Computation of zeros of the Zeta function
New Riemann Hypothesis verification record
Riemann Hypothesis verified until the 1013-th zero. (October 12th 2004), by Xavier Gourdon with the help of Patrick Demichel. Billions of zeros at very large height (around the 1024-th zero) have also been computed. Details can be found in The 1013 first zeros of the riemann zeta function, and zeros computation at very large height.


The Riemann Hypothesis (RH) is one the most important unsolved problem in mathematics (see Zeta generalities for details about the RH). It has naturally been numerically checked threw the ages, thanks to techniques about the Zeta functions evaluations (see Numerical evaluation of the Riemann Zeta-function for details).

1  History of numerical verification of the RH

Numerical computations have been made threw the ages to check the RH on the first zeros. Computer age, starting with Turing computations, permitted to perform verification higher than billions of zeros. An history of the RH verification on the first n zeros is given below.
Year n Author
1903 15 J. P. Gram [6]
1914 79 R. J. Backlund [1]
1925 138 J. I. Hutchinson [7]
1935 1,041 E. C. Titchmarsh [22]
1953 1,104 A. M. Turing [24]
1956 15,000 D. H. Lehmer [12]
1956 25,000 D. H. Lehmer [11]
1958 35,337 N. A. Meller [14]
1966 250,000 R. S. Lehman [10]
1968 3,500,000 J. B. Rosser, J. M. Yohe, L. Schoenfeld [21]
1977 40,000,000 R. P. Brent [2]
1979 81,000,001 R. P. Brent [3]
1982 200,000,001 R. P. Brent, J. van de Lune, H. J. J. te Riele, D. T. Winter [25]
1983 300,000,001 J. van de Lune, H. J. J. te Riele [8]
1986 1,500,000,001 J. van de Lune, H. J. J. te Riele, D. T. Winter [9]
2001 10,000,000,000 J. van de Lune (unpublished)
2004 900,000,000,000 S. Wedeniwski [26]
2004 10,000,000,000,000 X. Gourdon and Patrick Demichel [5]
Notice that the latest RH verification on the first 1013 zeros have been achieved thanks to the use of the fast Zeta multi-evaluation algorithm by Odlyzko and Schönhage, and that it is the first RH verification done with this technique. Details can be found in [5].

2  Numerical computations of the distribution of the zeros of the Zeta function

While numerical computations on zeros of the Zeta function have long been focused on RH verification only (to check the RH, isolating the zeros is sufficient so no precise computations of the zeros are needed) it was Odlyzko who the first, computed precisely large consecutive sets of zeros to observe their distribution. More precisely, Odlyzko made some empirical observations of the distribution on the spacing between zeros of z(s) in various zones and checked the correspondence with the GUE hypothesis, which conjectures that normalized spacing between zeros behaves like eigenvalues of random hermitian matrices (see [5] for more details). In 1987, Odlyzko computed numerically 105 zeros of the Riemann Zeta function between index 1012+1 and 1012+105 to the accuracy of 10-8 and was the first to observe a good agreement with the GUE hypothesis (see [16]). Later, in order to reach much higher heights, Odlyzko with Schönhage [20] developed a fast algorithm for multi-evaluation of z(s). After refinements to make this method efficient for practical purposes, Odlyzko was able to compute 70 million zeros at height 1020 in 1989 and then 175 million in 1992 at the same height (see [17]). Later he reached the height 1021 (see [18]), and in 2001 he computed ten billion zeros at height 1022 (see [19]). In a more recent unpublished work in 2002, Odlyzko computed twenty billion zeros at height 1023.

3  Numerical verification of the Riemann Hypothesis on the first 1013 zeros of Zeta

The algorithm of Odlyzko and Schönhage to perform efficient multi-evaluation of Zeta function was implemented by Xavier Gourdon and used to check numerically the Riemann Hypothesis on the first 1013 zeros. Details can be found in the following paper.

Statistics on the first 1013 zeros of the Zeta function

Close roots

The smallest normalized spacing between consecutive zeros rn=1/2+ign and rn+1=1/2+ign+1, denoted by
dn = (gn+1-gn)  log(gn/(2p)

2p
has been reached at gn=1034741742903.35376 (for root index n=4,088,664,936,217), with a value of d 0.00007025. Non normalized difference between those roots is equal to 0.00001709. This is the smallest known difference between roots of the Zeta function.
Other close roots have been detected while verifying the RH on the first 1013 zeros, but some of them may have been missed since they were computed only when separation of Zeta zeros were made difficult. A (non exhaustive) list of close zeros of Zeta until the 1013-th zeros in excel format can be found here : CloseRoots.xls.

