- Research Article
- Open Access

# Applications of Wirtinger Inequalities on the Distribution of Zeros of the Riemann Zeta-Function

- SamirH Saker
^{1, 2}Email author

**2010**:215416

https://doi.org/10.1155/2010/215416

© Samir H. Saker. 2010

**Received: **10 October 2010

**Accepted: **17 December 2010

**Published: **4 January 2011

## Abstract

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.

## Keywords

## 1. Introduction

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 [2] and Cheer and Goldston [3].

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 [7] of Ivić and the papers by Kim [8–11].

for positive constants and will be defined later.

where is the Barnes -function (for the definition of the Barnes -function and its properties, we refer to [5]).

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 [13] Hughes [5] evaluated the first four functions and then writes a numerical experiment suggesting the next three. The values of for have been collected in [6]. 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.

*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.

assuming the generalized Riemann hypothesis for the zeros of the Dirichlet -functions.

holds for any for more than proportion of the zeros with a computable constant .

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 [30].

Theorem 2.1.

Proof.

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 [31] and establish a new explicit formula for and then use it to find new lower bounds.

Theorem 2.2.

Proof.

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.

Theorem 2.3.

Proof.

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 [13] 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. [4]. 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.

## Declarations

### Acknowledgments

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.

## Authors’ Affiliations

## References

- Garunkštis R, Steuding J: Simple zeros and discrete moments of the derivative of the Riemann zeta-function.
*Journal of Number Theory*2005, 115(2):310–321. 10.1016/j.jnt.2004.12.006MathSciNetView ArticleMATHGoogle Scholar - Conrey JB: More than two fifths of the zeros of the Riemann zeta function are on the critical line.
*Journal für die Reine und Angewandte Mathematik*1989, 399: 1–26.MathSciNetMATHGoogle Scholar - Cheer AY, Goldston DA: Simple zeros of the Riemann zeta-function.
*Proceedings of the American Mathematical Society*1993, 118(2):365–372. 10.1090/S0002-9939-1993-1132849-0MathSciNetView ArticleMATHGoogle Scholar - Conrey JB, Rubinstein MO, Snaith NC: Moments of the derivative of characteristic polynomials with an application to the Riemann zeta function.
*Communications in Mathematical Physics*2006, 267(3):611–629. 10.1007/s00220-006-0090-5MathSciNetView ArticleMATHGoogle Scholar - Hughes CPThesis, University of Bristol, Bristol, UK, 2001 Thesis, University of Bristol, Bristol, UK, 2001Google Scholar
- Steuding J: The Riemann zeta-function and moment conjectures from random matrix theory.
*Mathematica Slovaca*2009, 59(3):323–338. 10.2478/s12175-009-0129-0MathSciNetView ArticleMATHGoogle Scholar - Ivić A:
*The Riemann zeta-function*. John Wiley & Sons, New York, NY, USA; 1985:xvi+517.MATHGoogle Scholar - Kim T: Euler numbers and polynomials associated with zeta functions.
*Abstract and Applied Analysis*2008, 2008:-11.Google Scholar - T. Kim, Barnes-type multiple
-zeta functions and
-Euler polynomials,
*Journal of Physics*, vol. 43,no. 25, 11 pages, 2010.Google Scholar - Kim T: Note on the Euler
-zeta functions.
*Journal of Number Theory*2009, 129(7):1798–1804. 10.1016/j.jnt.2008.10.007MathSciNetView ArticleMATHGoogle Scholar - Kim T: On
-adic interpolating function for
-Euler numbers and its derivatives.
*Journal of Mathematical Analysis and Applications*2008, 339(1):598–608. 10.1016/j.jmaa.2007.07.027MathSciNetView ArticleMATHGoogle Scholar - Keating JP, Snaith NC: Random matrix theory and
.
