Riemann Hypothesis and Random Walks: the Zeta case

Riemann Hypothesis and Random Walks: the Zeta case

André LeClair111andre.leclair@gmail.com Cornell University, Physics Department, Ithaca, NY 14850
Abstract

In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its -function is valid to the right of the critical line , and the Riemann Hypothesis for this class of -functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet -functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a 1-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely -th zero to over 100 digits, far beyond what is currently known.

I Introduction

There are many generalizations of Riemann’s zeta function to other Dirichlet series, which are also believed to satisfy a Riemann Hypothesis. A common opinion, based largely on counterexamples, is that the -functions for which the Riemann Hypothesis is true enjoy both an Euler product formula and a functional equation. However a direct connection between these properties and the Riemann Hypothesis has not been formulated in a precise manner. In EPF1 (); EPF2 () a concrete proposal making such a connection was presented for Dirichlet -functions, and those based on cusp forms, due to the validity of the Euler product formula to the right of the critical line. In contrast to the non-principal case, in this approach the case of principal Dirichlet -functions, of which Riemann zeta is the simplest, turned out to be more delicate, and consequently it was more difficult to state precise results. In the present work we address further this special case.

Let be a Dirichlet character modulo and its -function with . It satisfies the Euler product formula

(1)

where is the -th prime. The above formula is valid for since both sides converge absolutely. The important distinction between principal verses non-principal characters is the following. For non-principal characters the -function has no pole at , thus there exists the possibility that the Euler product is valid partway inside the strip, i.e. has abscissa of convergence . It was proposed in EPF1 (); EPF2 () that for this case. In contrast, now consider -functions based on principal characters. The latter character is defined as if is coprime to and zero otherwise. The Riemann zeta function is the trivial principal character of modulus with all . -functions based on principal characters do have a pole at , and therefore have abscissa of convergence , which implies the Euler product in the form given above cannot be valid inside the critical strip . Nevertheless, in this paper we will show how a truncated version of the Euler product formula is valid for .

The primary aim of the work EPF1 (); EPF2 () was to determine what specific properties of the prime numbers would imply that the Riemann Hypothesis is true. This is the opposite of the more well-studied question of what the validity of the Riemann Hypothesis implies for the fluctuations in the distribution of primes. The answer proposed was simply based on the multiplicative independence of the primes, which to a large extent underlies their pseudo-random behavior. To be more specific, let for . In EPF1 (); EPF2 () it was proven that if the series

(2)

is , then the Euler product converges for and the formula (1) is valid to the right of the critical line. In fact, we only need up to logs (see Remark 1); when we write write , it is implicit that this can be relaxed with logarithmic factors. For non-principal characters the allowed angles are equally spaced on the unit circle, and it was conjectured in EPF2 () that the above series with behaves like a random walk due to the multiplicative independence of the primes, and this is the origin of the growth. Furthermore, this result extends to all since domains of convergence of Dirichlet series are always half-planes. Taking the logarithm of (1), one sees that is never infinite to the right of the critical line and thus has no zeros there. This, combined with the functional equation that relates to , implies there are also no zeros to the left of the critical line, so that all zeros are on the line. The same reasoning applies to cusp forms if one also uses a non-trivial result of Deligne EPF2 ().

In this article we reconsider the principal Dirichlet case, specializing to Riemann zeta itself since identical arguments apply to all other principal cases with . Here all angles , so one needs to consider the series

(3)

which now strongly depends on . On the one hand, whereas the case of principal Dirichlet -functions is complicated by the existence of the pole, and, as we will see, one consequently needs to truncate the Euler product to make sense of it, on the other hand can be estimated using the prime number theorem since it does not involve sums over non-trivial characters , and this aids the analysis. This is in contrast to the non-principal case, where, however well-motivated, we had to conjecture the random walk behavior alluded to above, so in this respect the principal case is potentially simpler. To this end, a theorem of Kac (Theorem 1 below) nearly does the job: in the limit , which is also a consequence of the multiplicative independence of the primes. This suggests that one can also make sense of the Euler product formula in the limit . However this is not enough for our main purpose, which is to have a similar result for finite which we will develop.

