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ON THE DISTRIBUTION OF SPACINGS BETWEEN ZEROS OF THE ZETA FUNCTION. A. M. Odlyzko AT&T Bell Laboratories Murray Hill, New Jersey ABSTRACT
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1 ON THE DISTRIBUTION OF SPACINGS BETWEEN ZEROS OF THE ZETA FUNCTION A. M. Odlyzko AT&T Bell Laboratories Murray Hill, New Jersey ABSTRACT A numerical study of the distribution of spacings between zeros of the Riemann zeta function is presented. It is based on values for the first 10 5 zeros and for zeros number to that are accurate to within ± 10 8, and which were calculated on the Cray-1 and Cray X-MP computers. This study tests the Montgomery pair correlation conjecture as well as some further conjectures that predict that the zeros of the zeta function behave similarly to eigenvalues of random hermitian matrices. Matrices of this type are used in modeling energy levels in physics, and many statistical properties of their eigenvalues are known. The agreement between actual statistics for zeros of the zeta function and conjectured results is generally good, and improves at larger heights. Several initially unexpected phenomena were found in the data and some were explained by relating them to the primes.
2 ON THE DISTRIBUTION OF SPACINGS BETWEEN ZEROS OF THE ZETA FUNCTION A. M. Odlyzko AT&T Bell Laboratories Murray Hill, New Jersey 1. Introduction The Riemann Hypothesis (RH) has been of central interest to number theorists for a long time, and many unsuccessful attempts have been made to either prove or disprove it by analytic methods. A series of large computations have also been made, culminating in the recent verification [52] that the RH holds for the first zeros, an effort that involved over a thousand hours on a modern supercomputer. A proof of the RH would lead to tremendous improvements in estimates for various arithmetic functions, such as the difference between π(x), the number of primes x, and Li (x), the logarithmic integral of x. However, many other questions would remain open, such as the precise magnitude of the largest gap between consecutive primes below a given bound. Answers to such questions depend on a much more detailed knowledge of the distribution of zeros of the zeta function than is given by the RH. Relatively little work has been devoted to the precise distribution of the zeros. The main reason for the lack of research in this area was undoubtedly the feeling that there was little to be gained from studying problems harder than the RH if the RH itself could not be proved. A major step towards a detailed study of the distribution of zeros of the zeta function was made by H. L. Montgomery [56, 57]. Under the assumption of the RH, he showed that if we define
3 - 2 - F(α, T) = 2π(T log T) 1 Σ T iα (γ γ ) (γ γ )2 0 < 0 < γ γ T T (1.1) for α and T real, T 2, where 1/2 + iγ and 1/2 + iγ denote nontrivial zeros of the zeta function, then F(α, T) = ( 1 + o( 1 ) ) T 2α log T + α + o( 1 ) as T, (1.2) uniformly for 0 α 1. Montgomery also presented heuristic arguments which suggested that F(α, T) = 1 + o( 1 ) as T (1.3) uniformly for α ε [a, b], where 1 a < b < are any constants. If the conjecture (1.3) were true, then the following estimate, known as the Montgomery pair correlation conjecture [56] would follow: Conjecture. For fixed 0 < α < β <, {(γ, γ ) : 0 < γ, γ T, 2π α( log T) 1 γ γ 2π β( log T) 1 } (1.4) _ T log T 2π α β 1 _ sin πu 2 du πu as T. The above conjecture is very striking, particularly in predicting that small gaps between zeros of the zeta function occur very infrequently. This behavior is very different from that of many other number theoretic functions. The number of primes in short intervals, for example, is observed experimentally and is conjectured theoretically
4 - 3 - to be distributed like a Poisson random variable [29]. The conjecture (1.4) suggests some further topics for investigation. In the language of mathematical physics, this conjecture says that 1 (( sin πu)/(πu) ) 2 is the pair correlation function of zeros of the zeta function. F. J. Dyson pointed out [56] that the gaussian unitary ensemble (GUE) has the same pair correlation function. The GUE has been studied very extensively in mathematical physics together with the gaussian orthogonal ensemble (GOE) and the gaussian symplectic ensemble (GSE) as models for distribution of energy levels in many-particle systems. The GUE consists of n n complex hermitian matrices of the form A = (a j k ), where a jj = 2 1/2 σ j, j, a j k = σ j,k + i η j,k for j < k, and a j,k = a _ k, j = σ k, j i η k, j for j > k, where the σ j,k and η j,k are independent standard normal variables. When n and the eigenvalues of the matrices of the GUE are suitably normalized, their pair correlations becomes 1 (( sin πu)/(πu) ) 2. (The pair correlations of the GOE and GSE are different.) For more information about the GUE and the other ensembles, the reader is referred to [2, 4, 11, 41, 54, 59, 63]. The possible connection between zeros of the zeta function and eigenvalues of random matrices is of interest in number theory because of the Hilbert and Po lya conjectures [56, 59] which say that the zeros of the zeta function correspond to eigenvalues of a positive linear operator. If true, the Hilbert and Po lyaconjectures would imply the RH, and some people feel that the most promising way to prove the RH is by finding the right operator and establishing the necessary results about it. One can argue heuristically that if such an operator exists, its eigenvalues might behave like those of a random operator, which in turn might behave like the limit of a sequence of random
5 - 4 - matrices, and that therefore the zeros of the zeta function might be distributed like the eigenvalues of a large random matrix. For more on these very vague conjectures, see [3, 59]. Since Montgomery s result (1.1) is consistent with the GUE distribution of eigenvalues, one could then argue that the operator associated to the zeta function ought to be complex hermitian, and that GUE properties of eigenvalues might apply at least approximately to the zeros of the zeta function. Since Montgomery s proof of (1.2) (which depends on the RH) was published, several other results have been obtained. Ozluk [62] has considered a q-analogue of Montgomery s method and showed, roughly speaking, that if one considers a function similar to the F(α, T) of (1.1), but where one sums over zeros of many Dirichlet L- functions, then the analogue of Montgomery s conjecture (1.3) is true for 1 α 2. For other results about Montgomery s pair correlation conjecture and its consequences, see [30, 31, 34, 35, 36, 42, 44, 59]. Overall, though, there is very little solid theoretical evidence in favor of even the Montgomery pair correlation conjecture (1.4), and essentially none for the speculative conjectures discussed above that link zeros of the zeta function to eigenvalues of random hermitian matrices. One feature of the speculative arguments mentioned above is that they propose a connection between zeros of the zeta function, about which relatively little is known theoretically and the GUE, which has been investigated very extensively and abounds in specific predictions. This makes it possible to compare data for the zeros of the zeta function against the theoretical predictions of the GUE. The purpose of this paper is to report the results of such empirical tests of the pair correlation conjecture and other GUE predictions.
