= \sum_{n=1}^ N d_ k(n) P(\log n/\log N)\;n^{-s} \] and \[ J_{k- 1,N}(T)= \int_ 1^ T |\zeta({\textstyle {1\over 2}}+it) |^ 2 | A_{k-1}({\textstyle {1\over 2}}+it,P)|^ 2 dt \] for \(N=T^ \theta\), \(0<\theta<1/2\) and \(P\) a polynomial, then \(J_{k-1,N}(T)\sim C_ k\) \(T(\log T)^{k^ 2}\) where the formulas for \(C_ k\) are given in the paper. Using their results, they obtain some bounds of \(I_ k\) for \(k=3\), \(4\leq k\leq 6\) (under LH) and \(1<k\leq 2\) (under RH).
For the entire collection see [Zbl 0772.00021].
Let \(I_ k= I_ k(T)= \int_ 1^ T | \zeta({1\over 2}+it)|^{2k} dt\). The asymptotic formulas for \(I_ k\) are known for \(k=0,1\) and 2 only. It is also known for some other values of \(k\) that \(I_ k\) has order \(T(\log T)^{k^ 2}\), and there is a conjecture that \(I_ k\sim C_ k\) \(T(\log T)^{k^ 2}\).
In this paper the authors obtain some related results. If \[ A_ k(s,p)= \sum_{n=1}^ N d_ k(n) P(\log n/\log N)\;n^{-s} \] and \[ J_{k- 1,N}(T)= \int_ 1^ T |\zeta({\textstyle {1\over 2}}+it) |^ 2 | A_{k-1}({\textstyle {1\over 2}}+it,P)|^ 2 dt \] for \(N=T^ \theta\), \(0<\theta<1/2\) and \(P\) a polynomial, then \(J_{k-1,N}(T)\sim C_ k\) \(T(\log T)^{k^ 2}\) where the formulas for \(C_ k\) are given in the paper. Using their results, they obtain some bounds of \(I_ k\) for \(k=3\), \(4\leq k\leq 6\) (under LH) and \(1<k\leq 2\) (under RH).
For the entire collection see [Zbl 0772.00021].
Reviewer: G.Kolesnik (Los Angeles)
MSC:
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