This article is mainly based on our previous work EPF1 (); EPF2 () but provides a more detailed analysis and extends it in several ways. It was suggested in EPF1 () that one should truncate the series at an that depends on . First, in the next section we explain how a simple group structure underlies a finite Euler product which relates it to a generalized Dirichlet series which is a subseries of the Riemann zeta function. Subsequently we estimate the error under truncation, which shows explicitly how this error is related to the pole at , as expected. The remainder of the paper, sections IV-VI, presents various applications of these ideas. We use them to study the argument of the zeta function. We present an algorithm to calculate very high zeros, far beyond what is currently known. We also study the statistical fluctuations of individual zeros, in other words, a 1-point correlation function.

In many respects, our work is related to the work of Gonek et. al. Gonek1 (); Gonek2 (), which also considers a truncated Euler product. The important difference is that the starting point in Gonek1 () is a hybrid version of the Euler product which involves both primes and zeros of zeta. Only after assuming the Riemann Hypothesis can one explain in that approach why the truncated product over primes is a good approximation to zeta. In contrast, here we do not assume anything about the zeros of zeta, since the goal is to actually understand their location.

We are unable to provide fully rigorous proofs of some of the statements below, however we do provide supporting calculations and numerical work. In order to be clear on this, below “Proposal” signifies the most important claims that we could not rigorously prove.

Ii Algebraic structure of finite Euler products

The aim of this section is to define properly the objects we will be dealing with. In particular we will place finite Euler products on the same footing as other generalized Dirichlet series. The results are straightforward and are mainly definitions.


Definition 1.

Fix a positive integer and let denote the first primes where . From this set one can generate an abelian group of rank with elements

(4)

where the group operation is ordinary multiplication. Clearly where are the positive rational numbers. There are an infinite number of integers in which form a subset of the natural numbers . We will denote this set as , and elements of this set simply as .


Definition 2.

Fix a positive integer . For every integer we can define the character :

(5)

Clearly, for a prime , if .


Definition 3.

Fix a positive integer and let be a complex number. Based on we can define the infinite series

(6)

which is a generalized Dirichlet series. There are an infinite number of terms in the above series since is infinite dimensional.


Example 1.

For instance


Because of the group structure of , satisfies a finite Euler product formula:

Proposition 1.

Let be the abscissa of convergence of the series where , namely converges for . Then in this region of convergence, satisfies a finite Euler product formula:

(7)
Proof.

Based on the completely multiplicative property of the characters,

(8)

one has

The result follows then from the fact that if . ∎


Example 2.

Let , so that . Then the above Euler product formula (7) is simply the standard formula for the sum of a geometric series:

(9)

Here the abscissa of convergence is .

The series defined in (6) has some interesting properties:

(i)   For finite the product is finite for , thus the infinite series converges for for any finite .

(ii)   Since the logarithm of the product is finite, for finite , has no zeros nor poles for . Thus the Riemann zeros and the pole at arise from the primes at infinity , i.e. in the limit . In this limit all integers are included in the sum (6) that defines since . This is in accordance with the fact that the pole is a consequence of there being an infinite number of primes.

The property (ii) implies that, in some sense, the Riemann zeros condense out of the primes at infinity . Formally one has

(10)

However since is going to infinity, the above is true only where the series formally converges, which, as discussed in the Introduction, is . Nevertheless, for very large but finite , the function can still be a good approximation to inside the critical strip since for finite there is convergence of for . This is the subject of the next section, where we show that a finite Euler product formula is valid for in a manner that we will specify.

Iii Finite Euler product formula at large to the right of the critical liine.

In this section we propose that the Euler product formula can be a very good approximation to for and large if is chosen to depend on in a specific way which was already proposed in EPF1 (); EPF2 (). The new result presented here is an estimate of the error due to the truncation.

The random walk property we will build upon is based on a central limit theorem of Kac Kac (), which largely follows from the multiplicative independence of the primes:

Theorem 1.

(Kac) Let be a random variable uniformly distributed on the interval , and define the series

(11)

Then in the limit and , approaches the normal distribution , namely

(12)

where denotes the probability for the set.

We wish to use the above theorem to conclude something about for a fixed, non-random . Based on Theorem 1, we first conclude the following for non-random, but large :

Corollary 1.

For any ,

(13)
Proof.

This is straightforward: as , even though is random, all in the range are tending to . One then uses the normal distribution in Theorem 1. ∎

Remark 1.

The proof of convergence of the Euler product in EPF2 () is not spoiled if the bound on is relaxed up to logs. For instance, if in the limit , , as suggested by the law of iterated logarithms relevant to central limit theorems, this is fine, as is for any positive power .

