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Applications of Wirtinger Inequalities on the Distribution of Zeros of the Riemann Zeta-Function
Journal of Inequalities and Applications volume 2010, Article number: 215416 (2011)
On the hypothesis that the th moments of the Hardy -function are correctly predicted by random matrix theory and the moments of the derivative of are correctly predicted by the derivative of the characteristic polynomials of unitary matrices, we establish new large spaces between the zeros of the Riemann zeta-function by employing some Wirtinger-type inequalities. In particular, it is obtained that which means that consecutive nontrivial zeros often differ by at least 6.1392 times the average spacing.
The Riemann zeta-function is defined by
and by analytic continuation elsewhere except for a simple pole at . The identity between the Dirichlet series and the Euler product (taken over all prime numbers ) is an analytic version of the unique prime factorization in the ring of integers and reflects the importance of the zeta-function for number theory. The functional equation
implies the existence of so-called trivial zeros of at for any positive integer ; all other zeros are said to be nontrivial and lie inside the so-called critical strip . The number of nontrivial zeros of with ordinates in the interval , is asymptotically given by the Riemann-von Mangoldt formula (see )
Consequently, the frequency of their appearance is increasing and the distances between their ordinates is tending to zero as .
The Riemann zeta-function is one of the most studied transcendental functions, having in view its many applications in number theory, algebra, complex analysis, and statistics as well as in physics. Another reason why this function has drawn so much attention is the celebrated Riemann conjecture regarding nontrivial zeros which states that all nontrivial zeros of the Riemann zeta-function lie on the critical line . The distribution of zeros of is of great importance in number theory. In fact any progress in the study of the distribution of zeros of this function helps to investigate the magnitude of the largest gap between consecutive primes below a given bound. Clearly, there are no zeros in the half plane of convergence , and it is also known that does not vanish on the line . In the negative half plane, and its derivative are oscillatory and from the functional equation there exist so-called trivial (real) zeros at for any positive integer (corresponding to the poles of the appearing Gamma-factors), and all nontrivial (nonreal) zeros are distributed symmetrically with respect to the critical line and the real axis.
There are three directions regarding the studies of the zeros of the Riemann zeta-function. The first direction is concerned with the existence of simple zeros. It is conjectured that all or at least almost all zeros of the zeta-function are simple. For this direction, we refer to the papers by Conrey  and Cheer and Goldston .
The second direction is the most important goal of number theorists which is the determination of the moments of the Riemann zeta-function on the critical line. It is important because it can be used to estimate the maximal order of the zeta-function on the critical line, and because of its applicability in studying the distribution of prime numbers and divisor problems. For more details of the second direction, we refer the reader to the papers in [4–6] and the references cited therein. For further classical results from zeta-function theory, we refer to the monograph  of Ivić and the papers by Kim [8–11].
For completeness in the following we give a brief summary of some of these results in this direction that we will use in the proof of the main results. It is known that the behavior of on the critical line is reflected by the Hardy -function as a function of a real variable, defined by
It follows from the functional equation (1.2) that is an infinitely often differentiable function which is real for real and moreover . Consequently, the zeros of correspond to the zeros of the Riemann zeta-function on the critical line. An important problem in analytic number theory is to gain an understanding of the moments of the Hardy -function function and the moments of its derivative which are defined by
For positive real numbers , it is believed that
for positive constants and will be defined later.
Keating and Snaith  based on considerations from random matrix theory conjectured that
where is the Barnes -function (for the definition of the Barnes -function and its properties, we refer to ).
Hughes  used the Random Matrix Theory (RMT) and stated an interesting conjecture on the moments of the Hardy -function and its derivatives at its zeros subject to the truth of Riemann's hypothesis when the zeros are simple. This conjecture includes for fixed the asymptotic formula of the moments of the form
where is defined as in (1.8) and the product is over the primes. Hughes  was able to establish the explicit formula
in the range , where is an explicit rational function of for each fixed . The functions as introduced by Hughes  are given in the following:
where , This sequence is continuous, and it is believed that both the nominator and denominator are monic polynomials in . Using (1.10) and the definitions of the functions , we can obtain the values of for . As indicated in  Hughes  evaluated the first four functions and then writes a numerical experiment suggesting the next three. The values of for have been collected in . To the best of my knowledge there is no explicit formula to find the values of the function for . This limitation of the values of leads to the limitation of the values of the lower bound between the zeros of the Riemann zeta-function by applying the moments (1.9). To overcame this restriction, we will use a different explicit formula of the moments to establish new values of the distance between zeros.
