Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line
Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 2, pp. 285-305.

Nous établissons des bornes inférieures inconditionnelles pour certains moments discrets de la fonction zêta de Riemann et de ses dérivées dans la bande critique. Nous utilisons ces moments discrets pour donner des bornes inférieures inconditionnelles pour les moments continus I k,l (T)= 0 T |ζ (l) (1 2+it)| 2k dt, où l est un entier positif et k1 un nombre rationnel. En particulier, ces bornes inférieures sont de l’ordre de grandeur attendu pour I k,l (T).

We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments I k,l (T)= 0 T |ζ (l) (1 2+it)| 2k dt, where l is a non-negative integer and k1 a rational number. In particular, these lower bounds are of the expected order of magnitude for I k,l (T).

DOI : 10.5802/jtnb.836
Thomas Christ 1 ; Justas Kalpokas 2

1 Department of Mathematics Würzburg University Emil-Fischer-Str. 40, 97074 Würzburg
2 Faculty of Mathematics and Informatics Vilnius University Naugarduko 24, 03225 Vilnius, Lithuania
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Thomas Christ; Justas Kalpokas. Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line. Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 2, pp. 285-305. doi : 10.5802/jtnb.836. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.836/

[1] T. Christ, J. Kalpokas, Upper bounds for discrete moments of the derivatives of the riemann zeta-function on the critical line, Lith. Math. Journal 52 (2012) 233–248. | MR

[2] R.D. Dixon, L. Schoenfeld, The size of the Riemann zeta-function at places symmetric with respect to the point 1/2, Duke Math. J. 33 (1966), 291–292. | MR | Zbl

[3] H.M. Edwards, Riemann’s zeta function, Academic Press, New York 1974. | MR | Zbl

[4] S.M. Gonek, Mean values of the Riemann zeta-function and its derivatives, Invent. Math. 75:1 (1984), 123–141. | MR | Zbl

[5] J. Gram, Sur les zéros de la fonction ζ(s) de Riemann, Acta Math. 27 (1903), 289–304. | MR

[6] G.H. Hardy, J.E. Littlewood, Contributions to the theory of the Riemann zeta-function, Proc. Royal Soc. (A) 113 (1936), 542–569.

[7] D. R. Heath-Brown, Fractional Moments of the Riemann Zeta-Function J. London Math. Soc. 2-24 (1981) 65–78. | MR | Zbl

[8] A.E. Ingham, Mean-value theorems in the theory of the Riemann zeta-function, Proc. London Math. Soc. (2) 27 (1926), 273–300. | MR

[9] A. Ivić, The Riemann zeta-function, John Wiley & Sons, New York 1985. | MR | Zbl

[10] J. Kalpokas, J. Steuding, On the Value-Distribution of the Riemann Zeta-Function on the Critical Line, Moscow Jour. Combinatorcs and Number Theo. 1 (2011), 26–42. | MR

[11] J. Kalpokas, M. Korolev, J. Steuding, Negative values of the Riemann Zeta-Function on the Critical Line, preprint, available at arXiv:1109.2224.

[12] M.B. Milinovich, N. Ng, Lower bound for the moments of ζ (ρ), preprint, available at arXiv:0706.2321v1.

[13] M.B. Milinovich, Moments of the Riemann zeta-function at its relative extrema on the critical line, preprint, available at arXiv:1106.1154. | MR

[14] N. Ng, A discrete mean value of the derivative of the Riemann zeta function, Mathematika 54 (2007), 113–155. | MR | Zbl

[15] M. Radziwill, The 4.36th moment of the Riemann Zeta-function, Int. Math. Res. Not. 18 (2012), 4245–4259. | MR

[16] K. Ramachandra, Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series , Hardy-Ramanujan J. I 1 (1978), 1–15. | MR | Zbl

[17] Z. Rudnick, K. Soundararajan, Lower bounds for moments of L-functions, Proc. Natl. Sci. Acad. USA 102 (2005), 6837–6838. | MR | Zbl

[18] R. Spira, An inequality for the Riemann zeta function, Duke Math. J. 32 (1965), 247–250. | MR | Zbl

[19] K. Soundararajan Moments of the Riemann zeta function, Ann. Math. (2) 170, No. 2, 981-993 (2009) | MR | Zbl

[20] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge studies in advanced mathematics 46, Cambridge University Press 1995. | MR | Zbl

[21] E.C. Titchmarsh, The Riemann zeta-function, 2nd edition, revised by D.R. Heath-Brown, Oxford University Press 1986. | MR

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