A consequence of Theorem 1 and the Corollary 1 is that the Euler product formula is valid to the right of the critical line in the limit , at least formally. Namely for ,

(14)

As shown in EPF1 (); EPF2 () and discussed in the Introduction, this formally follows from the growth of . The problem with the above formula is that due to the double limit on the RHS, it is not rigorously defined. For instance, it could depend on the order of limits. It is thus desirable to have a version of (14) where and are taken to infinity simultaneously. Namely, we wish to truncate the product at an that depends on with the property that . One can then replace the double limit on the RHS of (14) with one limit , or equivalently .

There is no unique choice for , but there is an optimal upper limit, , with its integer part, which we now describe. We can use the prime number theorem to estimate :

where is the usual exponential-integral function, and we have used

(16)

The prime number theorem implies . Using this in (III) and imposing leads to .

Based on the above, henceforth we will always assume the following properties of :

(17)

and will not always display the dependence of . Equation (14) now formally becomes

(18)

Extensive and compelling numerical evidence supporting the above formula was already presented in EPF1 ().

Based on the above results we are now in a position to study the following important question. If we fix a finite but large , and truncate the Euler product at , which is finite, what is the error in the approximation to to the right of the critical line? We estimate this error as follows:

Proposal 1.

Let satisfy (17). Then for and large ,

(19)

where is the actual function defined by analytic continuation and

(20)

is finite (except at the pole ) and satisfies

(21)

namely the error goes to zero as .

We provide the following supporting argument, although not a rigorous proof, for this Proposal. From (18), one concludes that (19) must hold in the limit of large with satisfying (21). The logarithm of (19) reads

(22)

First assume . Then in the limit of large , the error upon truncation is the part that is neglected in (18):

(23)

Expanding out the logarithm, one has

(24)

where in the second line we again used the prime number theorem to approximate the sum over primes. Next using , one obtains (20). Finally, the above expression can be continued into the strip if since which goes to zero as if . The latter also implies (21).

Proposal 1 makes it clear that the need for a cut-off originates from the pole at , since as long as , the error in (20) is finite. The error becomes smaller and smaller the further one is from the pole, i.e. as . In Figure 1 we numerically illustrate Proposal 1 inside the critical strip.

Remark 2.

For estimating errors at large the following formula is useful:

(25)
Figure 1: The error term with for inside the critical strip as a function of . The fluctuating (blue) curve is computed directly from the definition (19) with the usual analytic continuation into the strip. The smooth (yellow) curve is the approximation based on (20).
Theorem 2.

Assuming Proposal 1, all non-trivial zeros of are on the critical line.

Proof.

Taking the logarithm of the truncated Euler product, one obtains (22). If there were a zero with , then . However the right hand side of (22) is always finite, thus there are no zeros to the right of the critical line. The functional equation relating to shows there are also no zeros to the left of the critical line. ∎

Remark 3.

Interestingly, Proposal 1 and Theorem 2 imply that proving the validity of the Riemann Hypothesis is under better control the higher one moves up the critical line. For instance, it is known that all zeros are on the line up to , and beyond this, the error is too small to spoil the validity of the Riemann Hypothesis. Henceforth, we assume the RH.

Iv The argument of the -function near the critical line

In our work Trans (), was defined as follows:

(26)

It is important in the above definition that is not allowed to be strictly zero. It will also be important that the limit approaches the critical line from the right because this is the region where the (truncated) Euler product formula is valid in the sense described above. The above definition for is not identical to that of the conventional , and one should not assume they are the same. For instance, it is well known that is not defined at the ordinate of a zero, whereas is. (More generally, the argument of any analytic function at a zero is well-defined once the contour by which it is approached is specified.) The behavior of would be completely different if it were defined as a limit from the left.


Proposal 2.

The function is well defined and finite for all , i.e. .


For the remainder of the section we provide arguments supporting this Proposal. Based on the Euler product formula (Proposal 1) one has

(27)

where the limit is implicit. Recall that as , actually goes to zero. One can check numerically that the above formula works rather well with disregarded; see Figure 2. From this figure one clearly sees that the above formula knows about all the Riemann zeros even if one neglects the error term, since it jumps by one at each zero.

Figure 2: The exact (blue line) verses calculated from the primes, i.e. the Euler product formula (27) (yellow line). Here we took and .