Gram blocks and violations of Rosser rule

Complete statistics between zero of index 100,002 and the 1013-th zeros can be found here : ZetaStatGlobal_1e5_1e13.txt.
The same statistics for each 1012 step are available here :

Large values of |Z(t)|

No particular search or reconvergence was performed to find maximum values of |Z(t)| during the computation. The largest values encountered during evaluation were just recorded, leading to the following table : Maximum values of abs(Z(t)) encountered.

4  Computations of zeros of the Riemann Hypothesis at very large height

Computation of two billion zeros of the Zeta function at each of the height 1013, 1014, , 1023 and 1024 were done. What follows is a synthetic view of what is contained in the paper paper, where the reader should refer for more details.

4.1  The GUE hypothesis

While many attempts to prove the RH had been made, a few amount of work has been devoted to the study of the distribution of zeros of the Zeta function. A major step has been done toward a detailed study of the distribution of zeros of the Zeta function by Hugh Montgomery [15], with the Montgomery pair correlation conjecture. Roughly speaking, this conjecture states that the density of normalized spacing between non-necessarily consecutive zeros is 1-(sin(pu)/pu)2. It was first noted by the Freeman Dyson, a quantum physicist, during a now-legendary short teatime exchange with Hugh Montgomery, that this is precisely the pair correlation function of eigenvalues of random hermitian matrices with independent normal distribution of its coefficients. Such random hermitian matrices are called the Gauss unitary ensemble (GUE). As referred by Odlyzko in [16] for example, this motivates the GUE hypothesis which is the conjecture that the distribution of the normalized spacing between zeros of the Zeta function is asymptotically equal to the distribution of the GUE eigenvalues. Under this conjecture, we might expect that the distribution of the dn itself satisfies
 1

M
# { n : N+1 n N+M, dn [a,b]} ~
b

a 
p(0,u) du
(1)
where p(0,u) is a certain probability density function, quite complicated to obtain. As reported by Odlyzko in [18], we have the Taylor expansion around zero
p(0,u) =  p2

3
u2 -  2p4

45
u4 +  p6

315
u6 +
which under the GUE hypothesis entails that the proportion of dn less than a given small value d is asymptotic to (p2/9) d3 + O(d5). Thus very close pairs of zeros are rare.
Previous computations by Odlyzko [16,17,18,19], culminating with the unpublished result of computations at height 1023, were mainly dedicated to the GUE hypothesis empirical verifications. As observed by Odlyzko using different statistics, agreement is very good. Our goal here was to compute some of the statistics observed by Odlyzko relative to the GUE hypothesis, at height at each power of ten from 1013 to 1024. Our statistics, systematically observed at consecutive power-of-ten heights, are also oriented to observe empirically how the distribution of the spacing between zeros of the Zeta function converges to the asymptotic expectation.

4.2  Statistics

The statistics here are extracted from the paper paper, with in addition the statistics files generated by our program.

4.2.1  Computation information

Computation was launched on spare time of several machines. Zeros were computed starting roughly from the 10n-th zero for 13 n 24. An amount of roughly 2×109 zeros was computed at each height. Physical memory requirement was less than 512 Mo, and in the case of large height (for height 1023 and 1024), an amount of 12 Go of disk space was necessary.
Table below gives some indications of timing and the value of R used (see section ). It is to notice that due to the difficulty to have some long spare times on the different computers used, we adapt values of R that is why it is not monotonous. Due also to different capacities of the machines, the amount of used memory were not always identical. Timings are not monotonous also but at least, the table is just here to fix idea about cost. Third and fourth columns relates to offset index, so the value 10n should be added to have the absolute index of first or last zero. First and last zeros are always chosen to be Gram points proved regular with Turing's method (see section ).
Height Total timing in hours offset index of first zero offset index of last zero Value of R
1013 33.1 1 2×109 16777216
1014 35.0 3 2×109 16777216
1015 38.3 0 2×109-1 8388608
1016 49.5 1 2×109-1 16777216
1017 46.9 0 2×109 16777216
1018 81.6 1 2×109-1 33554432
1019 65.9 0 2×109+1 33554432
1020 87.8 4 2×109-1 33554432
1021 139.9 0 2×109-1 33554432
1022 151.5 2 2×109-1 134217728
1023 219.0 100 2×109-1 268435456
1024 326.6 0 2×109+47 268435456
Additional timing information relates to the efficiency of our implementation, using Odlyzko-Schönhage algorithm, compared to the direct evaluation of the Zeta function using Riemann-Siegel formula (). At height 1024 for example, two third of the total time was spent in the multi-evaluation of F(t) (see section ) and a single evaluation of Zeta using the direct optimized evaluation of Riemann-Siegel formula () (we used it for verification) took 5 % of the total time. So globally, the time needed to compute all the 2×109 zeros at height 1024 in our implementation is approximately equal to 20 evaluations of Zeta using the direct Riemann-Siegel formula. This proves the very high efficiency of the method.

