A note on a conjecture of Gonek. (English) Zbl 1318.11109
Summary: We derive a lower bound for a second moment of the reciprocal of the derivative of the Riemann zeta-function over the zeros of \(\zeta(s)\) that is half the size of the conjectured value. Our result is conditional upon the assumption of the Riemann Hypothesis and the conjecture that the zeros of the zeta-function are simple.
MSC:
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
References:
| [1] | H. M. Bui, M. B. Milinovich, and N. C. Ng, A note on the gaps between consecutive zeros of the Riemann zeta-function , Proc. Amer. Math. Soc. 138 (2010), 4167-4175. · Zbl 1225.11108 · doi:10.1090/S0002-9939-2010-10443-4 |
| [2] | J. B. Conrey, A. Ghosh, and S. M. Gonek, A note on gaps between zeros of the zeta function , Bull. London Math. Soc. 16 (1984), no. 4, 421-424. · Zbl 0536.10033 · doi:10.1112/blms/16.4.421 |
| [3] | J. B. Conrey, A. Ghosh, D. A. Goldston, S. M. Gonek, and D. R. Heath-Brown, On the distribution of gaps between zeros of the zeta-function , Quart. J. Math. Oxford Ser. (2) 36 (1985), no. 141, 43-51. · Zbl 0557.10028 · doi:10.1093/qmath/36.1.43 |
| [4] | Harold Davenport, Multiplicative number theory. Third edition. Revised and with a preface by Hugh L. Montgomery , Graduate Texts in Mathematics, 74. Springer-Verlag, New York, 2000. · Zbl 1002.11001 |
| [5] | S. M. Gonek, On negative moments of the Riemann zeta-function , Mathematika 36 (1989), 71-88. · Zbl 0673.10032 · doi:10.1112/S0025579300013589 |
| [6] | S. M. Gonek, An explicit formula of Landau and its applications to the theory of the zeta function , Contemp. Math 143 (1993), 395-413. · Zbl 0791.11043 · doi:10.1090/conm/143/1210528 |
| [7] | S. M. Gonek, Some theorems and conjectures in the theory of the Reimann zeta-function , unpublished manuscript. · Zbl 0594.10028 |
| [8] | S. M. Gonek, The second moment of the reciprocal of the Riemann zeta-function and its derivative , Talk at Mathematical Sciences Research Institute, Berkeley, June 1999. |
| [9] | C. B. Haselgrove, A disproof of a conjecture of Pólya , Mathematika 5 (1958), 141-145. · Zbl 0085.27102 · doi:10.1112/S0025579300001480 |
| [10] | C. P. Hughes, J. P. Keating, and N. O’Connell, Random matrix theory and the derivative of the Riemann zeta-function , Proc. Roy. Soc. London A 456 (2000), 2611-2627. · Zbl 0996.11052 · doi:10.1098/rspa.2000.0628 |
| [11] | A.E. Ingham, On two conjectures in the theory of numbers , Amer. J. Math. 64 (1942), 313-319. · Zbl 0063.02974 · doi:10.2307/2371685 |
| [12] | M. B. Milinovich, Upper bounds for the moments of \(\zeta'(\rho)\) , Bull. London Math. Soc. 42 (2010), no. 1, 28-44. · Zbl 1223.11102 · doi:10.1112/blms/bdp096 |
| [13] | H. L. Montgomery, The pair correlation of zeros of the zeta function , in: Proc. Sympos. Pure Math. 24, Amer. Math. Soc., Providence, R. I., 1973, 181-193. · Zbl 0268.10023 |
| [14] | H. L. Montgomery and R.C. Vaughan, Hilbert’s inequality , J. London Math. Soc. (2) 8 (1974), 73-82. · Zbl 0281.10021 · doi:10.1112/jlms/s2-8.1.73 |
| [15] | N. Ng, The distribution of the summatory function of the Möbius function , Proc. London Math. Soc. 89 (2004), no. 2, 361-389. · Zbl 1138.11341 · doi:10.1112/S0024611504014741 |
| [16] | N. Ng, The fourth moment of \(\zeta'(\rho)\) , Duke Math. J. 125 (2004), no. 2, 243-266. · Zbl 1083.11056 · doi:10.1215/S0012-7094-04-12522-9 |
| [17] | A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture , J. Reine Angew. Math. 357 (1985), 138-160. · Zbl 0544.10047 · doi:10.1515/crll.1985.357.138 |
| [18] | Z. Rudnick and K. Soundararajan, Lower bounds for moments of L-functions , Proc. Natl. Sci. Acad. USA 102 (2005), 6837-6838. · Zbl 1159.11317 · doi:10.1073/pnas.0501723102 |
| [19] | K. Soundararajan, On the distribution of gaps between zeros of the Riemann zeta-function , Quart. J. Math. Oxford Ser. (2) 42 (1996), no. 3, 383-387. · Zbl 0858.11047 · doi:10.1093/qjmath/47.187.383 |
| [20] | H. M. Stark, On the asymptotic density of the \(k\)-free integers , Proc. Amer. Math. Soc. 17 (1966), 1211-1214. · Zbl 0144.28205 · doi:10.2307/2036123 |
| [21] | E. C. Titchmarsh, The Theory of the Riemann Zeta-function , Oxford University Press, 1986. · Zbl 0601.10026 |
| [22] | K. M. Tsang, Some \(\Omega\)-theorems for the Riemann zeta-function , Acta Arith. 46 (1986), no. 4, 369-395. · Zbl 0606.10031 |
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