6 - 5 - The agreement between the predictions of the GUE theory and actual computed behavior of the zeros of the zeta function is quite good, especially when one takes account of the fact that the zeta function on the critical line approaches its asymptotic behavior very slowly, as is explained in Section 2. However, there are also several features of the data for the zeros that are not predicted by the GUE theory. One is that there are long-range correlations between spacings of consecutive zeros. These can be explained in terms of the primes through the use of the explicit formulas of prime number theory. Another is that small spacings between zeros are somewhat more common among high zeros than is expected. The deviation between the predicted and observed numbers is not large, but it is surprising, since it shows up quite early. It would be very desirable to carry out further computations at much greater heights to see if this phenomenon persists there. This is the first study that computed large numbers of high zeros of the zeta function to substantial accuracy. The large scale efforts to verify the RH numerically were designed only to prove that the zeros under investigation lie on the critical line, and did not produce accurate values for their positions. Some very accurate values of zeros have also been computed in order to use them in disproving a variety of number theoretic conjectures, but they were limited to a small number of initial zeros (such as the roughly 100 decimal digit values for the first 2000 zeros of [60]). For a numerical investigation of the Montgomery pair correlation conjecture and the other conjectures alluded to above, it was desirable to obtain a larger sample of zeros but only to medium accuracy. Furthermore, since there are many properties of the zeta function that are true only asymptotically, and the asymptotic behavior is often approached very slowly, it seemed
7 - 6 - desirable to compute a sample of zeros very high up. The computations on which this paper are based determined the values of the first 10 5 zeros and also of zeros number n, n , each to within ± 10 8, using the Cray-1 and Cray X-MP computers, respectively. In the second case it was also verified that those zeros satisfy the RH. Zero number is at height approximately , and given the available computational resources and the method used it was about as high as one could go. (A more efficient algorithm for evaluating the zeta function was invented recently [61], but it is quite complicated and has not yet been implemented.) The computations used on the order 20 hours on the Cray X-MP, and the capabilities of this modern supercomputer were essential for the project. Several other samples of 10 5 zeros each at large heights had been computed earlier on a Cray-1, but due to a defect in the manufacturer s software, described at the end of Section 3, some of their accuracy was lost, and so they are used for only some of the statistical analyses. The comparison of the empirical data for the zeros of the zeta function to the results that are proved for the GUE is made in Section 2. Section 3 is devoted to a description of the computation of the zeros of the zeta function and the associated error analysis. Section 4 gives some statistical information about the zeros that were found, as well as of some other zeros. This information is not directly relevant for the purposes of Section 2, but are of interest because of the comparison with the earlier computations of [10, 52]. Section 5 reports on several alternative techniques for computing the zeros that might be of use for other computations. Finally, Section 6 describes the computation of the GUE predictions.
8 Zeros and eigenvalues Before presenting the results of the empirical study of the zeros, we first recall some standard notation, and then discuss some known results about the zeta function and how they might affect our interpretation of the data. As usual, we consider only zeros ρ of the zeta function with Im (ρ) > 0, and we number them ρ 1, ρ 2,... (counting each according to its multiplicity) so that 0 < Im (ρ 1 ) Im (ρ 2 ).... All the zeros ρ n that have been computed so far are simple and lie on the critical line, so we can write them as ρ n = 1/2 + i γ n, γ n ε R +. We have γ 1 = , γ 2 = , etc. We let θ(t) = arg [π it /2 Γ( 1/4 + it /2 ) ], (2.1) S(t) = π 1 arg ζ( 1/2 + it), (2.2) where in both cases the argument is defined by continuous variation of s in π s /2 Γ(s /2) or ζ(s), respectively, starting at s = 2, going up vertically to s = 2 + it, and then horizontally to s = 1/2 + it, and where we assume (in the case of S(t)) that there are no zeros ρ with Im (ρ) = t. If N(t) denotes the number of zeros ρ with 0 < Im (ρ) < t (counted according to their multiplicity), then [68; Ch. 9.3] N(t) = 1 + π 1 θ(t) + S(t). (2.3) By Stirling s formula [40] we have θ(t) = 2 1 t log (t /( 2πe) ) π/8 + O(t 1 ) as t, (2.4) so that
9 - 8 - N(t) = t log t + _ 7 _ + O(t 1 ) + S(t). (2.5) 2π 2πe 8 We will discuss the function S(t) at greater length below. For the moment, though, we will mention only that the best unconditional bound that is known for S(t) is [68; Thm. 9.4] S(t) = O( log t) as t. (2.6) In view of the bound (2.6), we see by (2.5) that N(t) is almost exactly equal to π 1 θ(t). In particular, zeros become denser and denser the higher up one goes in the critical strip, and the average vertical spacing between consecutive zeros at height t is 2π/( log (t/( 2π) )). Therefore, in order to study quantities that are largely invariant with height, we define the normalized spacing between consecutive zeros 1/2 + i γ n and 1/2 + i γ n + 1 (in cases where both zeros satisfy the RH) to be δ n = (γ n + 1 γ n ) log _ γ n 2π. (2.7) 2π It then follows from (2.5) and (2.6) that the δ n have mean value 1 in the sense that for any positive integers N and M, N + M Σ δ n = M + O( log NM). (2.8) n = N + 1 In this notation, Montgomery s pair correlation conjecture can be reformulated to say that for any fixed α, β, 0 < α < β <,
