On the self-replicating properties of Riemann zeta zeros: A statistical study
PPreprint manuscript No. (will be inserted by the editor)
On the self-replicating properties of Riemann zetazeros: A statistical study
Jouni J. Takalo the date of receipt and acceptance should be inserted laterSpace physics and astronomy research unit,University of Oulu, POB 3000, FIN-90014, Oulu, FinlandTel.: +358-40-7172358E-mail: jouni.j.takalo@oulu.fi;[email protected] ORCID ID 0000-0002-1807-6674 a r X i v : . [ n li n . C D ] J un Jouni J. Takalo
Abstract
We study distributions of differences of unscaled Riemann zeta ze-ros, γ − γ (cid:48) , at large distances. We show, that independently of the height, asubset of finite number of successive zeros knows the locations of lower level ze-ros. The information contained in the subset of zeros is inversely proportionalto ln ( γ/ (2 π )), where γ is the average zeta of the subset. Because the meandifference of the zeros also decreases as inversely proportional to ln ( γ/ (2 π )),each equally long segment of the line (cid:60) ( z ) = 1 / Keywords
Riemann zeta zeros · difference of zeta zeros · paircorrelation · prime numbers · statistical methods · Johnson distribution
It is well-known that the correlation between close pairs of nontrivial zeros ofthe Riemann zeta function (scaled to have unit average spacing) is [8] R ( x ) = 1 − (cid:18) sin ( π x ) π x (cid:19) . (1)Figure 1 shows the density of 5 million differences starting from billionthzero and the prediction of Eq.1 as a red curve. Odlyzko (1987) already showedthat this conjecture was supported also by larger heights of zeta zeros [9]. Theaforementioned papers are restricted to analyse only consecutive or locallyclose differences of zeros. P´erez-Marco (2011) has shown that the statistics ofvery large zeros do find the location the first Riemann zeros [10]. There havebeen several studies of the differences (pair correlations, triple correlations, n-point correlations) between zeta zeros. They predict troughs in the correlationfunctions at the sites of the zeros themselves [1], [2], [3], [13], [10], [11], [14].We study here distributions of differences of unscaled zeta zeros at longerdistances, and analyse in more detail the reason for the troughs in correlationsof zeta zeros using statistical methods. elf-replicating zeta zeros 3 Fig. 1
Distribution of locally close scaled differences of zeta zeros. Five million zeros tartingfrom billionth zero were used in the statistics. Red curve is the plot of function in Eq. 1. were kindly providedby Dr. A. Odlyzko [5].2.2 Johnson distributionJohnson distribution for the variable x is defined as z = λ + δ ln ( f ( u )) , (2)with u = ( x − ξ ) /λ, (3)Here z is a standardized normal variable and f ( u ) has three different formsthe lognormal distribution, S L : f ( u ) = u, (4)the unbounded distribution, S U : f ( u ) = u + (cid:0) u (cid:1) / , (5) Jouni J. Takalo and the bounded distribution, S B : f ( u ) = u/ (1 − u ) . (6)The supports for the distributions are S L : ξ < x, S U : −∞ < x < ∞ and S B : ξ < x < ξ + λ [6], [15]. With these definitions, the probability distributionsare for S L : P ( u ) = δ √ π × u × exp (cid:26) −
12 [ γ + δ ln ( u )] (cid:27) . (7)for S U P ( u ) = δ √ π × √ u + 1 × exp (cid:26) − (cid:104) γ + δ ln (cid:16) u + (cid:112) u + 1 (cid:17)(cid:105) (cid:27) . (8)and for S B P ( u ) = δ √ π × u/ (1 − u ) × exp (cid:40) − (cid:20) γ + δ ln (cid:18) u − u (cid:19)(cid:21) (cid:41) (9) δ ), of unscaled zeta zeros. We use thefollowing notation δ ( n ) = γ ( n + i ) − γ ( i ) , n = 1 , , , ... (10)Here i goes from j to j+5000000 in our analyses, where j is the ordinalnumber of the starting zeta zero. Because the zeros are not stabilized yet atthe height of millionth zero we study, in the next, zeros and their differencesat the height of 1, 10, 100 billionth zeta zero, and zeta zeros starting at 10 [14]. Figure 2a shows the distributions of five million δ s for n=1,2,3,..,159from the the height of one billionth zeta zero. The distributions with n=40,60, 71, 87, 94, 107, 117, 123, 137, 142 and 151 are the nearest distributions tothe sites of first zeta zeros (marked with red vertical dashed line). Figure 2bdepicts the integrated distribution of the δ -distributions in Fig. 2a. Note thatin Fig. 2a each distribution is separately normalized to unit area, while forFig. 