4.2.2  Distribution of spacing between zeros

Statistics were done to observe numerically the agreement of asymptotic formulas (3), where the function p(0,t) has been computed with modern techniques. In our statistical study to check the validity of the GUE hypothesis, we observed the agreement of the empirical data with formulas (3) on each interval [a,b), with a = i/100 and b = (i+1)/100 for integer values of i, 0 i < 300. In figure 1, in addition to the curve representing the density probability function p(0,t), points were plotted at abscissa (i+1/2)/100 and coordinate
ci = 100  1

M
# { n : N+1 n N+M,dn [i/100,(i+1)/100]},
for height N=1013 and number of zeros M @ 2 ×109. As we can see the agreement is very good, whereas the graphic is done with the lowest height in our collection : human eye is barely able to distinguish between the points and the curve. That is why it is interesting to plot rather the density difference between empirical data and asymptotic conjectured behavior (as Odlyzko did in [19] for example). This is the object of figure 2, and this time what is plotted in coordinate is the difference
di = ci -
(i+1)/100

i/100 
p(0,t) dt.
To make it readable, the graphic restricts on some family of height N even if the corresponding data were computed at all height.
Even if oscillations in the empirical data appear because the sampling size of 2×109 zeros is a bit insufficient, we clearly see a structure in figure 2. First, the form of the difference at each height has a given form, and then, the way this difference decreases with the height can be observed.
Figure 1: Probability density of the normalized spacing dn and the GUE prediction, at height 1013. A number of 2×109 zeros have been used to compute empirical density, represented as small circles.
Figure 2: Difference of the probability density of the normalized spacing dn and the GUE prediction, at different height (1014, 1016, 1018, 1020, 1022, 1024). At each height, 2×109 zeros have been used to compute empirical density, and the corresponding points been joined with segment for convenience.

4.2.3  Violations of Rosser rule

The table below lists statistics obtained on violations of Rosser rule (VRR). As we should expect, more and more violations of Rosser rule occurs when the height increases. Special points are Gram points which are counted in a VRR, so equivalently, they are points that do not lie in a regular Gram block.
Height VRR per million zeros Number of types of VRR Number of special points Average number of points in VRR
1013 37.98 68 282468 3.719
1014 54.10 86 418346 3.866
1015 72.42 109 581126 4.012
1016 93.99 140 780549 4.152
1017 117.25 171 1004269 4.283
1018 142.30 196 1255321 4.411
1019 168.55 225 1529685 4.538
1020 197.28 270 1837645 4.657
1021 225.80 322 2156944 4.776
1022 256.53 348 2507825 4.888
1023 286.97 480 2868206 4.997
1024 319.73 473 3262812 5.102

4.2.4  Behavior of S(t)

The S(t) function is defined in () and permits to count zeros with formula (). It plays an important role in the study of the zeros of the Zeta function, because it was observed that special phenomenon about the zeta function on the critical line occurs when S(t) is large. For example, Rosser rule holds when |S(t)| < 2 in some range, thus one needs to have larger values of S(t) to find more rare behavior.
As already seen before, it is known unconditionally that
S(t) = O(logt).
Under the RH, we have the slightly better bound
S(t) = O
 logt