10 - 9 - N 1 (n,k) : 1 n N, k 0, δ n + δ n δ n + k ε [α,β] (2.9) α β 1 _ sin πu 2 du πu as N. If the pair correlation conjecture holds, we might even expect something stronger to hold, namely that M 1 (n,k) : N + 1 n N + M, k 0, δ n δ n + k ε [α,β] α β 1 _ sin πu 2 du (2.10) πu as N, M with M not too small compared to N, say M N η for some η > 0. The definition of the δ n makes it easy to compare distribution of spacings between consecutive zeros of the zeta function and the normalized spacings between consecutive eigenvalues in the GUE, since the latter are also normalized to have mean value 1. The GUE hypothesis will be used to refer to the conjecture that the distribution of the δ n approaches asymptotically the distribution known to hold for the GUE eigenvalues. In particular, the GUE hypothesis leads to the prediction that the δ n should have a particular distribution function; for any α,β with 0 α < β <, M 1 n: N + 1 n N + M, δ n ε [α,β] α β p( 0,u) du (2.11) as M, N with M not too small compared to N, where p( 0,u) is a certain complicated probability density function discussed in Section 6 and graphed (with the
11 solid line curve) in figures 3 and 4. Similarly, one obtains the prediction that M 1 n: N + 1 n N + M, δ n + δ n + 1 ε [α,β] α β p( 1,u) du, (2.12) where p( 1,u) is another density function. More generally, one obtains the prediction that the δ n approach asymptotically the behavior of a stationary (but non-markovian) process, so that for any k, the empirical joint distribution function of δ n, δ n + 1,..., δ n + k for N + 1 n N + M approaches that of the GUE process. Most of the studies to be reported below are based on the δ n for 1 n 10 5 and n , although a few other sets of 10 5 consecutive δ n for n < will also be referred to occasionally. The letter N will be used to designate the starting point of a data set, so that, for example, a graph or a table entry labelled with N = will refer to data for δ n or δ n + δ n + 1, n The reason for selecting sets of size 10 5 was to obtain a sample so large that random sampling errors would not be very significant. The reason for not computing zeros higher than ρ n for n = (approximately) is that on the machine that was available to the author (a single-processor Cray X-MP with 2 million words of memory) such computations would have been prohibitively slow, since the author s program, described in Section 3, achieves high speed by using several auxiliary storage arrays, so that the size of available memory was a major limit. Most of the studies of the available δ n are concerned with the pair correlation conjecture (2.10) and the conjectured distributions (2.11) and (2.12) for δ n and δ n + δ n + 1, respectively. The reason for this is less the difficulty of computing the GUE
12 predictions for other quantities than the question of significance of the comparison of the experimental data to the theoretical predictions. By (2.5) and (2.6) we see that the interesting behavior of spacings between zeros is due to S(t). The bound (2.6) is the best that is known unconditionally. Even on the assumption of the RH, it is only known that S(t) = O log t as t. (2.13) log log t The true rate of growth is likely to be much smaller, though. It has been known [68; Ch ] for a long time that on the RH, S(t) = Ω ± ( ( log t) 1/2 ε ) as t for any ε > 0, and Montgomery [58] has shown more recently that S(t) = Ω ± ( ( log t) 1/2 ( log log t) 1/2 ) as t. (2.14) It is thought likely that S(t) = O( ( log t) 1/2 + ε ) as t (2.15) for every ε > 0 (cf. [49]) and Montgomery has even conjectured [58] that (2.14) represents the right rate of growth of S(t). For statistical studies of the kind we are undertaking, though, the typical size of S(t) is more important. Selberg [66, 68] has shown that for all k ε Z +, T S(t) 2k dt ( 2k)! 0 k! ( 2π) 2k T ( log log T) k as T. (2.16) This means that a typical value of S(t) is on the order of ( log log t) 1/2, an extremely small quantity. Experimentally, it is known that S(t) < 1 for 7 < t 280, and
13 S(t) < 2 for 7 < t The largest value of S(t) that has been observed so far appears to be S(t) = for some t near γ n, n = 1, 333, 195, 692 [R]. (Large values of S(t) are associated to violations of Gram s and Rosser s rules, see [10, 23, 49, 59], which explains why some statistics about them are available.) Aside from (2.6) and Selberg s estimate (2.16), several other estimates have been proved. For example, if S 1 (t) = 0 t S(u) du, then S 1 (t) = O( log t) unconditionally, and S 1 (t) = O( ( log t) ( log log t) 2 ) on the RH [68]. The true maximal order of magnitude is probably again around ( log t) 1/2. Therefore there is a lot of cancellation in the local behavior of S(t). Many results about the distribution of values of S(t + h) S(t) are also known [25, 26, 27, 32, 33, 44, 69]. For example, Theorem 13.2 of [69] says that if 1/2 < η 1, then for T η < H T, 0 < h < 1, T T + H S(t + h) S(t) 2k dt = HA k log ( 2 + h log T) k (2.17) + O H c k k k k k + ( log ( 2 + h log T) ) k 1/2, where A k = ( 2k)! / ( 2 k π 2k k! ), and c > 0 is a constant. There are also theorems about distribution of S 1 (t + h) S 1 (t), cf. [69]. Since S(t) grows very slowly, we can expect low zeros of the zeta function to be very
14 restricted in their behavior, and their asymptotic behavior to be approached quite slowly. Pseudo-random behavior of the zeros (like that predicted by the GUE hypothesis) can only be expected over ranges of about ( log T) 1/2 at height T; i.e., over a range of about ( log n) 1/2 normalized spacings from zero number n. Furthermore, we can expect gaps between next-to-nearest neighbors (i.e., δ n + δ n + 1 ) to be much more constrained than those between nearest neighbors (i.e., δ n ), and to approach their asymptotic behavior even more slowly. This is indeed what is observed, and it s the main reason for restricting most of the present investigation to nearest or next-to-nearest spacings. We will see, in fact, that when we consider very distant spacings, a totally different phenomenon occurs, whose explanation lies not in the GUE, but rather in prime numbers. We now proceed to a discussion of the evidence. Figures 1 and 2 show the data for the pair correlation conjecture for the two sets of zeros 1/2 + i γ n, N + 1 n N , and for N = 0 and N = 10 12, respectively. For each interval I = [α,β), with [α,β) = [ 0, ), [ 0. 05, 0. 1 ),..., [ 2. 95, 3 ), a star is plotted at the point x = (α + β)/2, y = a α,β, where 20 a α,β = 10 5 (n,k) : N + 1 n N , k 0, δ n δ n + k ε [α,β). The solid lines in both figures are the GUE prediction y = 1 (( sin π x)/(πx) ) 2. Note that for a typical value of α 1, a α, β is derived from about 5000 points δ n δ n + k so the random sampling error is expected to be on the order of All of the plots in this paper, as well as most of the statistical computations, were done using the S system [1].