2b of integrated distribution the whole area is one unit (we have cut partof the distribution to better show the troughs). We note that the troughs areclearer in Fig. 2a, where the distributions are the lower the nearer they are tothe zeta zeros. When the distributions are summed together the troughs arepartly filled with the tails of nearby distributions.The distributions near the zeros have also characteristic features. Figure 3adepicts the variances of the distributions shown in Fig. 2, and Fig. 3b second(discrete) derivative of the variance function (we calculated these values a little elf-replicating zeta zeros 5 Fig. 2 a) The distributions from the the height of one billionth zeta zero using δ ( n ) forn=1,2,3,..,159 . Note that the sites of first zeta zeros (marked with red vertical dashed line)are located near distributions with n=40, 60, 71, 87, 94, 107, 117, 123, 137, 142 and 151. b)The integrated distribution of the δ -distributions of figure a. further, i.e. for n=1-400). It is evident that variances have local maxima (orat least a turning point) at zeta zero. This is more clearly seen in Fig. 3b asa minimum of second derivative at each zeta zero. It, indeed, seems that thefirst zeros are encoded to the zeros (or their differences) at higher level.Figure 4a depicts the kurtoses of the distributions for the same interval asFig. 3. We note that the kurtoses have minima at (or near) every zeta zero.These are, however, not so clear than the maxima of the variances in Fig. 3a.Let us define the second difference, i.e. difference of the first difference δ ( i )such that δ = δ ( i + 1) − δ ( i ) , i = 1 , , , ... − δ ( i + n ) − δ ( i ) instead, the minima are muchclearer, as seen in the Fig. 4b. Now the local minima, which are not sites ofzeta zeros, are tiny and clearly separable from the minima at the sites of zetazeros. Note also the similarity of Figs. 3b and 4b, only the vertical scale isdifferent. Again the information of the lower zeros is somehow contained inthe differences of higher level zeros.Figure 5 shows that the awareness of the other zeros is not restricted onlyto very first zeros. The variances of this figure are calculated from 5 milliondifferences starting from 10 rd zero. The upper panel of Fig. 5 depicts thesecond (discrete) derivative of δ (n)s with n=5500-5999. These distributions arelocated at height of the zeros between 504-568, and the zeros in this interval Jouni J. Takalo
Fig. 3 a) The variances of the distributions of δ ( n ), n=1-400 from billionth zero.b) Second(discrete) derivative of the variance function of figure a. are shown as vertical dashed lines. The lower panel of Fig. 5 depicts the secondderivative of δ (n)s with n=1000001-1000499. These distributions are locatedat height of the zeros between 128002-128066. Notice that the intervals areequally long, although the latter has twice as many zeros as the first interval.The minima of the derivatives of the variances coincide quite well with thezeros at those intervals. However, the resolution is too poor to distinguish thetwo, very nearby, pairs. e.g. (728.405, 728.759) and (750.656, 750.966) in theupper panel or the triple peak (128043.51, 128043.85, 128044.12) in the lowerpanel as separate peaks [14].3.2 Skewnesses of the δ -distributionsFigure 6 shows more detailed pattern of the δ -distributions of Fig. 2a for n=37-65. The Johnson distribution fits are also shown in the figure. The red dottedvertical lines show the two first zeta zeros, and blue thinner dotted line locatesin the halfway between these zeros. The decimal numbers tell the skewnessesof each distribution, and their sign is also shown as text above them. Notethat the skewness changes sign at the zeros, and furthermore, at the halfwaybetween the zeros such that δ -distribution is here always skewed towards the elf-replicating zeta zeros 7 Fig. 4 