loglogt

.
However, it is thought that the real growth of rate of S(t) is smaller. First, it was proved that unconditionally, the function S(t)/(2p2 loglogt)1/2 is asymptotically normally distributed. So in some sense, the "average" order of S(t) is (loglogt)1/2. As for extreme values of S(t); Montgomery has shown that under the RH, there is an infinite number of values of t tending to infinity so that the order of S(t) is at least (logt/loglogt)1/2. Montgomery also conjectured that this is also an upper bound for S(t). As described in section 4.2.6 with formula (5), the GUE suggests that S(t) might get as large as (logt)1/2 which would contradict this conjecture.
As explained in [18,P. 28], one might expect that the average number of changes of sign of S(t) per Gram interval is of order (loglogt)-1/2. This is to be compared with the last column of the table below, which was obtained thanks to the statistics on Gram blocks and violations of Rosser rule.
As it is confirmed in heuristic data in the table below, the rate of growth of S(t) is very small. Since exceptions to RH, if any, would probably occur for large values of S(t), we see that one should be able to reach much larger height, not reachable with today's techniques, to find those.
Height Minimum of S(t) Maximum of S(t) Number of zeros with S(t) < -2.3 Number of zeros with S(t) > 2.3 Average number of change of sign of S(t) per Gram interval
1013 -2.4979 2.4775 208 237 1.5874
1014 -2.5657 2.5822 481 411 1.5758
1015 -2.7610 2.6318 785 760 1.5652
1016 -2.6565 2.6094 1246 1189 1.5555
1017 -2.6984 2.6961 1791 1812 1.5465
1018 -2.8703 2.7141 2598 2743 1.5382
1019 -2.9165 2.7553 3487 3467 1.5304
1020 -2.7902 2.7916 4661 4603 1.5232
1021 -2.7654 2.8220 5910 5777 1.5164
1022 -2.8169 2.9796 7322 7359 1.5100
1023 -2.8178 2.7989 8825 8898 1.5040
1024 -2.9076 2.8799 10602 10598 1.4983

4.2.5  Estimation of the zeros approximation precision

As already discussed in section , a certain proportion of zeros were recomputed in another process with different parameters in the implementation and zeros computed twice were compared. Table below list the proportion of twice computed zeros per height, mean value of absolute value of difference and maximal difference.
Height Proportion of zeros computed twice Mean difference for zeros computed twice Max difference for zeros computed twice
1013 4.0% 5.90E-10 5.87E-07
1014 6.0% 6.23E-10 1.43E-06
1015 6.0% 7.81E-10 1.08E-06
1016 4.5% 5.32E-10 7.75E-07
1017 8.0% 5.85E-10 9.22E-07
1018 7.5% 6.59E-10 1.88E-06
1019 11.0% 5.15E-10 3.07E-06
1020 12.5% 3.93E-10 7.00E-07
1021 31.5% 5.64E-10 3.54E-06
1022 50.0% 1.15E-09 2.39E-06
1023 50.0% 1.34E-09 3.11E-06
1024 50.0% 2.68E-09 6.82E-06

4.2.6  Extreme gaps between zeros

The table below lists the minimum and maximal values of normalized spacing between zeros dn and of dn+dn+1, and compares this with what is expected under the GUE hypothesis (see section 4.1). It can be proved that p(0,t) have the following Taylor expansion around 0
p(0,u) =  p2

3
u2 - 2  p4

45
u4 +
so in particular, for small delta
Prob(dn < d) =
d

0 
p(0,u) du ~  p2

9
d3
so that the probability that the smallest dn are less than d for M consecutive values of dn is about
1-
1-  p2

9
d3
M

 
@ 1-exp
-  p2

9
d3 M
.
This was the value used in the sixth column of the table. The result can be also obtained for the dn+dn+1
Prob(dn+dn+1 < d) ~  p6

32400
d8,
from which we deduce the value of the last column.
Height Mini dn Maxi dnMini dn+dn+1 Maxi dn+dn+1 Prob min dn in GUE Prob min dn+dn+1 in GUE
1013 0.0005330 4.127 0.1097 5.232 0.28 0.71
1014 0.0009764 4.236 0.1213 5.349 0.87 0.94
1015 0.0005171 4.154 0.1003 5.434 0.26 0.46
1016 0.0005202 4.202 0.1029 5.433 0.27 0.53
1017 0.0006583 4.183 0.0966 5.395 0.47 0.36
1018 0.0004390 4.194 0.1080 5.511 0.17 0.67
1019 0.0004969 4.200 0.0874 5.341 0.24 0.18
1020 0.0004351 4.268 0.1067 5.717 0.17 0.63
1021 0.0004934 4.316 0.1019 5.421 0.23 0.50
1022 0.0008161 4.347 0.1060 5.332 0.70 0.61
1023 0.0004249 4.304 0.1112 5.478 0.15 0.75
1024 0.0002799 4.158 0.0877 5.526 0.05 0.19
For very large spacing in the GUE, as reported by Odlyzko in [18], des Cloizeaux and Mehta [4] have proved that
logp(0,t) ~ -p2 t2/8        (t),
which suggests that

max
N+1 n N+M 
dn ~  (8logM)1/2

p
.
(2)
This would imply that S(t) would get occasionally as large as (logt)1/2, which is in contradiction with Montgomery's conjecture about largest values of S(t), discussed in section 4.2.4.