It is clear that based on (27), because it is finite for all . Let us try to be more specific based on our results thus far. Under the assumption of Proposal 1, which implies the Euler product formula (27) for , then is well-defined for all . Let us fix satisfying (17). Expanding the logarithm, one has

(28)

We neglected the error since it is also by (25). The first term is finite, thus is finite.

As for other functions defined by sums over primes, such as the prime number counting function , there is a leading smooth part which is determined by the prime number theorem, and a sub-leading fluctuating part that depends on the exact locations of the primes. We can therefore write

(29)

where is the smooth part coming from the prime number theorem, and are the fluctuating corrections. Consider first the smooth part:

For :

(31)

Thus . Now, as , in (IV) one can replace with , and the two terms cancel:

(32)

Let us now turn to the fluctuating term which actually knows about the locations of the zeros since at each zero it jumps by its multiplicity. Since the leading contribution goes to zero, has no growth and consists only of these jumps, all occurring around , which is consistent with the average of being zero.

If one assumes all zeros of are simple, as Theorem 3 below would imply, then one can further argue that is nearly always on the principal branch: . If all zeros are simple, then jumps by only at each zero. Thus the largest value of is approximately corresponding to a jump beginning at . In other words, is never very far from zero so that most of the jumps pass through as seen in Figure 2.

Figure 2 provides numerical evidence for the above statements. Simply stated, the Proposal 2 says that there is no change in behavior of as increases to infinity, such that the pattern in Figure 2 persists. We checked its validity all the way up to . Only rarely is slightly above . Over this whole range we found .

V -point correlation function of the Riemann zeros

Montgomery conjectured that the pair correlation function of ordinates of the Riemann zeros on the critical line satisfy GUE statistics Montgomery (). Being a 2-point correlation function, it is a reasonably complicated statistic. In this section we propose a simpler 1-point correlation function that captures the statistical fluctuations of individual zeros.

Let be the exact ordinate of the -th zero on the critical line, with and so forth. The single equation is known to have an infinite number of non-trivial solutions . In Trans (), by placing the zeros in one-to-one correspondence with the zeros of a cosine function, the single equation was replaced by an infinite number of equations, one for each that depends only on :

(33)

where is the Riemann-Siegel function:

(34)

The term equals discussed in the last section. It is important that the approaches the critical line from the right, since this is where the Euler product formula is valid in the sense described above. This equation was used to calculate zeros very accurately in Trans (), up to thousands of digits. There is no need for a cut-off in the above equation since the term is defined for arbitrarily high by standard analytic continuation. One aspect of this equation is the following theorem:

Theorem 3.

(França-LeClair) If there is a unique solution to the equation (33) for every positive integer , then the Riemann Hypothesis is true, and furthermore, all zeros are simple.

Remark 4.

Details of the proof are in Trans (). The main idea is that if there is a unique solution, then the zeros are enumerated by the integer and can be counted along the critical line, and the resulting counting formula coincides with a well known result due to Backlund for the number of zeros in the entire critical strip. The zeros are simple because the zeros of the cosine are simple. The above theorem is another approach towards proving the Riemann Hypothesis, however it is not entirely independent of the above approach based on the Euler product formula, in particular Theorem 2. In Trans (), we were unable to prove there is a unique solution because we did not have sufficient control over the relevant properties of the function . The previous section helps close this gap in our understanding of by showing that is indeed well defined at the zeros and consequently there should be a unique solution to (33) for all .

If the term is ignored, then there is indeed a unique solution for all since is a monotonically increasing function of . Using its asymptotic expansion for large , equation (40) below, and dropping the term, then the solution is

(35)

where is the Lambert -function. The only way there would fail to be a solution is if is not well defined for all . However, as discussed above, this would appear to contradict the analysis of the last section, in particular Proposal 2.

The fluctuations in the zeros come from since is a smooth function of . These small fluctuations are shown in Figure 3. Let us define . One needs to properly normalize , taking into account that the spacing between zeros decreases as . To this end we expand the equation (33) around . Using , one obtains where is the derivative with respect to . Using , this leads us to define

(36)

The probability distribution of the set

(37)

for large is then an interesting property to study. Here “probability” is defined as frequency of occurrence. The equation (36) together with (27) makes it clear that the origin of the statistical fluctuations of is the fluctuations in the primes.

