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 z(s) are simple and lie on the line Re(s) = 1
2. 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

        V~ ---
       |_  t sum /2p _| 
L(t) :=        1 V~ k .cos(l(t,k)), l(t,k) := (tln(k)- h(t)) mod 2p
        k=1

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 ~l (t,k) be the computed value of l(t,k) and let ~c (d) be the approximated cosine-values of cos(2p . d/220) for 0 < d < 220 then the value of L(t) is between

 V~ ----
 |_  sum t/2p _| 
       V~ 1-.min{~c( |_ ~l(2t,2k0) _| - 1),~c( |_ ~l(t22,k0) _| + 1)}
 k=1    k

and

 |_  V~ t/2p _| 
   sum    1 V~ --        ~l(t,k)        ~l(t,k)
        k .max{~c( |_ 220  _|  - 1),~c( |_ 220  _| + 1)}.
  k=1

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 Bn-1, Bn-2 and the next two Rosser blocks Bn+1, Bn+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 Bn. 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:

Year
Author
Source
n
T(n)





1903 J. P. Gram [6] 15 65.801
1914 R. J. Backlund [1] 79 199.649
1925 J. I. Hutchinson [7] 138 300.468
1935 E. C. Titchmarsh [22] 1, 041 1, 467.477
1953 A. M. Turing [23] 1, 104 1, 539.742
1956 D. H. Lehmer [10] 15, 000 14, 041.137
1956 D. H. Lehmer [11] 25, 000 21, 942.593
1958 N. A. Meller [16] 35, 337 29, 750.168
1966 R. S. Lehman [9] 250, 000 170, 570.745
1968 J. B. Rosser, J. M. Yohe,
L. Schoenfeld [17] 3, 500, 000 1, 893, 193.452
1977 R. P. Brent [2] 40, 000, 000 18, 114, 537.803
1979 R. P. Brent [3] 81, 000, 001 35, 018, 261.243
1982 R. P. Brent, J. van de Lune,
H. J. J. te Riele, D. T. Winter [4] 200, 000, 001 81, 702, 130.190
1983 J. van de Lune, H. J. J. te Riele [13] 300, 000, 001 119, 590, 809.282
1986 J. van de Lune, H. J. J. te Riele,
D. T. Winter [14] 1, 500, 000, 001 545, 439, 823.215
2001 J. van de Lune [15] 10, 000, 000, 0003, 293, 531, 632.415
Where the function T(n) gives a positive upper bound on |Im(rn)| for which the Riemann Hypothesis is true, implied by the statement H(n) that the first n nontrivial zeros of z(s) are simple and lie on the critical line. We take T(n) equal to the (n-1)st Gram point if this Gram point seperates the imaginary part of two consecutive nontrivial zeros of the zeta function.

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 Bn 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,g75,039,937,803)1. The distance between these two zeros is about 0.0001008. Comparing their distance with the quantity 2p/ln(tn) which is a natural measure on the distance of two consecutive zeros tn and tn+1 of Z(t), we have

(tn+1-tn)ln(tn)-~= 0.00038.
     2p

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

PICT

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

------n--------
  4,020,755,336
  4,219,726,751
  6,811,982,058
 16,265,396,005
 16,513,952,045
 17,003,807,753
 24,081,166,838
 25,304,589,019
 48,348,026,882
 50,740,831,997
 71,117,177,807
 72,264,805,717
 73,780,104,948

where the zero-pattern “0 0 0 0” is in the interval [gn,gn+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 [gn-2,gn+5) for n = 4,020,755,336:

PICT
The following figure presents a graph of Z(t) in the interval [gn-1,gn+5) for n = 4,219,726,751:
PICT

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

Z(15458728299.51038) ~= 192.64

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

                 ~
Z(7965404181.3057)= -176.842

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

PICT
The following figure presents a graph of Z(t) in the interval [gn-2,gn+5) for n = 25,304,589,019:
PICT

2.4 Extreme values of S(t)

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

PICT
The following figure presents a graph of Z(t) in the interval [gn-2,gn+4) for n = 67,976,501,145 where a negative extreme value S(t) = -3.2281 exist:
PICT