4.2.7  Moments of spacings

The table below list statistical data about moments of the spacing dn-1 at different height, that is mean value of
Mk = (dn-1)k,
together with the GUE expectations.
Height M2 M3 M4 M5 M6 M7 M8 M9
1013 0.17608 0.03512 0.09608 0.05933 0.10107 0.1095 0.1719 0.2471
1014 0.17657 0.03540 0.09663 0.05990 0.10199 0.1108 0.1741 0.2510
1015 0.17697 0.03565 0.09710 0.06040 0.10277 0.1119 0.1759 0.2539
1016 0.17732 0.03586 0.09750 0.06084 0.10347 0.1129 0.1776 0.2567
1017 0.17760 0.03605 0.09785 0.06123 0.10407 0.1137 0.1789 0.2590
1018 0.17784 0.03621 0.09816 0.06157 0.10462 0.1145 0.1803 0.2613
1019 0.17805 0.03636 0.09843 0.06189 0.10511 0.1152 0.1814 0.2631
1020 0.17824 0.03649 0.09867 0.06215 0.10553 0.1158 0.1824 0.2649
1021 0.17839 0.03661 0.09888 0.06242 0.10595 0.1165 0.1836 0.2668
1022 0.17853 0.03671 0.09906 0.06262 0.10627 0.1169 0.1842 0.2678
1023 0.17864 0.03680 0.09922 0.06282 0.10658 0.1174 0.1850 0.2692
1024 0.17875 0.03688 0.09937 0.06301 0.10689 0.1179 0.1859 0.2708
GUE 0.17999 0.03796 0.10130 0.06552 0.11096 0.1243 0.1969 0.2902
In the next table we find statistical data about moments of the spacing dn+dn+1-2 at different height, that is mean value of
Nk = (dn+dn+1-2)k,
together with the GUE expectations.
Height N2 N3 N4 N5 N6 N7 N8 N9
1013 0.23717 0.02671 0.16887 0.06252 0.2073 0.1530 0.3764 0.4304
1014 0.23846 0.02678 0.17045 0.06301 0.2099 0.1550 0.3827 0.4388
1015 0.23956 0.02688 0.17181 0.06349 0.2122 0.1568 0.3880 0.4458
1016 0.24050 0.02700 0.17299 0.06396 0.2142 0.1585 0.3927 0.4523
1017 0.24132 0.02713 0.17404 0.06446 0.2159 0.1601 0.3970 0.4583
1018 0.24202 0.02726 0.17494 0.06488 0.2175 0.1614 0.4005 0.4630
1019 0.24264 0.02740 0.17574 0.06530 0.2188 0.1627 0.4036 0.4672
1020 0.24319 0.02753 0.17645 0.06569 0.2201 0.1639 0.4065 0.4713
1021 0.24366 0.02766 0.17709 0.06609 0.2212 0.1651 0.4092 0.4753
1022 0.24409 0.02778 0.17765 0.06643 0.2222 0.1660 0.4114 0.4780
1023 0.24447 0.02790 0.17819 0.06679 0.2232 0.1671 0.4140 0.4821
1024 0.24480 0.02801 0.17863 0.06709 0.2240 0.1679 0.4158 0.4846
GUE 0.249 0.03 0.185 0.073 0.237 0.185 0.451 0.544
The last table below is about mean value of logdn, 1/dn and 1/dn2.
Height logdn 1/dn 1/dn2
1013 -0.101540 1.27050 2.52688
1014 -0.101798 1.27124 2.53173
1015 -0.102009 1.27184 2.54068
1016 -0.102188 1.27235 2.54068
1017 -0.102329 1.27272 2.54049
1018 -0.102453 1.27308 2.54540
1019 -0.102558 1.27338 2.54906
1020 -0.102650 1.27363 2.54996
1021 -0.102721 1.27382 2.54990
1022 -0.102789 1.27401 2.54783
1023 -0.102843 1.27415 2.55166
1024 -0.102891 1.27427 2.55728
GUE -0.1035 1.2758 2.5633

4.2.8  Statistic files

The following files contains, in a "brute" form, the global statistics of our computation of two billion non trivial zeros of the Riemann zeta function, at different height (zero number 10n for 13 n 24). Each file corresponds to one height, and contains data like Gram blocks, types and number of violations of Rosser rule, extremal values of deltan and dn+dn+1 (dn is the normalized gap between consecutive zeros), data on S(t), moments on dn, and observed distributions on gaps.

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