Figure 3: The first 30 Riemann zeros . The smooth curve is the approximation in (35), whereas the dots are the actual zeros .

In Figure 4 we plot the distribution of for . It closely resembles a normal distribution, however as we will argue, we believe it is not exactly normal. Let us first suppose does satisfy a normal distribution . Using the properties of described in the last section, together with the equation (36), we can propose then the following. First, one expects that the average of is zero since it is known that the average of is zero, thus . Secondly, if is nearly always on the principal branch, as argued in the last section, then at each jump by at , on average passes through zero. This implies that the average . For a normal distribution . Thus one expects the standard deviation of to be . In Figure 4 we present results for the first -th known exact zeros. The distribution function fits a normal distribution with rather well. Performing a fit, one finds . For higher values of around , a fit gives , which is closer to the predicted value.

Figure 4: The probability distribution for the set defined in (37) for . The smooth curve is the normal distribution with .

The p-values for the fit to the normal distribution in Figure 4 are quite low. For instance for the Pearson test. Other tests have even worse p-values. This leads us to believe that the distribution is not precisely normal. If the distribution were exactly normal, then some would be arbitrarily large. Equation (36) would then imply that could also be arbitrarily large, which contradicts Proposal 2. These issues played an important role in our proposal that the normalized gaps between zeros have an upper bound ZeroGaps (). For these reasons, we believe that has a distribution that is only approximately normal.


If we approximate the distribution of as normal, then we can construct a simple probabilistic model of the Riemann zeros:


Definition 4.

A probabilistic model of the Riemann zeros.   Let be a random variable with normal distribution . Then a probabilistic model of the zeros can be defined as the set , where

(38)

and is defined in (35). In the above formula is chosen at random independently for each .


The statistical model (38) is rather simplistic since it is just based on a normal distribution for and is smooth and completely deterministic. A natural question then arises. Does the pair correlation function of satisfy GUE statistics as does the actual zeros ? We expect the answer is no, since the only correlation between pairs of ’s is the smooth, predictable part . Nevertheless, it is interesting to study the 2-point correlation function of . Montgomery’s pair correlation conjecture can be stated as follows. Let denote the number of zeros up to height , where . Let denote zeros in the range . Then in the limit of large :

(39)

where is a normalized distance between zeros .

In Figure 5 we plot the pair correlation function for the first -th ’s. We chose since in this range of this gives a better fit to the normal distribution of the 1-point function. The results are reasonably close to the GUE prediction (39), especially considering that for just the first true zeros the fit to the GUE prediction is not perfect; for much higher zeros it is significantly better OdlyzkoPair (). We interpret the deviation from the GUE prediction to be additional evidence that the distribution of the set is not exactly normal.

Figure 5: The pair correction function of defined in (38) for up to where the standard deviation of was taken to be . The solid curve is the GUE prediction. The parameters in (39) are with and the -axis is given by .

Vi Computing very high zeros from the primes

This section can be viewed as providing additional numerical evidence for some of the previous results. Since we will be calculating from the primes using (27), which requires , this is pushing the limit of the validity of the Euler product formula, nevertheless we will obtain reasonable results. We emphasize that this method has nothing to do with the random model for the zeros in Definition 4, but rather relies on the Euler product formula to calculate .

Many very high zeros of have been computed numerically, beginning with the work of Odlyzko. All zeros up to the -th have been computed and are all on the critical line Gourdon (). Beyond this the computation of zeros remains a challenging open problem. However some zeros around the -st and -nd are known Odlyzko (). In this section we describe a new and simple algorithm for computing very high zeros based on the results of Section IV. It will allow us to go much higher than the known zeros since it does not require numerical implementation of the function itself, but rather only requires knowledge of some of the lower primes.

Let us first discuss the numerical challenges involved in computing high zeros from the equation (33) based on the standard Mathematica package. The main difficulty is that one needs to implement the term. Mathematica computes , i.e. on the principal branch, however near a zero this is likely to be valid based on the discussion in section IV. The main problem is that Mathematica can only compute for below some maximum value around . This was sufficient to calculate up to the -th zero from (33) in Trans (). The term must also be implemented to very high , which is also limited in Mathematica.

