Prime zeta function statistics and Riemann zero-difference repulsion
PPrime zeta function statistics andRiemann zero-difference repulsion
Gordon Chavez ∗ and Altan Allawala Abstract
We present a derivation of the numerical phenomenon that differences between the Riemannzeta function’s nontrivial zeros tend to avoid being equal to the imaginary parts of the zerosthemselves, a property called statistical “repulsion” between the zeros and their differences.Our derivation relies on the statistical properties of the prime zeta function, whose singularitystructure specifies the positions of the Riemann zeros. We show that the prime zeta function onthe critical line is asymptotically normally distributed with a covariance function that is closelyapproximated by the logarithm of the Riemann zeta function’s magnitude on the 1-line. Thiscreates notable negative covariance at separations approximately equal to the imaginary partsof the Riemann zeros. This covariance function and the singularity structure of the prime zetafunction combine to create a conditional statistical bias at the locations of the Riemann zerosthat predicts the zero-difference repulsion effect. Our method readily generalizes to describesimilar effects in the zeros of related Dirichlet L-functions.
AMS 2010 Subject Classification: 11M06, 11M26
Contents
List of Theorems ∗ [email protected] a r X i v : . [ m a t h . N T ] F e b . Chavez and A. Allawala Prime zeta function statistics and Riemann zero-difference repulsionFigure 1: A histogram of the differences for the first 100,000 Riemann zeros. Vertical lines arepositioned at values of ∆ such that ζ (1 / i ∆) = 0. Zeros were computed by Odlyzko (see e.g.1988). Numerical evidence from multiple sources has shown a notable phenomenon in the famous nontrivialzeros of the Riemann zeta function. It appears that differences between the Riemann zeros tend toavoid being equal to the imaginary parts of the Riemann zeros themselves. This “repulsion effect”has been presented in Snaith’s (2010) review and it was recently discovered in histograms of Riemannzero differences by Perez Marco (2011). Clear numerical evidence for this repulsion effect is visible inFigure 1, where a histogram of differences for the first 100,000 Riemann zeros clearly shows troughsaround the imaginary parts of the Riemann zeros themselves.The first predictions of this effect came from research on the 2-point correlation function of theRiemann zeros, following the influential tradition initiated by Montgomery (1973). The effect wasoriginally predicted by Bogolmny & Keating (1996), who gave a heuristic argument based on ideasfrom quantum chaos. Conrey & Snaith (2008) then provided support for this argument by showingits derivation from a conjecture on the ratios of L-functions given by Conrey, Farmer, & Zirnbauer(2008). Rodgers (2013) gave further support for Bogolmny & Keating’s result under the RiemannHypothesis, and Ford and Zaharescu (2015) provided unconditional proof of the effect.In this paper we take a different approach and present an alternative, probabilistic derivation ofthe repulsion effect, using the statistical properties of prime Dirichlet series. Our argument reaches arelated conclusion to previous research, showing that the repulsion effect originates from the influence2. Chavez and A. Allawala Prime zeta function statistics and Riemann zero-difference repulsionof ζ (1 + i ∆). We show that the statistical properties of the prime zeta function on the critical line, P (1 / iτ ), are described by a covariance function closely approximated by log | ζ (1 + i ∆) | . Thisproduces negative covariance at separations ∆ approximately equal to the imaginary parts of theRiemann zeros. By an important relation between P ( s ) and log ζ ( s ), the singularity structure of P (1 / iτ ) determines the location of Riemann zeros on the critical line. We then show that thecombination of P (1 / iτ )’s singularity structure and covariance structure creates a conditional biasat the Riemann zeros, which predicts the zero-difference repulsion effect. We close by briefly notinghow our methodology may be readily generalized to predict the same effect in the complex zeros ofrelated Dirichlet L-functions. We first study the statistical properties of prime Dirichlet series with the form X t ( τ ) = (cid:88) p ≤ t a p e i ( τ log p + θ p ) (2.1)where the a p > p , τ ∈ R , and θ p ∈ [0 , π ). It isclear that EX t ( τ ) = 0 for τ uniformly distributed in R . We will focus on (2.1)’s covariance function,which is given by another prime Dirichlet series. Theorem 2.1.