15 Figures 3 and 4 show the data on the distribution of the normalized spacings δ n. The stars again correspond to a histogram of the δ n ; for each interval [α,β), α = k /20, β = α + 1/20, a star is plotted at x = (α + β)/2, y = b α,β, where 20 b α,β = 10 5 n: N + 1 n N , δ n ε [α,β). The solid lines are the GUE predictions, y = p( 0,x). (As in explained in Section 6, no rigorous error analyses is available for the computed values of p( 0, x), but they are thought to be quite accurate.) Similarly, figures 5 and 6 show the data on the distribution of δ n + δ n + 1. The graphs at first glance show a satisfying amount of agreement between the experimental data and the GUE predictions. Graphs have roughly the same shape, and agreement between data and prediction improves dramatically as one goes from the first 10 5 zeros to the 10 5 zeros with n Moreover, the disagreement between data and prediction is concentrated precisely where our discussion of the effect of the small size of S(t) would lead one to expect it; there are fewer very large or very small values of δ n and δ n + δ n + 1 in the data than predicted, and more values near the mean. Further comparison of the distribution of zeros to the GUE prediction is provided by Table 1, which shows values of 10 5 N Σ (δ n 1 ) k (2.18) n = N + 1 and
16 N Σ (δ n + δ n ) k (2.19) n = N + 1 for the two sets N = 0 and N = 10 12, as well as the GUE values. (Numbers in Table 1, as well as others in this paper that don t have "..." at the end are usually rounded to the number of digits shown, whereas those with "..." are truncated.) Table 2 shows moments of log δ n, δ 1 n, and δ 2 n. The histograms of figures 1-6 and the moments of tables 1-2 provide a very rough measure of the degree to which zeros of the zeta function match the GUE predictions. One can obtain a better quantitative measure of the degree of fit between the observed and the predicted distributions using the histogram data (e.g., one can use the χ 2 -test [47] or the asymptotic results of [24]). In our case, since we know the predicted distribution quite well (at least numerically), we will use the quantile-quantile (q q) plot to detect deviations between the empirical and theoretical distributions. Given a sample x 1,..., x n, and a continuous cumulative distribution function F(z) for some distribution, the theoretical q q plot is obtained by plotting x ( j) against q j, where x ( 1 ) x ( 2 )... x (n) are the x i sorted in ascending order, and the q j are the theoretical quantiles defined by F(q j ) = ( j 1/2 )/ n [13]. The q q plot is a very sensitive tool for detecting differences among distributions, especially near the tails. If the x i are drawn from a distribution corresponding to F(z), and the sample size n is large, the q q plot will be very close to the straight line y = x. Figures 7-10 show segments of the q q plots of the δ n and δ n + δ n + 1 for N = 0 and N = To obtain a quantitative measure of the agreement between the observed distributions of the δ n and the GUE predictions, we use the Kolmogorov test [47; Section 30.49].
17 Given a sample x 1,..., x n drawn from a distribution with the continuous cumulative distribution function F(z), we let F e (z) be the sample distribution function: F e (z) = n 1 k: 1 k n, x k z. The Kolmogorov statistic is then D = sup F e (z) F(z). (2.20) z The asymptotic distribution of D is known [47; Eq ]; if the x i are drawn from the distribution defined by F(z), then lim Prob (D > un 1/2 ) = g(u), (2.21) n where g(u) = 2 Σ ( 1 ) r 1 exp ( 2 r 2 u 2 ). (2.22) r = 1 Table 3 gives the Kolmogorov statistic D for δ n and δ n + δ n + 1 for several blocks of 10 5 consecutive zeros. Also given is an estimate of the probability that this statistic would arise if the δ n (δ n + δ n + 1, resp.) were drawn from the GUE distribution. This estimate is obtained by evaluating g(d 10 5/2 ). The data presented so far are fairly consistent with the GUE predictions. The differences between computed values and the predicted ones diminish as one studies higher zeros, and they are more pronounced for δ n + δ n + 1 than for δ n. Moreover, the computed spacings δ n and δ n + δ n + 1 are more concentrated near their means than the GUE predictions, which is again to be expected on the basis of the small size of S(t).
18 However, there are some indications, based on the tails of the distribution of the δ n and δ n + δ n + 1, that asymptotically the zeros of the zeta function might not obey the GUE predictions. Already in Table 2 we see that the mean value of δ 2 n for N = exceeds the GUE prediction, which indicates an excess of small δ n. The q q plot of Fig. 8 (for N = ) is much closer to a straight line than that of Fig. 7 (for N = 0), just as that of Fig. 10 is closer to a straight line than that of Fig. 9. However, the graphs in figures 9 and 10 are both below the straight line y = x (which corresponds to perfect agreement between theory and prediction), which indicates that in both cases there are fewer large spacings than the GUE hypothesis predicts, which was to be expected. The graph of Fig. 7 is above the straight line y = x, which indicates fewer small spacings than predicted, which again was to be expected, while on the other hand the graph of Fig. 8 is below the line y = x, which indicates an excess of small spacings, which is quite unexpected. The small size of S(t) leads one to expect a substantial deficiency in the number of small δ n and δ n + δ n + 1, since the graph of S(t) is an uneven sawtooth curve, with S(t) jumping up by 1 at t = γ n, then decreasing essentially linearly up to t = γ n + 1, jumping up again by 1 at t = γ n + 1, and so on, so that small gaps between zeros also correspond to large values of S(t). Fig. 11 shows an initial part of the q q plot for the δ n for N = which indicates behavior very similar to that of Fig. 8. The q q plot for N = is very similar to those for N = and N = 10 12, while that of N = 10 9 lies on the other side of the y = x line. (We should note again that the data for N = 10 9, 10 11, and are not very accurate, and so have to be treated with caution. In fact, Fig. 11 provides a way to gauge the inaccuracy introduced into those data sets by the bug described in Section 3. The q q plot of Fig. 11 has in places the appearance of a
19 staircase, with flat horizontal segments. These flat segments arise from identical computed values of the δ n. The correct values of the δ n are expected to be all distinct and much more evenly distributed, as is the case for N = 0 and N = ) Table 4 shows the number of δ n and δ n + δ n + 1 that are very small or very large, as well as the number that would be expected under the GUE for a sample of size Large δ n and δ n + δ n + 1 are generally less numerous than the GUE predicts, which is to be expected. The data for the high blocks of zeros show that the number of δ n and of δ n 0. 1 is in excess of that predicted. (We would have obtained the same results if we considered the pair correlation conjecture (2.10) with α = 0. 01, β = 0. 05, say.) The excess is not very large, given that the expected number is fairly small, but it may be significant that while the first few sets (for N = 0, , and 10 9 ) all show deficiencies, all the high sets of zeros (N = 10 11, , and ) show an excess of δ n and of δ n Another way to investigate this question is to look at the minimal values of δ n, shown in Table 5. These values include some very small δ n that were found by Brent [10] and van de Lune et al. [52]. These investigators were not computing the γ n, but occasionally, when they had difficulty separating a close pair of zeros, they noted the approximate locations of them. In particular, Brent [10] found that δ n = for n = 41, 820, 581, and van de Lune et al. [52] found that δ n = for n = 1, 048, 449, 114. Thus these values provide upper bounds for the minimal δ n in the intervals investigated by those authors, and it is likely that they are the minimal values.