2.5 Some statistics concerning Rosser blocks

The following two tables give the number of Rosser blocks of length k in the interval [g0,g75,039,937,803) for strings of 1010 successive Gram intervals J(k,n + 1010) - J(k,n):

  |_ (n- 1)/1010 _|    k = 1         2          3         4        5
------------------------------------------------------------------
           01   66870207800670350644  11006163716013778429  225675048925340413   6618314425214805   1103445421540113
           2   6701634358  1063334116  268191756   70453258   14572811
           3   6685595060  1063414660  269870001   71887721   15324503
           4   6673792597  1063484462  271072687   72934993   15887117
           5   6664616290  1063528524  272010773   73740299   16324301
           6   6657063763  1063498345  272784334   74414225   16692731

        Total%s 469105698923.563  7442125136.3089  187650732.6830  492918019.817  102677307.178

- |_ (n--1)/1010 _| --k-=6------7------8------9----10---11-
           0   1031509   77384   5895   327    3
           1   1594818   127769   10833   731   16
           2   1871372   156581   13902  1010   45
           3   2067716   178307   15845  1157   60
           4   2220246   195702   17571  1377   51
           5   2342769   212679   18967  1524   79    3
           6   2449752   226836   20453  1670   95    2
       Totals  13578182  1175258  103466  7796  349    5
          %       0.02     0.00    0.00  0.00 0.00 0.00

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,g75,039,937,803)3:

  P  min(M(P ))      |M(P )|         P   min(M(P ))    |M(P )|
-----------------------------------------------------------
  13    139995-217  50262555871616464       00111102   282860637047353  2484823186327
 00    13999525      689927       0114  35787927025         6
 02        125   3988674865       0130       18243   16851637
 04   237516723        2192       0132   2622913795        45
 20        133   3988616830       0310       39889   16848453
 22   182105255      155321       0312   5255811310       219
 40    61331768        2283       2110       83701  248453529
 010   207482540       63036       2112   6525989682       166
 012       4921    960246633       2130   3624306354       294
 014  2920372778         131       2310   9562776948        66
 030       2144     93712289       4110   6123046568         9
 032120   5261963233536    9602113732548
 212  1710461477        9552
 230  1070232756        1820
 410   983377344         103

    P   min(M(P ))   |M(P )|            P   min(M(P ))  |M(P )|
-------------------------------------------------------------
 0011111102   8130841545829566 5017242211       00111111111320   19552621609623872  226581725774
 01114  42986435283        1       0111310    20641464   98570
 01130       68084  4245005       0113110    37091042   56958
 01132  35758065236        1       0131110    17121221   98748
 01310      601944  2382998       0311110    13869654  251757
 01312  43074781029        5       2111110   258779892  267841
 03110      243021  4244224       2131110 18993705653       1
 03112  18677602197        8      01111112  5987600114    2904
 21110     1833652 50168978      01111130   165152519   26782
 21130   9302708401       15      01111310   145659810   20539
 21310  11993665779       21      01113110   454025825    6671
 2431111100  1658934457894919870539       121      0011133111111100   711775537342083094   260673990
 011112    19986469  5804318      03111110   112154948   26826
 011130     2842089  1161317      21111110  2894423255    2855
 011310     1181229   458172
 013110     4718714   459470
 031110     2656216  1162742
 211110    20046223  5802000
 211130  24317005112        1
 211310  25297162824        1
 213110  17509921902        2
 231110  10747518961        2

       P   min(M(P ))  |M(P)|               P   min(M(P ))  |M(P )|
-----------------------------------------------------------------
 011111112 61261479098       3        0111111310  8098448992     131
 001111111111133100  2689118208343598321    2785062        01011111111331111100 296043900630247514567      702
 011113110   978739921     693        0111311110 22194047158       6
 011131110 12459572359     247        0113111110 12250043299      78
 011311110  1331284715     675        0131111110 14325990226     120
 013111110   542964969    2781        0311111110 64906424945       2
 031111110  3877899334     735       01111113110 58203743827       3
 211111110 33635792395       3       01131111110 50366441415       5

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

[1]    R. J. Backlund, ¨ Uber die Nullstellen der Riemannschen Zetafunktion, Dissertation, Helsingfors, 1916.