Suppose τ is uniformly distributed in R . Then (2.1)’s summands are independentand for all ∆ ∈ R R t (∆) = (cid:88) p ≤ t a p cos (∆ log p ) (2.2) where R t (∆) = E { Re X t ( τ + ∆)Re X t ( τ ) } = E { Im X t ( τ + ∆)Im X t ( τ ) } . (2.3) Proof.
We first derive the characteristic function of (2.1)’s imaginary part. Recall that a randomvariable x ’s characteristic function is defined ϕ x ( λ ) = E (cid:8) e iλx (cid:9) . (2.4)We substitute (2.1)’s imaginary part into (2.4) to write ϕ X t ( λ ) = E (cid:40)(cid:89) p ≤ t exp ( ia p λ sin ( τ log p + θ p )) (cid:41) . (2.5)We expand (2.5) using the Bessel function identity e ix sin φ = ∞ (cid:88) n = −∞ J n ( x ) e inφ (2.6)where J n ( . ) is the n th-order Bessel function of the first kind (Laurinˇcikas 1996). This gives ϕ X t ( λ ) = E (cid:40) (cid:88) n ,...,n N J n ( a p λ ) ...J n N ( a p N λ ) e i ( n θ p + ... + n N θ pN ) e iτ ( n log p + ... + n N log p N ) (cid:41) . (2.7)3. Chavez and A. Allawala Prime zeta function statistics and Riemann zero-difference repulsionThe exponential terms on (2.7)’s far right-hand side are unit circle rotations with τ . Therefore takingthe expected value will cause all terms to vanish except those for which n log p + ... + n N log p N = 0 . (2.8)However, by unique-prime-factorization, the log p ’s are linearly independent over the rational num-bers. Therefore the only solution to (2.8) is given by n = n = ... = n N = 0 . (2.9)This simplifies (2.7) to give ϕ Im X t ( λ ) = (cid:89) p ≤ t J ( a p λ ) . (2.10)We next note from (2.4) and (2.6) that the characteristic function for a single summand in (2.1)’simaginary part, a p sin( τ log p + θ p ), is given by ϕ p ( λ ) = J ( a p λ ) . (2.11)Therefore, by (2.10) and (2.11), ϕ Im X t ( λ ) = (cid:89) p ≤ t ϕ p ( λ ) . (2.12)This proves that the summands of (2.1)’s imaginary part are statistically independent. Essentiallyequivalent reasoning using the identity e ix cos φ = ∞ (cid:88) n = −∞ i n J n ( x ) e inφ (2.13)gives equivalent results for (2.1)’s real part.We next compute the covariance function for (2.1)’s imaginary part. We first writeIm X t ( τ + ∆)Im X t ( τ ) = (cid:88) p ≤ t a p sin (( τ + ∆) log p + θ p ) sin ( τ log p + θ p ) + 2 (cid:88) p (cid:54) = qp,q ≤ t a p a q sin (( τ + ∆) log p + θ p ) sin ( τ log q + θ q ) . (2.14)By the independence of (2.1)’s summands from (2.10)-(2.12), the expected value of (2.14)’s lastsummation vanishes. We then note thatsin (( τ + ∆) log p + θ p ) sin ( τ log p + θ p ) = 12 (cos (∆ log p ) − cos ((2 τ + ∆) log p + 2 θ p )) . (2.15)The expected value of (2.15)’s second term vanishes. Applying this reasoning to (2.14)’s first sum-mation gives the result (2.2). Essentially equivalent reasoning with the identitycos (( τ + ∆) log p + θ p ) cos ( τ log p + θ p ) = 12 (cos (∆ log p ) + cos ((2 τ + ∆) log p + 2 θ p )) (2.16)gives the result (2.2) for (2.1)’s real part as well.Theorem 2.1 shows that, for each prime p , the p iτ are independent random variables. Thisindependence property enables the straightforward evaluation of (2.1)’s covariance function. Nextwe apply Theorem 2.1’s results to study the repulsion effect in the nontrivial zeros of the Riemannzeta function. 4. Chavez and A. Allawala Prime zeta function statistics and Riemann zero-difference repulsion We use the notation P t ( s ) = (cid:80) p ≤ t p − s and we denote the case with infinite t as simply P ( s ), whichis the prime zeta function. We consider the particular series P t (1 / iτ ) = (cid:88) p ≤ t p / iτ , (3.1)which is an important object for research on the Riemann zeta function’s behavior on the criticalline (see e.g. Fyodorov & Keating 2014, Arguin & Tai 2019). We will apply the previous section’sresults to study (3.1)’s statistical properties. We first use the results of Section 2 to show that (3.1)’sreal and imaginary parts are asymptotically normally distributed. Lemma 3.1.