20 Since it is known [54, 55] that p( 0,t) has the Taylor series expansion near t = 0 given by p( 0,t) = _ π 2 t 2 _ 2π 4 t 4 + _ π 6 t , (2.23) (for comparison, we note that p( 1, t) = π 6 t 7 / [ 55 ] ), the GUE prediction is that among M consecutive δ n s, the probability that the minimal δ n is α M 1/3 is approximately 1 1 _ π 2 _ α 3 M 9 M 1 exp π2 α 3 /9. (2.24) The function on the right side of (2.24) was used to compute the last column in Table 5. The probabilities of obtaining the minimal δ n that are observed from the GUE distribution are rather low, in some cases very low. (We would have obtained comparable inconsistencies between data and predictions had we worked with the pair correlation function instead of the distribution of the δ n.) It is hard to draw any firm conclusions from the data that is available. It would be very desirable to obtain data from much higher sets of zeros to see whether they also have an excess of smaller spacings compared to the GUE hypothesis. If they do, this would raise doubt not only about the GUE hypothesis but even about the Montgomery pair correlation conjecture. This is an important possibility because the GUE hypothesis is very speculative, as it relies on the hope that the eigenvalues of the hypothetical operator associated to the zeta function would behave like those of a random operator. Thus it is easy to conceive of the GUE hypothesis being false even when there is an appropriate operator. On the other hand, the pair correlation conjecture depends only on
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text:
Definition Edit
Proactive network defense
|consists of deploying protective and detective measures that disrupt an adversary before an exploit has been successfully executed. By recognizing and engaging the adversary during the reconnaissance, weaponization, and delivery phases of the cyber attack life cycle, network defenders are able to deploy mitigations or take some other course of action to ensure that mission-critical assets are protected prior to an adversary successfully executing an exploit.[1]
References Edit
- ↑ NIST Special Publication 800-150, at 10.

subset: commoncrawl


text:
Mike Rivers is no stranger to Naked Kombat, he's wrestled three previous times on the gay wrestling site. He's lost two matches and won one, so he's gotten his ass fucked on the mats twice.
But this wrestling match is a little different because this time Mike Rivers is wrestling with his real-life partner, Alexander. Standing 5' 11" and weighing 170 pounds, Alexander is a little taller and heavier than Mike who only stands 5' 8", and Alexander's got 10 pounds on his lover.
Mike does well against his partner and he actually wins the match, which means that he's going to get to fuck his lover on the mats. But it doesn't always work out that way at Naked Kombat. The winner doesn't always have to fuck the loser's ass (I didn't know that until now) - the winner has a choice. And Mike Rivers chooses to get down on all fours and let his lover fuck his ass. And Alexander thanks him by giving him one, hard, aggressive butt fucking. Almost as if to teach him a lesson for humiliating him on the mats.


text:
Retail landlords and tenants may have battled through the warm start to winter, which saw the traditional mid-year sales start earlier, but the move into spring will bring some relief, according to the latest sales data.
Sentiment is starting to improve, albeit slowly, but not for department chains.However, with the spring racing carnivals not far away, shoppers are already picking out outfits and accessories, which is welcome news for retailers.
In the latest figures, Australian retail sales grew 4.9 per cent in June, compared with the same period a year ago. Sales trends have stabilised following the budget-induced drop in sentiment.
Citi retail analyst Craig Woolford said sales growth remained widely divergent between categories, with department stores, discount department stores and electronics retailers losing their share of consumers' wallets.
''For the 12 months to June 2014, retail spending rose 4.6 per cent. This is an improvement on the 2013 financial year result of 2.7 per cent growth,'' Mr Woolford said.
''Retail sales conditions remain below trend and our view is that sales will become tougher later in calendar 2014. The most recent upswing in retail was closely tied to house price growth.
''As we move through October 2014 onwards, we are lapping 12 months of solid house price growth. We expect retail spending to remain below 5 per cent in 2015 financial year estimates and prefer retailers with store rollout or margin expansion opportunities.''
Retailers will soon start putting spring and summer fashions into the shops and put the older winter clothes into the discount bin.
In the ABS data the growth came from hardware stores, as people renovate the house to sell or work on the new home, and cafes, which is putting continued pressure on apparel retailers.
Recent data from Jones Lang LaSalle for the June quarter showed retail vacancy rates decreasing in the first six months of the year, as retail market conditions begin to improve.
Jones Lang LaSalle's Australian head of retail, property & asset management, Tony Doherty, said the average retail vacancy rate for Australia fell to 3.6 per cent in June from 3.8 per cent in December 2013 on average across all retail formats.
''The city and sub-regional malls (over 20,000 square metres in size) were the main drivers, with the sub-regional vacancy rate falling from 3 per cent to 2.6 per cent and CBD vacancies falling from 6.6 per cent to 6.2 per cent,'' Mr Doherty said.
''Vacancies in regional shopping centres fell marginally (down 0.1 per cent to 1.4 per cent) and neighbourhood centres increased marginally (up 0.1 per cent to 4.2 per cent).''
Mr Doherty said the figures were encouraging, particularly for Brisbane and Perth. As of December last year, both markets had the highest rate of retail vacancy in the country, but conditions are improving, predominantly as high CBD vacancy rates begin to ease. This modest improvement in the first half of the year now puts them ahead of Adelaide.