a) The kurtoses of the distributions of δ ( n ), n=1-400 from billionth zero.b) Thekurtoses of the distributions of δ ( n ), n=1-400 from billionth zero. nearest zeta zero. It seems that the δ s of the zeros are avoiding the zerosthemselves, i.e., Riemann zeros repel their δ s [10]. Figure 7 shows the intervalof δ -distribution for n=570-640 starting from 10 rd zero. The situation isnow more complicated, because the zeros are nearer to each other. In spiteof this, the distributions are still lower at or near the sites of zeta zeros. Thedistributions in the left start with negative value for skewness, because thereis a nearby zeta zero at 72.067 (not seen in the figure). The skewness changesfrom negative to positive at the blue dashed line, because of the next zerolocated at 75.705. After the zero the skewness, however, does not change tonegative, but only decreases somewhat. It is evident that the two next zeros atright side 77.145 and 79.337 repel together more strongly than the only zeroon the left side. The skewness changes sign then at 77.145, except that thelast distribution at the left side is already slightly negative. Note also, that theskewness stays negative while passing the zero at 79.337, and only decreasesagain somewhat. This lasts until the blue dashed line, after which the skewnesschanges to positive due to the lurking zero at 82.910. Jouni J. Takalo
Fig. 5 a) Second derivative of the variances of the δ ( n )-distributions for n=5500-5999calculated for 5 million zeta zeros starting from zero 10 as a function of mean( δ ( n )).b)Same as a, but for n=1000001-1000499. Zeta zeros are shown with vertical dashed red lines. Fig. 6
The δ ( n )-distributions at height of billionth zeta zero using n=37-65 and fitted withJohnson probability density function. The decimal numbers are skewnesses of the separatedistributions and the text tells their signs in each region.elf-replicating zeta zeros 9 Fig. 7
The δ ( n )-distributions calculated using 5 million zeta zeros with n=280-350 startingfrom 10 rd zero. Distributions are fitted with Johnson distributions. Fig. 8
A fit of the function f ( t ) = (2 π ) / ln( T/ (2 π )) to the mean differences of zeta zeros;0.351, 0.313, 0.282 and 0.128 for intervals at 1, 10, 100 billionth and 10 rd zeta zero,respectively. δ -distributionsFigure 8 shows a fit of the function f ( t ) = (2 π ) / ln( T / (2 π )) to the meandifferences of zeta zeros; 0.351, 0.313, 0.282 and 0.128 for intervals of zetazeros starting at 1, 10, 100 billionth and 10 rd zero, respectively. Note thatthe function is the inverted normalizing function of the first differences of thezeta zeros [9]. It is also clear that the mean difference of unscaled zeta zerosapproaches zero, when the height of the zeros increase without bounds (atleast for zeros on the line (cid:60) ( z ) = 1 / δ -distributions at 1 and 100 billionth zetazeros, and Fig. 9b δ -distributions starting at 10 rd zero for n=1-499 as a Fig. 9 a) Variances of δ -distributions at 1 and 100 billionth zeta zeros, and b) of δ -distributions starting at 10 rd zero for n=1-499 as a function of corresponding mean( δ ( n )). function of corresponding mean( δ ( n )). The patterns of the variances are sim-ilar, except that their total length changes such that the same amount ofdistributions makes shorter variance curve when going higher level of the ze-ros. We plotted the first zeros (red vertical dashed lines) only to the Fig.9b for clarity. The last distributions seen in the variance plots are at meanvalue of δ (499), i.e. at values 63.873, 140.779 and 175.171 for 10 rd, 100billionth and billionth zeta zeros, respectively. The ratios of these numbersare 1:2.204:2.743. These numbers are inversely proportional to ln ( γ/ (2 π )),where γ is the value of average zeta zero in the corresponding zero interval(because the zeros are at so height level, we can here use the first zeros of elf-replicating zeta zeros 11 the interval). The first zeros are γ =371870203.837, γ =29538618431.613 and γ =13066434408794275940027.301 for billionth, 100 billionth and 10 rd zetazero. The ratios 1 /ln ( γ / (2 π )) : 1 /ln ( γ / (2 π )) : 1 /ln ( γ / (2 π )) are 1:2.204:2.743,which are same as aforementioned ratios of the lengths of the patterns of dis-tributions. We could say that the higher we go in the zeros the less informationthe same amount of zeros contains about the lower zeros; the information con-tent of the zeros at height γ is inversely proportional to ln ( γ/ (2 π )). However,while the mean difference of the zeros decreases also as inversely proportionalto ln ( γ/ (2 π )), this means that equally long segments of the line (cid:60) ( z ) = 1 / δ -distributions can be fitted very well with the JohnsonSB and SU PDFs (very rarely SL is needed). Figure 10 shows the distributionsin the skewness-kurtosis -plane [4] for zeros at 1 billion, 100 billion and 10 .The red curve in the figure shows the border between Johnson SU and SBdistributions (this is also region of SL distribution). It is notable that δ (1)s arealmost in the same place in the plane for all groups. Otherwise, the points seemto be more compactly located when going to higher levels of zeta zeros. Note,that all groups of distributions with n=2-999 form a heart-shaped pattern suchthat the point (0,3), which is the site of normal distribution in the skewness-kurtosis -plane, is located in the trough of the heart. We suppose that theaverage kurtosis is approaching the value 3, when going still to upper levels ofzeta zeros, and the average skewness approaches zero from the positive side, i.e.distributions approach to normal distribution [14]. Furthermore, when going tothe higher zeros the δ -distributions are thinner and thinner, i.e, the standarddeviation decreases. (We find that the standard deviation decreases somewhatslower than the mean value, i.e. proportional to about 2 . / ( ln ( γ/ (2 π ))) . ).We can then approximate the distributions (at very high levels of zeta zeros)with a normal distribution whose standard deviation approaches to zero, i.e. ittends to Dirac delta function at the mean value of the distribution. In this casewe need more and more very nearby distributions to extract the informationabout the earlier zeros of the zeta function. We calculate the function of prime powers from the cosine series [7] P ( x ) = − N (cid:88) i =1 cos ( γ i × ln ( x )) , (12)where λ i is the first element in the sequence of the zeta zeros and N is thenumber of zeros in the sequence. Figure 11 depicts the primes and their powersbetween 200-300 calculated from 12 using 2 million zeta zeros starting at 10 Fig. 10
The points of distributions starting from 1 billionth (black square), 100 billionth(light blue star) and 10 rd (red circle) zero in the Skewness-kurtosis -plane. The blue curveis Johnson SL distribution, which divides the plane to Johnson SU and SB regions. Cyandot at (0,3) is the site of normal distribution in the skewness-kurtosis -plane. millionth (red curve), billionth (blue curve) and 100 billionth zeta zero (greencurve). There are four twin primes (227/229, 239/241, 269/271 and 281/283),three prime powers (3 , 2 and 17 ) and eight single primes in this interval.The interesting thing here is that the two million zeros at height 10 millionthzero give stronger peaks at the sites of the primes than same amount of zerosat height 1 billionth zero, which in turn give stronger peaks than two millionzeros at height 100 billionth zero. The ratios of the heights are 1:1.244:1.629,which is same ratio as 1 /ln ( γ / (2 π )) : 1 /ln ( γ / (2 π )) : 1 /ln ( γ / (2 π )), where γ , γ and γ are the average zeros at height 100 billion, billion and 10 million,respectively, i.e, the strength of the line is (again) inversely proportional toln( γ /(2 π ). Note that the peaks of prime powers are lower than the other primepeaks. This is because peaks are scaled with the power such that the strength ofthe peak decreases to 1 /p ( n − / , where n=2,3,4,...is the power of the originalprime p. elf-replicating zeta zeros 13 Fig. 11
The primes and their powers between 200-300 calculated from Eq. 12 using 2 millionzeta zeros starting at ten millionth (red curve), billionth (blue curve) and 100 billionth zetazero (green curve).