For τ uniformly distributed in R , as t → ∞ , (cid:113) log log t Re P t (1 / iτ ) and 1 (cid:113) log log t Im P t (1 / iτ ) d −→ N (0 , . (3.2) Proof.
We note from (2.2) that the variance of (3.1)’s real and imaginary parts is given byvar { Re X t ( τ ) } = var { Im X t ( τ ) } = R t (0) = 12 (cid:88) p ≤ t a p . (3.3)Setting a p = 1 /p / then shows thatvar { Re P t (1 / iτ ) } = var { Im P t (1 / iτ ) } = 12 P t (1) = 12 log log t + O (1) (3.4)by Mertens’ 2nd theorem. We next note that since P t (1 / iτ )’s summands are independent byTheorem 2.1, we may apply Lyapunov’s Central Limit Theorem (see e.g. Billingsley 1995) with δ = 2, using (3.4) to give1 (cid:0) P t (1) (cid:1) (cid:88) p ≤ t E (cid:40)(cid:18) cos( τ log p ) p / (cid:19) (cid:41) = 1 (cid:0) P t (1) (cid:1) (cid:88) p ≤ t E (cid:40)(cid:18) sin( τ log p ) p / (cid:19) (cid:41) = C (cid:0) P t (1) (cid:1) (cid:88) p ≤ t p = O (1) log log t + O (log log t ) = O (cid:18) log t (cid:19) , (3.5)where C = E cos ( ωτ ) = E sin ( ωτ ) = 12 /
32 for arbitrary ω ∈ R . This completes the proof of (3.2),since (3.5) vanishes as t → ∞ .The result (3.2) is quite similar to the famous central limit theorem proven by Selberg (1946).This can be explained for the real part with the following important fact, which is used in proofs ofSelberg’s limit theorem, and which we will use later.Re P t X ( t ) (1 / iτ ) d −→ Re P (1 / iτ ) (3.6)as t → ∞ , where t ≤ τ ≤ t and X ( t ) = 1 / (log log log t ) (Radziwill & Soundararajan 2017). Since(3.1) is asymptotically normally distributed, its statistical dependence structure is entirely describedby its covariance function as t → ∞ . By (2.2), the covariance function for (3.1) is given by2 R t (∆) = (cid:88) p ≤ t cos(∆ log p ) p = Re P t (1 + i ∆) . (3.7)5. Chavez and A. Allawala Prime zeta function statistics and Riemann zero-difference repulsionFigure 2: A graph of (3.7)’s 2 R t (∆) with t = p , , (Red) along with log | ζ (1 + i ∆) | (Green) for0 ≤ ∆ ≤ ζ (1 / i ∆) = 0.Numerical evidence suggests that the behavior of the series (3.7) is closely connected to the imaginaryparts of the Riemann zeros. This connection is visually evident in Figure 2, where we have computed(3.7) using the first 1 million primes. It is clear that (3.7) has minima approximately positioned atvalues of ∆ such that ζ (1 / i ∆) = 0. These minima correspond to notable anti-correlations oversuch distances ∆. It is also clear that (3.7) very closely follows the function log | ζ (1 + i ∆) | .To explain this phenomenon, we first note that there is no requirement in Theorem 2.1 that t be finite. We therefore conclude that the covariance function for the full prime zeta function on thecritical line, P (1 / iτ ), is given by the prime zeta function on the 1-line, P (1 + i ∆), i.e.,2 R (∆) = 2 lim t →∞ R t (∆) = (cid:88) p cos(∆ log p ) p = Re P (1 + i ∆) . (3.8)We next prove the following relation between P (1 / iτ )’s covariance function and the Riemannzeta function. Lemma 3.2.