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text: Still bemoaning the fact that XF 1.3 will not get a navigation manager... So heres my question. How do i add, remove menu items from addons that appear on my navigation? Is there an easy way to change the order? Can someone please provide some guidance. For example, on my site : http://www.nzwarriors.com/ i want to remove the following bolded sub items : Player Profile -> Most Active Members Do i need to contact the addon developer in each instance?


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This city hotel was opened in 2003. Behind a restored 18th century fa�ade, the hotel comprises 22 rooms spread over 6 floors of which 19 are doubles, 2 are singles and 1 is a suite. The main building offers a lobby with 24-hour reception, lift, a conference and lounge room, a currency exchange desk, a bar and an air-conditioned restaurant. There are also a laundry service and medical assistance on offer as well as a public Internet terminal. The hotel also offers car parking and garage facilities to its guests.
This hotel is located at the centre of Toledo. The immediate vicinity offers a wide range of shopping and entertainment venues and many of the sights of the area, for example, the Puerta del Sol that is only a few minutes walk/ 500m from the hotel. There is a stop for the public transport to be found in the vicinity lying approximately 100 m from the hotel.
The light and welcoming rooms come with a private bathroom, a hairdryer and a king-sized bed. They also include a direct dial telephone, satellite TV, a radio, an Internet connection, a hire safe and individually controllable air conditioning/ central heating.
The nearest golf course is 20 km away.
Guests may select their breakfast from a buffet.
American Express, Diners Club, MasterCard and VISA are accepted as methods of payment.
Room Facilities:
Direct dial telephone
Satellite cable TV
HiFi stereo system
Internet access
Central airconditioned
Central heating
Individual heating


text:
Aging would not be so bad if it did not explain to on us at both fight. As we hug one another, we written account the respect handles on the other's waist, the fat on their munition and the sag on their cheeks. We gaze into respectively other's faces. After all, sole saints discovery no satisfaction in the wrinkles of their peers. We are not inexcusable sinners; we are checking to see how cured we are retaining up. Estimating someone's age and determinative how healthy that person has old tells us a lot. Looking and emotion little than written account age has increasingly becomes a criterion of glory. Preoccupation near age is a trueness for our apace senescent society. It is sector of our survey and both bit as far-reaching as our state collection.
We may be sound in our revulsion at aging, even scientists are puzzled by the purposelessness of the act. Every biologic case in the quality body, from construct to time of life to maturity, has a end bar ageing. Aging makes no import. Scientists are mazed as to why mammals have more than shorter go spans than more rude species. We don't in performance near as long-dated as the Galapagos chelonian reptile but we in the main crisscross up superficial similar one.
Nowadays, researchers pre-raphaelite for the gene that causes ageing. If they breakthrough it, they will facial expression for a thing to change it, as but it were a infectious agent. They human face an horrible project. A cell, the simplest gel of life, is more mazy than Mexico City, the greatest built-up in the planetary. No one sequence or any one internal secretion is amenable for eudaimonia and juvenile person. It is the just what the doctor ordered labour of all the cells and variety meat effective equally in sheer chord to rule out the disruption of the thing. What makes us deduce we can contemplate restrict aging only just by turn a electric switch in the body?
The signs of disintegration that we ponder the signs of senescent are the proceed of surplus impose on the cellular rank. When our engagements are in doubt to nature, the results are divers types of organic structure ills, deformities, and repulsiveness. Eating denaturized and devitalized nutrient leaves residues the natural object cannot utilize, and they are deposited in places our Creator never witting. Whatever the article cannot use, becomes harmful accumulations that shoplift our wellbeing and our juvenile.
Body acids get into in chemic reactions beside discarded products causing chemical change of the brushed body part. Now tissues of essential variety meat and glands go burdened with granite salts and crystallized deposits. Overloaded with waste, they not merely can no long activate typically but also become more phrase to gravity's lug. As all body part elongates and sags, flagging jowls and flaccid cheeks instigate to furnishings on both sides of the mouth, and the eyebrows start off to endowment ended the opinion.
External deformities are funnel manifestations of middle pathologies. Ugly chains of unhealthy veins, bouffant faces, and fat are telling tales around your on the inside clause. Every pimple, psoriasis, or colour metamorphosis on your buckskin is in certainty a rumination of whatever body part struggling to do its job. Every bulge, boil, or injured is a mark that the body is pushing out a number of toxins in its application to guard itself.
Most of us quality our excuse lacks thing. In reality, appearance is much active overload. Toxic accumulations in our bodies are answerable for stealing our condition and attractiveness. Beauty lies potential low cushions of retained fluids, deposits of fat, and sick-abed tissues. Your good looks is hidden alive, but in record cases it can be revived in a altered copy that will be copacetic to you. You must hold contiguous act to meliorate it. When you do, your bare good looks will amazement and hearten you
Some environmental characteristics of our frontage and organic structure we cannot change-they were resolute prior to our get-go. But the intake of the raw plant fare as an big will clear a disparity in the tactile property of shell and hair, the wellbeing of nails, weight and complexion. All of these traits and more are strong-willed by day by day choices, next to supplies someone one of the furthermost of value and, luckily, the one we can full power.
Since there can be no inbred condition minus consumption 100 percent automatic food, peak of citizens have ne'er explored best eudaimonia. In my narrative Your Right to Be Beautiful, I proponent a fare ready-made up entirely of raw fruits, vegetables, loony and seeds-Rawsome Diet. No, you will not be asked to scratch the garden, dig up the squirrel's nuts or eat out on the birdfeeder, but you will be introduced to an undiscovered global of preparation techniques that permit you to craft not sole the maximum alimentary meals, but also dishes that are delicious, filling and satisfying. Some raw supplies recipes even ape usual cuisine and create the change of state to the Rawsome fare easier. There are 100s of opposite kinds of fruits and veggies, kooky and seeds near a mixed bag of tastes and flavors, which awakened nibble buds will statesman to recognize. If you unite solely two or cardinal ingredients at all meal, next the figure of plate combinations is continuous.
The flair to regenerate cells is improved into your body, and the body itself is dexterous of devising you outward show more tempting. After all, we were designed to biological process unceasingly as old cells get replaced by new ones. The Rawsome diet makes the most of this rejuvenating flair of our bodies. Rawsome good looks is not factory-made from the out-of-doors but cultivated from the enclosed by nutritionally negatively charged stores. Whole raw substance restores the state of both cell and facilitates the optimal operation of every body part during and out. The body, having been aroused by the improved diet, in its wisdom, will reach out to the places where on earth welfare was lacking, and you will unhurriedly see your good looks egression. The Rawsome diet will do away with all the toxins on a deeper plane and bring on astonishing changes to your visual aspect.
