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On the zeros of the Riemann zeta-function in the critical strip. IV. (English) Zbl 0585.10023

Summary: This paper is a continuation of three papers by R. P. Brent [ibid. 33, 1361–1372 (1979; Zbl 0422.10031)], R. P. Brent and the authors [ibid. 39, 681–688 (1982; Zbl 0486.10028)], and the first two authors [ibid. 41, 759–769 (1983; Zbl 0521.10030)]. For corrections to the last two papers, see corrigenda in ibid. 46, 771 (1986).

Very extensive computations are reported which extend and, partly, check previous computations concerning the location of the complex zeros of the Riemann zeta-function. The results imply the truth of the Riemann hypothesis for the first $1,500,000,001$ zeros of the form $\sigma +it$ in the critical strip with $0 i.e., all these zeros have real part $\sigma =1/2$. Moreover, all these zeros are simple. Various tables are given with statistical data concerning the numbers and first occurrences of Gram blocks of various types; the numbers of Gram intervals containing $m$ zeros, for $m=0,1,2,3$ and 4; and the numbers of exceptions to “Rosser’s rule” of various types (including some formerly unobserved types). Graphs of the function $Z\left(t\right)$ are given near five rarely occurring exceptions to Rosser’s rule, near the first Gram block of length 9, near the closed observed pair of zeros of the Riemann zeta- function, and near the largest (positive and negative) found values of $Z\left(t\right)$ at Gram points. Finally, a number of references are given to various number-theoretical implications.

##### MSC:
 11M06 $\zeta \left(s\right)$ and $L\left(s,\chi \right)$ 11-04 Machine computation, programs (number theory) 11Y35 Analytic computations