On the connection of fractional moments of the Riemann zeta-function with its zeros
Let, as usual, \(N_0(T)\) denote the number of zeros of the Riemann zeta-function \(\zeta(s)\) of the form \(1/2+it\) such that \(0 < t \leq T\). The classical result of Hardy and Littlewood that \(N_0(T) \gg T\) was improved by \textit{A. Selberg} in 1942 [see e.g., his Collected papers. Volume I. Berlin etc.: Springer-Verlag (1989; Zbl 0675.10001)] to \(N_0(T) \gg T\log T\), which is one of the most important results of zeta-function theory, since \(N_0(T) \ll T\log T\) holds. The present authors prove the weaker result that \[ N_0(T) \gg_\varepsilon T(\log T)^{1/4-\varepsilon}.\leqno(1) \] The merit of (1) is that its proof is considerably less difficult than Selberg's proof. The proof of (1) is based on a variation of a method used by \textit{A. A. Karatsuba} [Math. Notes 72, No. 4, 466--472 (2002); translation from Mat. Zametki 72, No. 4, 502--508 (2002; Zbl 1137.11333)], who proved that \(N_0(T) \gg T(\log T)^{1/8}\). The present improvement comes primarily from the use of sharper estimates for fractional moments of \(|\zeta(1/2+it)|\), obtained by \textit{D. R. Heath-Brown} [J. Lond. Math. Soc., II. Ser. 24, 65--78 (1981; Zbl 0431.10024)]. Reviewer's remark: In the middle of p. 191 the choice of \(l\) was \(l = 2/(1-4\varepsilon)\), but \(l\) should be an integer, so e.g., \(l = [2/(1-4\varepsilon)]\) should be taken.
scientific article; zbMATH DE number 5924762