For all ∆ ∈ R , R (∆) = log | ζ (1 + i ∆) | − ε (∆) , (3.9) where | ε (∆) | < − γ ≈ . .
6. Chavez and A. Allawala Prime zeta function statistics and Riemann zero-difference repulsion
Proof.
We first note the following result relating P ( s ) and log ζ ( s ) for Re( s ) >
0, Im( s ) (cid:54) = 0, whichcan be derived by using the Euler product representation of ζ ( s ), taking the logarithm, and applyingM¨obius inversion (see Glaisher 1891, Fr¨oburg 1968). P ( s ) = log ζ ( s ) + ∞ (cid:88) n =2 µ ( n ) n log ζ ( ns ) , (3.10)where µ ( n ) denotes the M¨obius function. We next note from the Dirichlet series representation of ζ ( n + in ∆) that for n ≥ | ζ ( n + in ∆) − | ≤ | ζ ( n ) − | <
1. Therefore by the alternating seriesexpansion of the logarithm, | log ζ ( n + in ∆) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:88) k =1 ( − k +1 k ( ζ ( n + in ∆) − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < | ζ ( n + in ∆) − | ≤ ζ ( n ) − . (3.11)Then by (3.11) and since µ ( n ) ∈ {− , , } for all n , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:88) n =2 µ ( n ) n log ζ ( n + in ∆) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ (cid:88) n =2 | log ζ ( n + in ∆) | n < ∞ (cid:88) n =2 ζ ( n ) − n = 1 − γ. (3.12)(3.12)’s last equality can be shown by using the integral definition of ζ ( n ) for n > n )’sfactorial and integral definition to write ∞ (cid:88) n =2 ζ ( n ) − n = ∞ (cid:88) n =2 n ! (cid:90) ∞ x n − (cid:18) e x − − e x (cid:19) dx = (cid:90) ∞ e x − − xxe x ( e x − dx = (cid:90) ∞ dxe x − (cid:90) ∞ (cid:18) e x − − xe x (cid:19) dx = 1 − γ (3.13)since (3.13)’s last integral is an identity for γ (Whittaker & Watson 1990). Applying (3.12) in (3.10)with s = 1 + i ∆ shows that P (1 + i ∆) = log ζ (1 + i ∆) + f (∆) (3.14)where | f (∆) | < − γ . Combining (3.8) and (3.14) with Re f (∆) = − ε (∆) completes the proof.This shows that (3.8)’s R (∆) is highly dependent on log | ζ (1 + i ∆) | , which produces negativeminima at ∆ near values such that ζ (1 / i ∆) = 0. This behavior is also exhibited by (3.7), whichcan be explained by showing that (3.7) converges in mean square and hence converges in probabilityto (3.8) very quickly as t increases. Proposition 3.1.
Suppose ∆ is uniformly distributed in R , then for all t ≥ , E (cid:8) (Re P t (1 + i ∆) − Re P (1 + i ∆)) (cid:9) = P (2) − P t (2)2 ≤ . ... (3.15) Proof.