Fig. 12
The primes and prime power 5 between 78100-78200 calculated from first tenmillion zeta zeros. We note also that the peaks are decreasing when going further in the realaxis. This is probably due to the logarithmic dependence of the x (see Eq.12) such that while period in constantly increasing the amplitude decreasesand eventually dies out [12]. Also the extra fluctuation around (and between)the lines of primes increases when going upper in the real axis. Figure 12depicts the primes between 78100-78200 calculated from first ten million zetazeros. There are two twin primes, seven single primes and one prime power(5 ) between this interval. Although we used so many zeta zeros the result isnot anymore satisfying. In order to plot large primes we need a huge amountof zeros and capacity in calculation. The extra fluctuation can be diminishedusing windowing, but anyhow this method is impractical for finding new primes[12]. We have studied the δ -distributions of zeros of Riemann zeta function atheights 10 millionth, billionth 100 billionth and 10 -rd zero such that wecalculate distribution for each difference, δ ( n ), separately. We used 5 million δ s for these analyses, and showed that statistical properties are very similarfor all intervals. It is interesting that a finite subset of successive zeros, inde-pendently of its height, from the infinite sequence of nontrivial zeros knowsthe location of other zeros. The information contained in the subset of zerosin inversely proportional to ln ( γ/ (2 π )), where γ is the average of the zeros inthe subset. However, while the density of the zeros increases as proportionalto ln ( γ/ (2 π )), each equally long segment of the line (cid:60) ( z ) = 1 / ln ( γ/ (2 π )) exists also in thestrength of the lines of the prime powers calculated from subset of zeta zeros.The skewness of the δ -distributions change sign, when crossing zeta zeroor, at least, decreases when passing the zero in increasing direction. The vari-ance has local maximum or turning-point and the kurtosis local minimumat zeta zero. We also plotted δ ( n )-distributions (for n=1-999) of zeta zeros atheights 1 billion, 100 billion and 10 on the skewness-kurtosis -plane. All thesegroups of points form a heart-shaped pattern such that the higher the zeros themore compactly the corresponding points are located in the skewness-kurtosis-plane. The site of normal distribution in the plane, i.e. point (0,3) is in thetrough of this pattern. We believe, that going still higher levels in the zerosthe average kurtosis approaches value 3, which is the kurtosis of normal dis-tribution. The δ (1) is located almost at the same point for all groups apartfrom other points of the corresponding group. Acknowledgements
We acknowledge LMFDB and A.M. Odlyzko for the zeta zero data.
Conflict of interest
The authors declare that they have no conflict of interest.
References
1. Bogomolny E.B. and Keating J.P., Random matrix theory and the Riemann zeros I:three and four-point correlations ,
Nonlinearity , 8, pp. 11151131, 19952. Bogomolny E.B. and Keating J.P. , Random matrix theory and the Riemann zeros II:n-point correlations,
Nonlinearity , 9 , pp. 911935, 1996.3. Conrey J.B. and N Snaith, Triple correlation of the Riemann zeros,
Journal de Thoriedes Nombres de Bordeaux , 20, 61-106, 2008.4. Cugerone K. and De Michele C., Johnson SB as general functional form for raindrop sizedistribution,
Water Resources Research , 51, pp. 6276-6289, 2015.5. Hiary, G.A. and Odlyzko, A.M., The zeta function on the critical line: Numerical evidencefor moments and random matrix theory models,
Mathematics of Computation , 81 (279),pp. 1723-1752, 2012.elf-replicating zeta zeros 156. Johnson N.L., Systems of frequency curves generated by methods of translation,
Biometrika , 36, pp. 149-176, 1949.7. Mazur, B. and Stein, W. Prime Numbers and the Riemann Hypothesis, pp. 108-110,Cambridge University Press, Cambridge CB2 8BS, United Kingdom (2016).8. Montgomery H.L., The pair correlation of zeros of the zeta function, Analytic numbertheory, Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St.Louis, Mo., (1973).9. Odlyzko, A.M., On the Distribution of Spacings Between Zeros of the Zeta Function,
Mathematics of Computation , 48 (177), pp. 273-308, (1987).10. P´erez-Marco R., Statistics of Riemann zeros , arXiv:1112.0346v1 [math.NT] , 2011.11. Rogers. B, Macroscopic pair correlation of the Riemann zeroes for smooth test functions,
The Quarterly Journal of Mathematics , 64 (4), pp. 11971219, 2013.12. Sakhr, Jamal and Bhaduri, Rajat and van Zyl, Brandon, Zeta Function Zeros, Powersof Primes, and Quantum Chaos,
Phys. Rev. E, Statistical, nonlinear, and soft matterphysics , vol. 68, 026206, (2003). DOI:10.1103/PhysRevE.68.026206.13. Snaith N.C., Riemann Zeros and Random Matrix Theory,