We deal with these difficulties first by computing from the formula (27) involving a finite sum over primes. We will neglect the term at first in (27) since it vanishes in the limit ; however we will return to it when we will estimate the error in computing zeros this way. Then, the term can be accurately computed using corrections to Stirling’s formula:

(40)

Let denote the ordinate of the -th zero computed using the first primes based on (33). For high zeros, it is approximately the solution to the following equation

(41)

where it is implicit that . The important property of this equation is that it no longer makes any reference to itself. It is straightforward to solve the above equation with standard root-finder software, such as FindRoot in Mathematica.

One can view the computation of as a kind of Markov process. If one includes no primes, i.e. , and drops the next to leading corrections, then the solution is unique and explicitly given by in terms of the Lambert -function in (35). One then goes from to by finding the root to the equation for in the vicinity of , then similarly is calculated based on and so forth. At each step in the process one includes one additional prime, and this slowly approaches , so long as . In practice we did not follow this iterative procedure, but rather fixed and simply solved (41) in the vicinity of .

We can estimate the error in computing the zero from the primes using equation (41) as follows. As in Section V, we expand the equation (33) now around rather than . One obtains

where is the error in computing from the primes (27). Using (24), we have

Now from the prime number theorem, . Recall is cut off at , which cancels the in the previous formula. Finally it is meaningful to normalize the error by the mean spacing . The result is

(42)

where we have used . The left hand side represents the ratio of the error to the mean spacing between zeros at that height. Again, it is implicit that . The interesting aspect of the above formula is that the relative error decreases with , although rather slowly. The cosine factor also implies there are large scale oscillations around the actual .

For very high , is extremely large and it is not possible in practice to work with such a large number of primes. This is the primary limitation to the accuracy we can obtain. We will limit ourselves to the relatively small primes. Let us verify the method by comparing with some known zeros around and . The results are shown in Table 1. Equation (42) predicts for these and , and inspection of the table shows this is a good estimate. Odlyzko was of course able to calculate more digits; our accuracy can be improved by increasing in principle. We also checked some zeros around the -rd computed by Hiary Hiary (), again with favorable results.

     (Odlyzko)
Table 1: Zeros around the -st and -nd computed from (41) with primes. We fixed . Above, denotes the integer part of the second column.

Having made this check, let us now go far beyond this and compute the -th zero by the same method. Again using only primes, we found the following :

Obtaining this number took only a few minutes on a laptop using Mathematica. We are confident that the last digits are correct since we checked that they didn’t change between and . Furthermore, digits is consistent with (42), which predicts that for these and , . We calculated the next zero to be .

We were able to extend this calculation to the -th zero without much difficulty. As equation (42) shows, the relative error only decreases as one increases . It is also straightforward to extend this method to all primitive Dirichlet -functions and those based on cusp forms using the transcendental equations in Trans () and the results in EPF2 ().

Acknowledgments

We wish to thank Denis Bernard, Guilherme França, Ghaith Hiary, Giuseppe Mussardo, and German Sierra for discussions. We also thank the Isaac Newton Institute for Mathematical Sciences for their hospitality in the final stages of this work (January 2016).

References

  • (1) G. França and A. LeClair, “On the validity of the Euler product inside the critical strip”, arXiv:1410.3520 [math.NT].
  • (2) G. França and A. LeClair, “Some Riemann Hypotheses from Random Walks over Primes”, arXiv:1509.03643 [math.NT].
  • (3) M. Kac, Statistical Independence in Probability, Analysis and Number Theory, The Mathematical Association of America, New Jersey, 1959.
  • (4) S. M. Gonek, C. P. Hughes, and J. P. Keating, Duke Math. J 136 (2007) 507.
  • (5) S. M. Gonek, Trans. Amer. Math. Soc. 364 (2011) 2157.
  • (6) G. França and A. LeClair, Commun. Number Theory and Phys. 9 (2015) 1-50.
  • (7) A. LeClair, On large gaps between zeros of L-functions from branches, arXiv:1704.05834 [math.NT].
  • (8) H. Montgomery, in Analytic number theory, Proc. Sympos. Pure Math. XXIV (Providence, RI: AMS, 1973).
  • (9) A. M. Odlyzko, Math. Comp. 48. 273 (1987).
  • (10) X. Gourdon, 2004, http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf.
  • (11) A. M. Odlyzko, The -st zero of the Riemann zeta function, www.research.att.com/amo, 1998.
  • (12) G. Hiary, Ann. Math., 174-2 (2011) 891;
    https://people.math.osu.edu/hiary.1/outd3/out.88837796029624663862630219091085.zeros
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