We recall the independence of (2.1)’s summands and that E cos ( ω ∆) = 1 / ω to write E (cid:8) (Re P (1 + i ∆) − Re P t (1 + i ∆)) (cid:9) = E (cid:32)(cid:88) p>t cos (∆ log p ) p (cid:33) = E (cid:40)(cid:88) p>t cos (∆ log p ) p (cid:41) + 2 E (cid:88) p (cid:54) = qp,q>t cos (∆ log p ) cos (∆ log q ) pq = 12 (cid:88) p>t p , which completes the proof of (3.15). 7. Chavez and A. Allawala Prime zeta function statistics and Riemann zero-difference repulsionOverall, the covariance structure of P t (1 / iτ ), with finite or infinite t , is heavily influenced bylog | ζ (1 + i ∆) | , which creates negative covariance like that depicted in Figure 2.We next apply the formula (3.10) to the critical line to note thatRe P (1 / iτ ) = log | ζ (1 / iτ ) | + ∞ (cid:88) n =2 µ ( n ) n log | ζ ( n (1 / iτ )) | . (3.16)Recall that ζ ( s ) has no zeros with Re( s ) ≥ s = 1. Also, one can easily show that the terms of the series on (3.16)’s far right-hand sideare O (cid:0) n n/ (cid:1) as n → ∞ and hence the series is convergent for any τ (cid:54) = 0. Therefore, for τ (cid:54) = 0, thesingularities of (3.16) unambiguously define the positions of ζ (1 / iτ )’s zeros. With this context,we next consider the following limit theorem, which nearly completes our derivation of ζ (1 / iτ )’szero-difference repulsion effect. Theorem 3.1.
For all < ∆ ≤ τ , as τ → ∞ , p Re P (1 / i ( τ + ∆)) (cid:113) log log τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ (1 / iτ ) = 0 → N − log | ζ (1 + i ∆) | + ε (∆) (cid:113) log log τ , (3.17) where p ( . | . ) denotes the conditional probability density function.Proof. By Lemma 3.1, both Re P t (1 / iτ ) and Re P t (1 / i ( τ + ∆)) are normally and identicallydistributed as t → ∞ . We therefore use the following formula for normally and identically distributed x and y with E { x } = E { y } = 0. E { x | y } = corr { x, y } × var { x } var { y } × y = cov { x, y } var { y } × y (3.18)Substituting (3.8) into (3.18) gives E { Re P t (1 / i ( τ + ∆))) | Re P t (1 / iτ ) } = R t (∆) R t (0) × Re P t (1 / iτ ) (3.19)and taking t → ∞ gives E { Re P (1 / i ( τ + ∆))) | Re P (1 / iτ ) } = R (∆) R (0) × Re P (1 / iτ )= 2 R (∆)log ζ (1) + O (1) × Re P (1 / iτ ) = 2 R (∆) Re P (1 / iτ )lim x → + log x + O (1) , (3.20)where the second line is reached by recalling from (3.9) that R t (0) → R (0) = log ζ (1) + O (1) andnoting from the Laurent expansion of ζ ( x ) that ζ (1) = lim x → + x − x → + x . For any finite value of Re P (1 / iτ ), (3.19) will vanish. However, by (3.16), at the Riemann zeroswe have E { Re P (1 / i ( τ + ∆))) | ζ (1 / iτ ) = 0 } = 2 R (∆) lim x → + log x + l.t. − lim x → + log x + O (1)= − R (∆) = − log | ζ (1 + i ∆) | + ε (∆) (3.21)8. Chavez and A. Allawala Prime zeta function statistics and Riemann zero-difference repulsionfrom Lemma 3.2. We next note the bivariate normal conditional variance formula,var { x | y } = var { x } (cid:34) − (cid:18) cov { x, y } var { x } (cid:19) (cid:35) = var { x } − (cov { x, y } ) var { x } . (3.22)Again noting Lemma 3.1 and applying (3.22) to Re P t (1 / iτ ) and Re P t (1 / i ( τ + ∆)) with∆ (cid:54) = 0 as t → ∞ then givesvar { Re P t (1 / i ( τ + ∆)) | Re P t (1 / iτ ) } = 12 P t (1) − R t (∆) P t (1) = 12 P t (1) + o (1) = 12 log log t + O (1)(3.23)by (3.8)-(3.9) and Merten’s 2nd theorem. We lastly note from (3.6) that for 0 < ∆ ≤ τ and t = τ / (log log log τ ) , P t (1 / i ( τ + ∆)) d −→ P (1 / i ( τ + ∆))as τ → ∞ . We therefore conclude from (3.21) and (3.23) that as τ → ∞ p (Re P (1 / τ + ∆)) | ζ (1 / iτ ) = 0) → N (cid:18) − log | ζ (1 + i ∆) | + ε (∆) ,
