On the zeros of the Riemann zeta function in the critical strip. IV

Abstract

Abstract. 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 a + it in the critical strip with 0 < t < 545,439,823.215, i.e., all these zeros have real part a = 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, form = O, l, 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( t) are given near five rarely occurring exceptions to Rosser's rule, near the first Gram block of length 9, near the clpsest observed pair of zeros of the Riemann zeta function, and near the largest (positive and negative) found values of Z(t) at Gram points. Finally, a number of references are given to various number-theoretical implications.

On the Zeros of the Riemann Zeta Function in the Critical Strip. IV

“…£5,000,000) and [g20o,ooo,ooo. g2io,ooo,ooo)-We found exact agreement with the corresponding results in [1] and [7]. The CPU time needed on the CYBER 205 was about 2800 seconds and 5 hours, respectively.…”

“…Introduction. This paper is a continuation of three papers by Brent [1], Brent et al [2]*, and van de Lune et al [7]*. The computations reported there (up to £300,000,000) have Deen extended up to £1,500,000,000 m order to show that the first 1,500,000,001 zeros of Riemann's zeta function in the critical strip are simple and lie on the vertical with real part 1/2.…”

“…Moreover, with the aid of our CYBER 205 program we checked some of the computations reported on in [1] and [7], viz., those concerning the intervals [gi.ooo. £5,000,000) and [g20o,ooo,ooo.…”

On the Zeros of the Error Term for the Mean Square of | ζ(½ + it) |

“…Then \E(t,)\ > \Ef(tt)\ -\f(tj)\ >V-CxTl/4-" > V/2 for i = 1, ... , R0. Analogously as in (13.66) of [11], we obtain (14) Ä0< TE(TV~i+ R0T¡/9Tin8V~2) for any given e > 0 . This gives for (16) r/>Ci7£/36r7/36+£/2 (C,>0).…”

“…Practically speaking, the bounds in (27) and (28) are weak: in [14] we have maxZ(i) = 116.88 ; the corresponding right-hand sides (27) and (28) yield 4.39 and 6.77 . respectively!…”