Conrey et al.  established the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and used this to formulate a conjecture for the moments of the derivative of the Riemann zeta-function on the critical line. Their method depends on the fact that the distribution of the eigenvalues of unitary matrices gives insight into the distribution of zeros of the Riemann zeta-function and the values of the characteristic polynomials of the unitary matrices give a model for the value distribution of the Riemann zeta-function. Their formulae are expressed in terms of a determinant of a matrix whose entries involve the -Bessel function and, alternately, by a combinatorial sum. They conjectured that
where is the arithmetic factor and defined as in (1.8) and
and denotes the set of partitions of into nonnegative parts. They also gave some explicit values of for . These values will be presented in Section 2 and will be used to establish the main results in this paper.
The third direction in the studies of the zeros of the Riemann zeta-function is the gaps between the zeros (finding small gaps and large gaps between the zeros) on the critical line when the Riemann hypothesis is satisfied. In the present paper we are concerned with the largest gaps between the zeros on the critical line assuming that the Riemann hypothesis is true.
Assuming the truth of the Riemann hypothesis Montgomery  studied the distribution of pairs of nontrivial zeros and and conjectured, for fixed α, satisfying , that
This so-called pair correlation conjecture plays a complementary role to the Riemann hypothesis. This conjecture implies the essential simplicity hypothesis that almost all zeros of the zeta-function are simple. On the other hand, the integral on the right hand side is the same as the one observed in the two-point correlation of the eigenvalues which are the energy levels of the corresponding Hamiltonian which are usually not known with uncertainty. This observation is due to Dyson and it restored some hope in an old idea of Hilbert and Polya that the Riemann hypothesis follows from the existence of a self-adjoint Hermitian operator whose spectrum of eigenvalues correspond to the set of nontrivial zeros of the zeta-function.
Now, we assume that are the zeros of in the upper half-plane (arranged in nondecreasing order and counted according multiplicity) and are consecutive ordinates of all zeros and define
These numbers have received a great deal of attention. In fact, important results concerning the values of them have been obtained by some authors. It is generally believed that and . Selberg  proved that
and the average of is 1. Note that is the average spacing between zeros. Fujii  also showed that there exist constants and such that
for a positive proportion of . Mueller  obtained
assuming the Riemann hypothesis. Montgomery and Odlyzko  showed, assuming the Riemann hypothesis, that
Conrey et al.  improved the bounds in (1.21) and showed that, if the Riemann hypothesis is true, then
Conrey et al.  obtained a new lower bound and proved that
assuming the generalized Riemann hypothesis for the zeros of the Dirichlet -functions. Bui et al.  improved (1.23) and obtained
assuming the Riemann hypothesis. Ng in  improved (1.24) and proved that
assuming the generalized Riemann hypothesis for the zeros of the Dirichlet -functions.
Hall in  (see also Hall ) assumed that is the sequence of distinct positive zeros of the Riemann zeta-function arranged in nondecreasing order and counted according multiplicity and defined the quantity
and showed that , and the lower bound for bear direct comparison with such bounds for dependent on the Riemann hypothesis, since if this were true the distinction between and would be nugatory. Of course and the equality holds if the Riemann hypothesis is true. Hall  used a Wirtinger-type inequality of Beesack and proved that
In  Hall proved a Wirtinger inequality and used the moment
due to Ingham , and the moments
due to Conrey , and obtained
Hall  proved a new generalized Wirtinger-type inequality by using the calculus of variation and obtained a new value of which is given by
Hall  employed the generalized Wirtinger inequality obtained in , simplified the calculus used in  and converted the problem into one of the classical theory of equations involving Jacobi-Schur functions. Assuming that the moments in (1.9) are correctly predicted by RMT, Hall  proved that
In  the authors applied a technique involving the comparison of the continuous global average with local average obtained from the discrete average to a problem of gaps between the zeros of zeta-function assuming the Riemann hypothesis. Using this approach, which takes only zeros on the critical line into account, the authors computed similar bounds under assumption of the Riemann hypothesis when (1.9) holds. They then showed that for fixed positive integer
holds for any for more than proportion of the zeros with a computable constant .