12 log log τ + o (log log τ ) (cid:19) (3.24)This completes the proof of (3.17).(3.19)-(3.21) shows that Re P (1 / i ( τ + ∆))’s conditional expectation asymptotically vanishesat all τ (cid:54) = 0 unless ζ (1 / iτ ) = 0, where it then becomes closely approximated by − log | ζ (1 + i ∆) | .At such values of τ , this results in Re P (1 / i ( τ + ∆))’s conditional expectation having maxima at∆ approximately equal to the imaginary parts of Riemann zeros. The positions ∆ of these maximain the means of (3.17) and (3.24) imply a decreased probability that Re P (1 / i ( τ + ∆)) < L forany L <
0. By (3.16), this implies a decreased probability that | ζ (1 / i ( τ + ∆)) | = 0. The results(3.17) and (3.24) thus predict the zero-difference repulsion effect.A visualization is given in Figure 3, where (3.24)’s probability that Re P (1 / i ( τ + ∆)) takesan extreme negative value is graphed for τ given by the hundred thousandth Riemann zero ordinate.It is clear that the likelihood of such an extreme negative value, which is a necessary condition for ζ (1 / i ( τ + ∆)) to vanish by (3.16), is much higher for regions of ∆ away from the imaginary partsof the Riemann zeros. We lastly note that (3.24) becomes the uniform distribution, and equivalently,(3.17)’s mean vanishes as τ → ∞ . Hence this effect weakens higher up the critical line. Perez Marco (2011) showed that a very similar effect occurs in many L-functions. In particular, heshowed that the differences of the complex zeros of the L-functions L ( χ , s ), L ( χ , s ), and L ( χ , , s )also seem to repel the Riemann zeros, indicating a connection between Riemann zeta and L-functionzeros. We may explain this using our methodology by defining the following prime Dirichlet series P χ ( s ) = (cid:88) p χ ( p ) p s (4.1)for a given Dirichlet character χ s.t. χ ( p ) = e iθ p . With s = 1 / iτ , Theorem 2.1 and Lemma 3.1’sreasoning may be applied to show that the real and imaginary parts of (4.1) are normally distributed.9. Chavez and A. Allawala Prime zeta function statistics and Riemann zero-difference repulsionFigure 3: A graph of P { Re P (1 / i ( τ + ∆)) ≤ − σ | ζ (1 / iτ ) = 0 } , evaluated by integrating(3.24), with 0 < ∆ ≤
100 and σ = log log τ , where τ is the imaginary part of the 100,000thRiemann zero.We next note that for any Dirichlet character modulo n , χ n , where n has prime factorization, n = (cid:89) p p m p , (4.2) χ n ( p ) = 0 for any p such that m p (cid:54) = 0, while for any p with m p = 0, | χ n ( p ) | = 1. Therefore wemay apply Theorem 2.1 to show that, for s = 1 / iτ , the covariance function for (4.1)’s real andimaginary parts corresponding to the Dirichlet character χ n have the same form as (3.8)-(3.9) with | ε χ n (∆) | < − γ + (cid:88) pm p (cid:54) =0 p . (4.3)For χ , χ , and χ , these error bounds are relatively small since the only prime factors of thesecharacters’ moduli are 3, 2, and 7 respectively. However, the error bound increases for moduli n witha larger number of prime factors. We lastly note that (3.10) may be generalized to give the followingrelationship between Dirichlet L-functions L ( χ, s ) and corresponding prime Dirichlet L-functions P χ ( s ): P χ ( s ) = ∞ (cid:88) n =1 µ ( n ) n log L ( ns, χ n ) . (4.4)Then a generalization of (3.6) for series of the form (4.1), such as that provided in Hsu & Wong (2019),enables derivation of an essentially equivalent result to (3.17), which, by (4.4) and its correspondinggeneralization of (3.16), has the same interpretation.Data used in this paper is accessible at and https://primes.utm.edu/lists/small/millions/ .10. Chavez and A. Allawala Prime zeta function statistics and Riemann zero-difference repulsion References [1] Arguin, L.P. & Tai, W. (2019). Is the Riemann-Zeta Function in a Short Interval a 1-RSB SpinGlass? In: Sidoravicius V. (ed.) Sojourns in Probability Theory and Statistical Physics-I. SpringerProceedings in Mathematics & Statistics, vol 298. Springer, Singapore.[2] Billingsley, P. (1995). Probability and Measure. New Jersey: Wiley.[3] Bogolmny, E. & Keating, J. (1996). Gutzwiller’s trace formula and spectral statistics: beyond thediagonal approximation.