Only the unit unrelenting on raw supplies will adult earthy beauty, or should we say Rawsome charm. The body becomes transcended and will blossom from the filling out. While your non-raw consumption peers unearth new blemishes, blotches, and moles on a just about day after day basis, you will see your own tegument irregularities unhurriedly fade or evaporate. Feeding your unit raw food will receive eyes, onetime hollow in bulblike flesh, look large and much swollen by eliminating the about symptom and by firming the eyelids. Eyebrows that were germ to silhouette an shelter ended the view will get their juvenile cloisters. As intuitive albuminoid productivity improves, it will cram in the places wherever it is requisite as in hollow cheeks. It will not lately dapple your face, but all one of your 3,000 court inches of bark will raise.
The outlook of the unit will translation. The plane of the skin texture will become yielding and sleek but still set and lithe. Visible pores will bring down. A sickly skin beside a pallid wanness will twist into a porcelain-like color. The whites of eyes, past red, will become dramatic beside a blueish jot. Raw feed consumption will clarify and better your features and send fineness to your frontage.
People who have been on the raw feed fashion for respective geezerhood instigate to have a glow, the good not habitually seen in middle-aged grouping. Optimal robustness is celebrated by an emerging ruddiness. Glow is knotty to misrepresent because it is internecine. It comes from an bounty of clear, pink, virtually semi-transparent cells that reading light up the facade. Only quality blood circulation can bring up this transfiguring shine. Several geezerhood on the raw matter diet will engineer you outward show as still you only stepped out of a graphic art by Renoir-the art movement select few particular for his preoccupation near restrained.
Colette, France's maximum adult female writer, quondam wrote: "The unease of aging, a ordinary neurosis, does not normally loaf for age and spares neither sex." But senescent is brimful of compassion towards general public on the raw silage fare. This fare will allow you to ripen, not to deterioration. Beauty does not have to crumble or disappear beside age. You will be flexible and flexible and bread and butter your prettiness for a lengthy case. It is like triumphant the Jack Pot in the Age Game. You go age-proof. When you discover The Most Beautiful You, old age will misplace its sting, and you will have achievement over and done with the ravages of event.
Even on the 100 proportionality raw sustenance diet, no one lives for ever and a day. For individuals who shadow this diet for oodles years, senescent comes a period or two back decease. Deterioration takes pop apace. Perennial young person is a perpetual mental imagery. The raw food lifestyle is the nearest somebody can travel to motility the maximum existence span near the prize of energy entire. The fare allows the article to mellowed gracefully. It produces an fusion of guise and vigour. The Rawsome Diet allows every person to have your home as daylong as God well-meaning and stare as God envisioned.
Excerpted from the narrative "Your Right to Be Beautiful: How to Halt the Train of Aging and Meet the Most Beautiful You" by Tonya Zavasta. The transcript is free at:
© 2004 Tonya Zavasta
This nonfictional prose may be of your own accord reprinted in its totality as long-lived as the full article, byline and URL are built-in.


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Apocalypse Knights - Endless Fighting with Blessed Weapons and Sacred Steeds
By InterServ International Inc.
Open iTunes to buy and download apps.
※※※※※ Congratulations! Over three million downloads worldwide! ※※※※※
※※※※※ Recommended most by international game sites! ※※※※※
※※※※※ Pros and gamers rave about its art and diverse game play! ※※※※※
"Apocalypse Knights" is a free Action Game. It's up to you, the heroic knight, to take back lands tainted with darkness. You shall save the world with blessed weapons, armors, ultimate skills, and holy horses. Come, your destiny awaits!
● Background
In the coming future, you are the knight faced with demons released from the melted Arctic / Antarctic zones. With refined weapons, biochemical mounts, and skills, you can conquer the demons and devils in all waves.
‧You can get more attacks with progression.
‧Unleash the ultimate skill to kill demons.
‧Each weapon has a unique ultimate skill. Collect as many weapons as possible to see all the cool moves and defeat bosses.
‧Your mount is your partner. You can active mount mode to enhance attributes.
‧Fight with mounts and enjoy the blazing speed while you smash barriers.
‧Collect all kinds of mounts and impress your friends.
‧The end of the world in "Apocalypse Knights" is coming and you from the end of world.
‧You can experience 10 levels of difficulty, 6 different scenes, and many kinds of monsters.
‧Knockout endless waves of monsters and earn special rewards in the ultimate instance mode.
‧ You can find ultimate weapons and armors from the shop.
‧Look closely to find legendary weapons and armors.
‧Take the mystic potions and improve your character ability.
‧Combing a modern, medieval influenced visual style. You're the ultimate knight with the mission to fight the demons.
‧Enjoy the astounding audio experience of your spear piercing through a zombie's gut or your broadsword slashing through witches.
‧We utilize the best performing engine to power up visual, resolution, sound and game controls.
※※※※ Strongly recommend to enjoy the ultimate gaming experience on iPad 2 or above.
Compatible with iPhone 4, iPhone 4S, iPod touch (4th generation) and iPad. Requires iOS 4.2 or later.
What's New in Version 1.0.13
New Content: Chinese God of War
1. Add New Equipments : Suits of Holy General Guan / Chinese God of War
2. Add New Weapons : Golden Crescent Blade / Green Dragon Crescent Blade
3. Add New Mounts : Red Hare
Customer Reviews
Great game, needs some tweaking
I never review games so read this! In the game, there's a quest that gives you a gem if you give it a five-star rating. I think it deserves all five of those stars because many aspects of the game are good, such as the art, the armor choices, and the combat styles. This game could use some improvements mainly because it might get boring killing people, but why else would this game be called "endless hack and slash"? Also the lag times should be deceased when hitting many mobs simultaneously and for some reason it won't connect to the Game Center sometimes when I load the game. Make these improvements and this game will be legend or editors choice or something.
A little something to be desired
At first play, this game seemed interesting because of the story and fairly decent graphics. However, when actually playing the game, I found the character response times to control input were markedly slow. Then came the moment when enemies became a little stronger, and standard potions became less able to replenish health health: as with many freemium games, I found (much to my disappointment) the purchase of better potions was limited to diamond use; of which can be purchased, or you can slowly acquire through game play.
