RESULTS CONNECTED WITH THE FIRST 100 BILLION ZEROS OF THE RIEMANN ZETA FUNCTION (DRAFT)

SEBASTIAN WEDENIWSKI

ABSTRACT. This report presents the results of recent and ongoing internet computing inside the intranet of IBM Corporation which required 1.3 ^. 10¹⁸ floating-point operations: the first 100 billion nontrivial zeros of the Riemann zeta function (s) are simple and lie on the line Re(s) = . Thus, the Riemann Hypothesis is true at least for all |Im(s)| < 29, 538, 618, 432.236.

Date: August 1, 2002.

Key words and phrases. Riemann zeta function, Riemann Hypothesis, zero computations.

1 Introduction

This report presents specific results which were obtained during the numerical verification of the Riemann Hypothesis. Therefore, we recommend the reader to study the four papers by Brent [3], Brent et al. [4], and van de Lune et al. [13], [14] for an easy understanding of this report and the notation.

Starting in Febrary 2001, we transferred the source code of van de Lune, te Riele, and Winter [12] from FORTRAN/COMPASS to C++, enabling us to use the program on different workstations and PCs. Similarly, our program used a very fast and efficient method and a 120 times slower but very accurate method. Especially for the x86 processor and the fast method, the evaluation of the sum

was written in Assembler which is part of the Riemann-Siegel formula for the real-valued function Z(t) with accurate IEEE floating-point arithmetic, since this needs about 78.7% of the running time. About 21% of the running time was spent for the slower method on evaluating Z(t) with twice the precision of IEEE double (106 mantissa bits) using Briggs’ C++ doubledouble package [5]. Our main difference in the fast method is that we evaluate a rigorous lower and upper bound for Z(t) simultaneously, which made the error analysis more simple than the error analysis of the paper [12]. Let (t,k) be the computed value of l(t,k) and let (d) be the approximated cosine-values of cos(2 ^. d/2²⁰) for 0 < d < 2²⁰ then the value of L(t) is between

and

Mainly, the strategy in our program of searching for sign changes of Z(t) in Rosser blocks of length greater than or equal to 2 based on the method of Rosser et al. [17] where many details are described in [12]. Therefore, the average number of Z-evaluations, needed to separate a zero from its predecessor, is about 1.26 which is nearly the same as in [12]. The different search routines which determine the missing zeros in a Rosser block by Rosser’s rule are improved by recognizing a local minimum and maximum value of Z(t) in the search interval before using a grid of increasing refinement to search again in the same interval, if the “missing two” (zeros) are not found. Moreover, to handle the failures to Rosser’s rule, we used a simple method which searches for two additional sign changes in the previous two Rosser blocks B_n-1, B_n-2 and the next two Rosser blocks B_n+1, B_n+2 in zig-zag manner - moving from the center of the blocks towards their periphery - if an exception to Rosser’s rule occurs in the Rosser block B_n. If the method looks just in one previous Rosser block, then we have the first exception for n = 53,365,784,979 (S(t) = 3.0214). Also, if the method looks just in one next Rosser block, then we have the first exception for n = 67,976,501,145 (S(t) = -3.2281).

In August 2001, we started the project “ZetaGrid” as Internet Computing inside the Intranet of IBM Corporation [24], and in August 2002, we extended this project for the Internet. So that there are more than 800 office computers/notebooks involved in this computation. These computers spend all of their idle time to the computation. Beside the mathematical correctness of this computation, the main challenges in the ZetaGrid computations were data management, data analysis, and social factors.

1.1 History

An historical overview of the number of computed zeros gives the following table:

In 1988, a faster method for simultaneous computation of large sets of zeros of the zeta function was invented by Odlyzko and Sch¨onhage [21]. It has been implemented and used to compute 175 ^. 10⁶ zeros near zero number 10²⁰, 10 billion zeros near zero number 10²² and about 20 billion zeros near zero number 10²³ (see [18], and [20]).

2 Summary of the Computational Results

During the course of the verification of H(75,039,937,803) we accumulated various statistics which are a continuation of the four papers by Brent [3], Brent et al. [4], and van de Lune et al. [13], [14].
Some of these results are important for other papers, e.g. Odlyzko [19].

2.1 Closest pair of zeros observed by us.

Our program did not explicitly search for pairs of close zeros of Z(t), but all small values of Z(t) which were computed by the more accurate method are printed out in logging files. Parsing all these logging files, we observed that the Rosser block B_n of length 2 and of zero-pattern “0 2” contains the closest pair of zeros for n = 60,917,681,408 in the interval [g₀,g_{75,039,937,803})¹. The distance between these two zeros is about 0.0001008. Comparing their distance with the quantity 2/ln(t_n) which is a natural measure on the distance of two consecutive zeros t_n and t_n+1 of Z(t), we have

The following figure presents a graph of Z(t) in the interval [g_n,g_n+1):

2.2 Largest distance between two zeros

Our program prints out all zero-patterns in files. Parsing all these files for the exceptions to Rosser’s rule with zero-pattern “0 0 0 0,” we found the following Gram indices

where the zero-pattern “0 0 0 0” is in the interval [g_n,g_n+4). Observing these numbers we found the largest distance between two zeros with a large positive value in the middle of these two zeros for n = 4,020,755,336 which is greater than 1.33805, and with a large negative value in the middle of these two zeros for n = 4,219,726,751 which is greater than 1.34254, respectively. A zero-pattern “0 0 0 0 0” has not yet been found, neither 5 zeros in 1 Gram interval. The following figure presents a graph of Z(t) in the interval [g_n-2,g_n+5) for n = 4,020,755,336:

2.3 Largest absolute value observed by us

Our program did not explicitly search for large absolute values of Z(t), although we searched for large values in the intervals with zero-pattern “0 0 0 0.” In these intervals, we found the largest positive value

for n = 63,052,468,321, and the largest negative value

for n = 25,304,589,019.² The following figure presents a graph of Z(t) in the interval [g_n-1,g_n+5) for n = 50,740,831,994:

2.4 Extreme values of S(t)

There exist a positive extreme value S(t) = 3.0214 for n = 53,365,784,979.

2.5 Some statistics concerning Rosser blocks

The following two tables give the number of Rosser blocks of length k in the interval [g₀,g_{75,039,937,803}) for strings of 10¹⁰ successive Gram intervals J(k,n + 10¹⁰) - J(k,n):

Let M(P) be the set of all Gram indices where the Rosser block has the zero-pattern P. Then the following three tables give the number and the first occurrences of a Rosser block with zero-pattern P in the interval [g₀,g_{75,039,937,803})³:

The heuristic of van de Lune, te Riele, Winter [12] is also valid for higher ranges, i.e. the tendency of the so-called “missing two zeros” in Rosser blocks of length between 3 and 7, to lie in one of the two outer Gram intervals of the Rosser block.

3 Acknowledgements

First and foremost, I thank my managers Herbert Kircher, J¨org Thielges, Reinhold Krause, and Ralf Grohmann at IBM. They have given me all support needed to make this computing effort possible. I also wish to thank Andrew M. Odlyzko, Herman J. J. te Riele and Jan van de Lune for many useful suggestions. Finally, I gratefully acknowledge the participation of more than 380 persons in ZetaGrid [24].

References

E-mail address: wedeniws@de.ibm.com