Physical Review Letters 77 (8), 1472-1475.[4] Conrey, J.B., Farmer, D.W., & Zirnbauer, M.R. (2008). Autocorrelation of ratios of L-functions.
Communications in Number Theory and Physics 2 (3), 593-636.[5] Conrey, J.B. & Snaith, N.C. (2008). Correlations of eigenvalues and Riemann zeros.
Communi-cations in Number Theory and Physics 2 (3), 477-536.[6] de la Vall´ee Poissin, C.J. (1896). Recherches analytiques de la th´eorie des nombres premiers.
Annales de la Soci´et´e scientifique de Bruxelles 20 , 183-256.[7] Ford, K. & Zaharescu, A. (2015). Unnormalized differences between zeros of L-functions.
Com-positio Mathematica 151 , 230-252.[8] Fr¨oberg, C-E. (1968). On the Prime Zeta Function.
Nordisk Tidskr. Informationsbehandling (BIT)8 (3), 187-202.[9] Fyodorov, Y.V. & Keating, J.P. (2014). Freezing transitions and extreme values: random ma-trix theory, and disordered landscapes.
Philosophical Transactions of the Royal Society A 372 :20120503.[10] Glaisher, J.W.L. (1891). On the Sums of Inverse Powers of the Prime Numbers.
QuarterlyJournal of Mathematics 25 , 347-362.[11] Hadamard, J. (1896). Sur la distribution des z´eros de la fonction ζ ( s ) et ses cons´equences arith-metiques. Bulletin de la S.M.F. 24 , 199-220.[12] Hsu, P. & Wong, P. (2019). On Selberg’s Central Limit Theorem for Dirichlet L-Functions.
Jour-nal de Th´eorie des Nombres de Bordeaux . .[13] Laurinˇcikas, A. (1996). Limit Theorems for the Riemann-Zeta Function. Dordrecht: KluwerAcademic Publishers.[14] Montgomery, H. (1973). The pair correlation of zeros of the zeta function. Proceedings of theSymposium on Pure Mathematics 24 , 181-193. Providence, R.I.: American Mathematical Society.[15] Odlyzko, A.M. & Sch¨onhage, A. (1988). Fast algorithms for multiple evaluations of the Riemannzeta function.
Transactions of the American Mathematical Society 309 (2), 797-809.[16] Perez Marco, R. (2011). Statistics on the Riemann zeros, https://arxiv.org/abs/1112.0346 .[17] Radziwill, M. & Soundararajan, K. (2017). Selberg’s central limit theorem for log | ζ (1 / it ) | . L’enseignement Mathematique 63 (1/2), 1-19.11. Chavez and A. Allawala Prime zeta function statistics and Riemann zero-difference repulsion[18] Rodgers, B. (2013). Macroscopic pair correlation of the Riemann zeroes for smooth test functions.
Quarterly Journal of Mathematics 64 (4), 1197-1219.[19] Selberg, A. (1946). Contributions to the theory of the Riemann zeta-function.
Arch. Math.Naturvid. 48 (5), 89-155.[20] Snaith, N.C. (2010). Riemann zeros and random matrix theory.