Overall, the game, its interactive capabilities, and coin / jewel rewards fall short of a half-decent rating. Keep up the work guys, and don't be so stingy with the rewards. You will still make your money without having to milk a player of his/her hard earned pay.
Amazing game, a few kinks but only the normal
This game is quite addicting and fun. It's control setup is similar to that of Glu's Eternity Warriors. The combat flows nicely and if you get the hang of it, you rarely need to use health potions and can save up for armor or weapons. My only issue is that like most games of this setup the money is difficult to build up. I've only played a little bit but I look forward to every minute of play and it feels like many games that I grew up playing. Overall, if you like the wave based, hack and slash, constant survival of stuff like horde modes, then this is the perfect mobile game for you. :D
- Free
- Category: Games
- Updated: Jan 22, 2014
- Version: 1.0.13
- Size: 406 MB
- Languages: English, Simplified Chinese, Thai, Traditional Chinese
- Seller: Interserv International Inc.
- © 2014 InterServ International Inc.
Compatibility: Requires iOS 4.2 or later. Compatible with iPhone 3GS, iPhone 4, iPhone 4s, iPhone 5, iPhone 5c, iPhone 5s, iPhone 6, iPhone 6 Plus, iPhone 6s, iPhone 6s Plus, iPhone SE, iPhone 7, iPhone 7 Plus, iPad, iPod touch (3rd generation), iPod touch (4th generation), iPod touch (5th generation), and iPod touch (6th generation).
Top In-App Purchases
- 140 Gems Pack$4.99
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- 10,000 Coins Pack$1.99
- 5,600 Gems Pack$99.99
- 28,000 Coins Pack$4.99
- 2700 Gems Pack$59.99


text:
Sally's chairman and CEO, John Wood, has been inducted into the International Association of Amusement Parks and Attractions Hall of Fame. This is a tremendous honor for the man who has dedicated thirty-six years of his life to the company and the development of animatronics, dark rides and interactivity. The IAAPA Hall of Fame honors pioneers for their significant and lasting contributions to the worldwide attractions industry. Among the legendary honorees are Walt Disney (Walt Disney Company); George Ferris (inventor of the Ferris Wheel) and Milton Hershey (Hersheypark).
Wood, who was honored during the kickoff event at the IAAPA Attractions Expo 2013 on November 19 in Orlando, was cited for his ideas and leadership in reviving family dark ride attractions around the world. It was noted that his creativity has encompassed adding extra elements of fun and competition to the rides with strong story lines, interactivity and modern technology, music, sound effects, and spectacular special effects.
Animatronic characters, the first creations of Sally Corporation, have been incorporated into the company's dark rides from the beginning, and the company mastered the art of black light painting–used in certain rides to create an additional exciting dimension–along with interactive laser targeting and ride car scoring consoles.
The company's most recent designs, including the world's first Justice League dark ride, Justice League: Alien Invasion 3D, at Warner Bros. Movie World in Australia, feature interactive 3D video technology blended seamlessly with 3D sets, scenery and animatronics.
Founded in Jacksonville in 1977 to design and build animatronic characters for marketing and entertainment, Sally Corporation's first foray into the world of dark rides was the renovation of an existing ride at Alton Towers, a famous amusement park in England, in 1986. Several more ride renovations followed, and Wood recognized that there was a huge gap between the high-quality, story-based rides at the Disney parks and most of the dark rides in the smaller markets. He was determined to offer better options: "We wanted to create something new," Wood says, "a highly-themed, affordable, interactive dark ride with a strong story line that would foster repeat business."
He broadened the company's in-house talent pool and began offering full design/build services. Sally's first full drawing board-to-installation project, Zombie Paradise, a classic haunted castle type ride, opened in Tokyo,Japan, in 1992. This successful ride was quickly followed by many rides, both large and small, featuring new interactive technology. These became big draws in locations from Myrtle Beach,SC, to Seoul,Korea. Today, a wide variety of Sally Corporation's dark rides can be found in theme and amusement parks from Sweden to Shanghai; Malaysia to Minnesota; Spain to Santa Cruz, CA; Windsor (U.K.) to Winter Haven, Florida.
Wood was joined by family entertainment center pioneers Jim and John Huish as the newest members of the IAAPA Hall of Fame. They were inducted by Ron Gustafson of Quassy Amusement Park, 2013 chairman of the IAAPA Hall of Fame and Archives Committee.
The annual IAAPA Attractions Expo is the largest conference and trade show in the world for the attractions industry. This year, over 29,000 industry professionals from more than one hundred countries attended.


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Fear for their own safety forced many residents in Northfield to break the rules and cut down potentially dangerous damaged trees after the June storms.
An ordinance that was introduced at the City Council meeting last Tuesday addresses the city's restrictions on tree-cutting on single family home lots.
In a situation like those storms, it was nearly impossible to enforce the ordinance that existed to limit tree-cutting in the city, so Mayor Vince Mazzeo introduced the ordinance, saying it was unfair to residents to have the rule and not enforce it.
"Some residents were scared of some of the damaged trees. If we get a good wind, their branches could come down and damage their properties," he said in a phone interview this week.
The proposed ordinance would do away with rules that say only three trees are allowed to be removed within a span of two years, and that the zoning department must be alerted each time, he said.
The new ordinance would reverse a 2008 amendment to the 1986 Land Use and Development Ordinance. The 2008 ordinance had been created after incidents of clear-cutting lots at that time, Mazzeo explained.
Councilman Tom Polistina said at the meeting the current proposal was a way to resolve residents being afraid to cut down trees on their own property, and that government should not be telling the residents what they can or cannot do on their own property.
"Residents will not have to worry about getting a permit to cut down their own tree," Mazzeo said.
Other council members raised concern over potential abuse of the law.
"I hope there is common sense when we are talking about trees, because it is a part of our community," Mazzeo said. "The derecho was a storm that damaged the community, and some neighborhoods looked like a war zone."
The ordinance will need to be reviewed by the Planning Board to determine how to enforce it while simultaneously discouraging abuse. It will be considered again by City Council at a meeting in November.
Contact Anjalee Khemlani:
Follow Anjalee Khemlani on Twitter @AnjKhem