Applications of Wirtinger Inequalities on the Distribution of Zeros of the Riemann Zeta-Function
© Samir H. Saker. 2010
Received: 10 October 2010
Accepted: 17 December 2010
Published: 4 January 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 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].
where is the Barnes -function (for the definition of the Barnes -function and its properties, we refer to ).
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.
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.
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.
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
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 .
and then we obtain the desired inequality (2.1). The proof is complete.
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.
which is the desired inequality (2.13). The proof is complete.
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.
which is the desired inequity (2.18). The proof is complete.
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 .
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.
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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