The improvement of this value as obtained in  is given by
In this paper, first we apply some well-known Wirtinger-type inequalities and the moments of the Hardy -function and the moments of its derivative to establish some explicit formulas for . Using the values of and , we establish some lower bounds for which improves the last value of . In particular it is obtained that which means that consecutive nontrivial zeros often differ by at least 6.1392 times the average spacing. To the best of the author knowledge the last value obtained for in the literature is the value obtained by Hall in (1.35) and nothing is known regarding for .
2. Main Results
In this section, we establish some explicit formulas for and by using the same explicit values of and we establish new lower bounds for . The explicit values of using the formula
are calculated in the following for :
The explicit values of the parameter that has been determined by Conrey et al.  for are given in the following:
Now, we are in a position to prove our first results in this section which gives an explicit formula of the gaps between the zeros of the Riemann zeta-function. This will be proved by applying an inequality due to Agarwal and Pang .
Assuming the Riemann hypothesis, one has
To prove this theorem, we employ the inequality
Now, we follow the proof of  and supposing that is the first zero of not less than and the last zero not greater than . Suppose further that for , we have
and apply the inequality (2.6), to obtain
Since the inequality remains true if we replace by , we have
Summing (2.9) over , applying (1.7), (1.12) and as in , we obtain
This implies that
and then we obtain the desired inequality (2.1). The proof is complete.
Using the values of and and (2.1) we have the new lower values for for in Table 1.
One can easily see that the value of in Table 1 does not improve the lower bound in (1.35) due to Hall, but the the approach that we used is simple and depends only on a well-known Wirtinger-type inequality and the asymptotic formulas of the moments. In the following, we employ a different inequality due to Brnetić and Pečarić  and establish a new explicit formula for and then use it to find new lower bounds.
Assuming the Riemann hypothesis, one has
where is defined by
To prove this theorem, we apply the inequality
that has been proved by Brnetić and Pečarić , where is continuous function on with . Proceeding as in the proof of Theorem 2.1 and employing (2.15), we may have
This implies that
which is the desired inequality (2.13). The proof is complete.
To find the new lower bounds for we need the values of for . These values are calculated numerically in Table 2.
Using these values and the values of ,, and the explicit formula (2.13) we have the new lower bounds for in Table 3.
We note from Table 3 that the value of improves the value that has been obtained by Hall.
Finally, in the following we will employ an inequality to Beesack [32, page 59] and establish a new explicit formula for and use it to find new values of its lower bounds.
Assuming the Riemann hypothesis, one has
To prove this theorem, we apply the inequality
that has been proved by Beesack [32, page 59], where is continuous function on with . Proceeding as in Theorem 2.1 by using (2.19), we may have
This implies that
which is the desired inequity (2.18). The proof is complete.
Using these values and the values of ,, and the explicit formula in (2.18) we have the new lower bounds for in Table 4.
We note from Table 4, that the values of for are compatible with the values of for that has been obtained by Hall [13, Table ] and since there is no explicit value of for , to obtain the values of for the author in  stopped the estimation for for .
We notice that the calculations can be continued as above just if one knows the explicit values of for where the values
are easy to calculate. Note that the values of that we have used in this paper are adapted from the paper by Conrey et al. . It is clear that the values of are increasing with the increase of and this may help in proving the conjecture of the distance between of the zeros of the Riemann zeta-function.
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The author is very grateful to the anonymous referees for valuable remarks and comments which significantly contributed to the quality of the paper. The author thanks Deanship of Scientific Research and the Research Centre in College of Science in King Saud University for encouragements and supporting this project.
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Saker, S. Applications of Wirtinger Inequalities on the Distribution of Zeros of the Riemann Zeta-Function. J Inequal Appl 2010, 215416 (2011). https://doi.org/10.1155/2010/215416
- Explicit Formula
- Critical Line
- Unitary Matrice
- Random Matrix Theory
- Riemann Hypothesis