[2]    R. P. Brent, The First 40, 000, 000 Zeros of z(s) Lie on the Critical Line, Notices of the American Mathematical Society 24 (1977), A-417.

[3]    R. P. Brent, On the Zeros of the Riemann Zeta Function in the Critical Strip, Mathematics of Computation 33 (1979), 1361-1372.

[4]    R. P. Brent, J. van de Lune, H. J. J. te Riele, D. T. Winter, On the Zeros of the Riemann Zeta Function in the Critical Strip II, Mathematics of Computation 39 (1982), 681-688.

[5]    K. M. Briggs, C++ software package ’doubledouble’, available at <http://www.btexact.com/people/briggsk2/doubledouble.html> or alternatively <http://more.btexact.com/people/briggsk2/doubledouble.html>.

[6]    J. P. Gram, Note sur les zéros de la fonction z(s) de Riemann, Acta Mathematica 27 (1903), 289-304.

[7]    J. I. Hutchinson, On the Roots of the Riemann Zeta Function, Transactions of the American Mathematical Society 27 (1925), 49-60.

[8]    T. Kotnik, Computational Estimation of the Order of z(1
2 + it), preprint, 2002.

[9]    R. S. Lehman, Separation of Zeros of the Riemann Zeta-Function, Mathematics of Computation 20 (1966), 523-541.

[10]    D. H. Lehmer, On the Roots of the Riemann Zeta-Function, Acta Mathematica 95 (1956), 291-298.

[11]    D. H. Lehmer, Extended Computation of the Riemann Zeta-Function, Mathematika 3 (1956), 102-108.

[12]    J. van de Lune, H. J. J. te Riele, D. T. Winter, Rigorous High Speed Separation of Zeros of Riemann’s Zeta Function, Report NW 113/81, Mathematical Centre, Amsterdam, October 1981.

[13]    J. van de Lune, H. J. J. te Riele, On the Zeros of the Riemann Zeta Function in the Critical Strip III, Mathematics of Computation 41 (1983), 759-767.

[14]    J. van de Lune, H. J. J. te Riele, D. T. Winter, On the Zeros of the Riemann Zeta Function in the Critical Strip IV, Mathematics of Computation 46 (1986), 667-681.

[15]    J. van de Lune, Private communication, October 27, 2001.

[16]    N. A. Meller, Computations Connected with the Check of Riemann’s Hypothesis, Doklady Akademii Nauk SSSR 123 (1958), 246-248.

[17]    J. B. Rosser, Y. M. Yohe, L. Schoenfeld, Rigorous Computation and the Zeros of the Riemann Zeta-Function, Information Processing 68, Proceedings of IFIP Congress (1968), 70-76.

[18]    A. M. Odlyzko, The 1020-th Zero of the Riemann Zeta Function and 175 Million of it Neighbors, available at <http://www.dtc.umn.edu/~odlyzko/>, 1990.

[19]    A. M. Odlyzko, An improved bound for the de Bruijn-Newman constant, Numerical Algorithms 25 (2000), 293-303.

[20]    A. M. Odlyzko, The 1022-nd zero of the Riemann zeta function, in M. van Frankenhuysen, M. L. Lapidus (Eds.): Dynamical, Spectral, and Arithmetic Zeta Functions, American Mathematical Society, Contemporary Mathematics, volume 290 (2001), 139-144.

[21]    A. M. Odlyzko, A. Sch¨onhage, Fast Algorithms for Multiple Evaluations of the Riemann Zeta Function, Transactions of the American Mathematical Society 309 (1988), 797-809.

[22]    E. C. Titchmarsh, The Zeros of the Riemann Zeta-Function, Proceedings of the Royal Society of London 151 (1935), 234-255.

[23]    A. M. Turing, Some Calculations of the Riemann Zeta-Function, Proceedings of the London Mathematical Society 3 (1953), 99-117.

[24]    S. Wedeniwski, ZetaGrid - Verification of the Riemann Hypothesis, available at <http://www.zetagrid.net/>.

IBM LABORATORY B¨OBLINGEN, SCH¨ONAICHERSTR. 220, 71032 B¨OBLINGEN, GERMANY

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