*Communications in Mathematical Physics*2000, 214(1):57–89. 10.1007/s002200000261MathSciNetView ArticleMATHGoogle Scholar - Hall RR: Large spaces between the zeros of the Riemann zeta-function and random matrix theory. II.
*Journal of Number Theory*2008, 128(10):2836–2851. 10.1016/j.jnt.2007.11.011MathSciNetView ArticleMATHGoogle Scholar - Montgomery HL: The pair correlation of zeros of the zeta function. In
*Analytic Number Theory (Proceedings of Symposia in Pure Mathematics )*.*Volume 24*. American Mathematical Society, Providence, RI, USA; 1973:181–193.Google Scholar - Selberg A: The zeta-function and the Riemann hypothesis.
*Skandinaviske Mathematiker-kongres*1946, 10: 187–200.MathSciNetGoogle Scholar - Fujii A: On the difference between
consecutive ordinates of the zeros of the Riemann zeta function.
*Proceedings of the Japan Academy*1975, 51(10):741–743. 10.3792/pja/1195518466MathSciNetView ArticleMATHGoogle Scholar - Mueller J: On the difference between consecutive zeros of the Riemann zeta function.
*Journal of Number Theory*1982, 14(3):327–331. 10.1016/0022-314X(82)90067-1MathSciNetView ArticleMATHGoogle Scholar - Montgomery HL, Odlyzko AM: Gaps between zeros of the zeta function. In
*Topics in Classical Number Theory, Vol. I, II*.*Volume 34*. North-Holland, Amsterdam, The Netherlands; 1984:1079–1106.Google Scholar - Conrey JB, Ghosh A, Gonek SM: A note on gaps between zeros of the zeta function.
*The Bulletin of the London Mathematical Society*1984, 16(4):421–424. 10.1112/blms/16.4.421MathSciNetView ArticleMATHGoogle Scholar - Conrey JB, Ghosh A, Gonek SM: Large gaps between zeros of the zeta-function.
*Mathematika*1986, 33(2):212–238. 10.1112/S0025579300011219MathSciNetView ArticleMATHGoogle Scholar - Bui HM, Milinovich MB, Ng NC: A note on the gaps between consecutive zeros of the Riemann zeta-function.
*Proceedings of the American Mathematical Society*2010, 138(12):4167–4175. 10.1090/S0002-9939-2010-10443-4MathSciNetView ArticleMATHGoogle Scholar - Ng N: Large gaps between the zeros of the Riemann zeta function.
*Journal of Number Theory*2008, 128(3):509–556. 10.1016/j.jnt.2007.03.011MathSciNetView ArticleMATHGoogle Scholar - Hall RR: The behaviour of the Riemann zeta-function on the critical line.
*Mathematika*1999, 46(2):281–313. 10.1112/S0025579300007762MathSciNetView ArticleMATHGoogle Scholar - Hall RR: Generalized Wirtinger inequalities, random matrix theory, and the zeros of the Riemann zeta-function.
*Journal of Number Theory*2002, 97(2):397–409. 10.1016/S0022-314X(02)00005-7MathSciNetView ArticleMATHGoogle Scholar - Hall RR: A Wirtinger type inequality and the spacing of the zeros of the Riemann zeta-function.
*Journal of Number Theory*2002, 93(2):235–245. 10.1006/jnth.2001.2719MathSciNetView ArticleMATHGoogle Scholar - Ingham AE: Mean theorems in the theorem of the Riemann zeta-function.
*Proceedings London Mathematical Society*1928, 27: 273–300. 10.1112/plms/s2-27.1.273MathSciNetView ArticleMATHGoogle Scholar - Conrey JB: The fourth moment of derivatives of the Riemann zeta-function.
*The Quarterly Journal of Mathematics*1988, 39(153):21–36.MathSciNetView ArticleMATHGoogle Scholar - Hall RR: Large spaces between the zeros of the Riemann zeta-function and random matrix theory.
*Journal of Number Theory*2004, 109(2):240–265. 10.1016/j.jnt.2004.01.007MathSciNetView ArticleMATHGoogle Scholar - Steuding R, Steuding J: Large gaps between zeros of the zeta-function on the critical line and moment conjectures from random matrix theory.
*Computational Methods and Function Theory*2008, 8(1–2):121–132.MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Pang PYH: Remarks on the generalizations of Opial's inequality.
*Journal of Mathematical Analysis and Applications*1995, 190(2):559–577. 10.1006/jmaa.1995.1091MathSciNetView ArticleMATHGoogle Scholar - Brnetić I, Pečarić J: Some new Opial-type inequalities.
*Mathematical Inequalities & Applications*1998, 1(3):385–390.MathSciNetView ArticleMATHGoogle Scholar - Beesack PR: Hardy's inequality and its extensions.
*Pacific Journal of Mathematics*1961, 11: 39–61.MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.