Explicit L 2 bounds for the Riemann ζ function
Daniele Dona1; Harald A. Helfgott1; Sebastian Zuniga Alterman2
1 Mathematisches Institut Georg-August-Universität Göttingen Bunsenstraße 3-5 37073 Göttingen, Germany
2 Institut de Mathématiques de Jussieu Université Paris Diderot P7, Bâtiment Sophie Germain 8 Place Aurélie Nemours 75013 Paris, France
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 91-133.

Explicit bounds on the tails of the zeta function ζ are needed for applications, notably for integrals involving ζ on vertical lines or other paths going to infinity. Here we bound weighted L 2 norms of tails of ζ.

Two approaches are followed, each giving the better result on a different range. The first one is inspired by the proof of the standard mean value theorem for Dirichlet polynomials. The second approach, superior for large T, is based on classical lines, starting with an approximation to ζ via Euler–Maclaurin.

Both bounds give main terms of the correct order for 0<σ≤1 and are strong enough to be of practical use for the rigorous computation of improper integrals.

We also present bounds for the L 2 norm of ζ in [1,T] for 0≤σ≤1.

Des bornes explicites pour la fonction zêta ζ loin de la droite réelle sont nécessaires pour des applications, notamment aux intégrales de ζ sur des lignes verticales ou bien sur d’autres chemins. Ici, nous bornons des normes L 2 ponderées de la fonction zêta loin de la droite réelle.

Nous suivons deux approches, chacune donnant le meilleur ré-sultat dans un certain rang. La première est inspirée par le théo-rème de la valeur moyenne pour les polynômes de Dirichlet. La deuxième, supérieure pour T grand, est basée sur des résultats classiques, en commençant par une approximation de ζ via la formule d’Euler–Maclaurin.

Ces bornes donnent toutes les deux des termes principaux d’or-dre correct pour 0<σ≤1. Elles sont assez fortes pour être d’utilité pratique dans le calcul numérique rigoureux d’intégrales impropres.

Nous présentons aussi des bornes pour la norme L 2 de ζ dans [1,T] pour 0≤σ≤1.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1194
Classification: 11M06
Keywords: Riemann zeta function, $L^2$ norm, mean square bounds, explicit bounds, mean value theorem.
Daniele Dona 1; Harald A. Helfgott 1; Sebastian Zuniga Alterman 2

1 Mathematisches Institut Georg-August-Universität Göttingen Bunsenstraße 3-5 37073 Göttingen, Germany
2 Institut de Mathématiques de Jussieu Université Paris Diderot P7, Bâtiment Sophie Germain 8 Place Aurélie Nemours 75013 Paris, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{JTNB_2022__34_1_91_0,
     author = {Daniele Dona and Harald A. Helfgott and Sebastian Zuniga Alterman},
     title = {Explicit $L^2$ bounds for the {Riemann} $\zeta $ function},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {91--133},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {34},
     number = {1},
     year = {2022},
     doi = {10.5802/jtnb.1194},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1194/}
}
TY  - JOUR
AU  - Daniele Dona
AU  - Harald A. Helfgott
AU  - Sebastian Zuniga Alterman
TI  - Explicit $L^2$ bounds for the Riemann $\zeta $ function
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2022
SP  - 91
EP  - 133
VL  - 34
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1194/
DO  - 10.5802/jtnb.1194
LA  - en
ID  - JTNB_2022__34_1_91_0
ER  - 
%0 Journal Article
%A Daniele Dona
%A Harald A. Helfgott
%A Sebastian Zuniga Alterman
%T Explicit $L^2$ bounds for the Riemann $\zeta $ function
%J Journal de théorie des nombres de Bordeaux
%D 2022
%P 91-133
%V 34
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1194/
%R 10.5802/jtnb.1194
%G en
%F JTNB_2022__34_1_91_0
Daniele Dona; Harald A. Helfgott; Sebastian Zuniga Alterman. Explicit $L^2$ bounds for the Riemann $\zeta $ function. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 91-133. doi : 10.5802/jtnb.1194. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1194/

[1] Handbook of mathematical functions with formulas, graphs, and mathematical tables. 10th printing, with corrections (Milton Abramowitz; Irene A. Stegun, eds.), National Bureau of Standards Applied Mathematics Series, John Wiley & Sons, 1972 | DOI | Zbl

[2] George E. Andrews; Richard Askey; Ranjan Roy Special Functions, Encyclopedia of Mathematics and Its Applications, 71, Cambridge University Press, 1999 | DOI

[3] Frederick V. Atkinson The mean value of the Riemann zeta function, Acta Math., Volume 81 (1949), pp. 353-376 | DOI | MR | Zbl

[4] Ralf J. Backlund Über die Nullstellen der Riemannschen Zetafunktion, Acta Math., Volume 41 (1918), pp. 345-375 | DOI | Zbl

[5] Ramachandran Balasubramanian An improvement on a theorem of Titchmarsh on the mean square of |ζ(1 2+it)|, Proc. Lond. Math. Soc., Volume 36 (1978), pp. 540-576 | DOI | MR

[6] Jörg Brüdern Einführung in die analytische Zahlentheorie, Springer, 1995 | DOI

[7] Yuanyou F. Cheng; Sidney W. Graham Explicit estimates for the Riemann zeta function, Rocky Mt. J. Math., Volume 34 (2004) no. 4, pp. 1261-1280 | MR | Zbl

[8] Kevin Ford Vinogradov’s integral and bounds for the Riemann zeta function, Proc. Lond. Math. Soc., Volume 85 (2002) no. 3, pp. 565-633 | DOI | MR | Zbl

[9] Anton Good Ein Ω-Resultat für das quadratische Mittel der Riemannschen Zetafunktion auf der kritischen Linie, Invent. Math., Volume 41 (1977), pp. 233-251 | DOI | MR | Zbl

[10] Godfrey H. Hardy; John E. Littlewood Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Math., Volume 41 (1918), pp. 119-196 | DOI | MR

[11] Godfrey H. Hardy; John E. Littlewood The zeros of Riemann’s zeta-function on the critical line, Math. Z., Volume 10 (1921), pp. 283-317 | DOI | MR | Zbl

[12] Godfrey H. Hardy; John E. Littlewood The approximate functional equation in the theory of the Zeta-function, with applications to the divisor problems of Dirichlet and Piltz, Proc. Lond. Math. Soc., Volume 21 (1922), pp. 39-74 | MR | Zbl

[13] David R. Heath-Brown The mean value theorem for the Riemann zeta-function, Mathematika (1978), pp. 177-184 | DOI | MR

[14] Harald A. Helfgott The ternary Goldbach conjecture to appear in Annals of Mathematics Studies, https://webusers.imj-prg.fr/~harald.helfgott/anglais/book.html (version 09/2019)

[15] Ghaith A. Hiary An explicit van der Corput estimate for ζ(1/2+it), Indag. Math., Volume 27 (2016) no. 2, pp. 524-533 | DOI | MR | Zbl

[16] Albert E. Ingham Mean-value theorems in the theory of the Riemann zeta-function, Proc. Lond. Math. Soc., Volume 27 (1928), pp. 273-300 | DOI | MR

[17] Henryk Iwaniec; Emmanuel Kowalski Analytic number theory, Colloquium Publications, 53, American Mathematical Society, 2004

[18] Fredrik Johansson Numerical integration in arbitrary-precision ball arithmetic, Mathematical software – ICMS 2018 (Lecture Notes in Computer Science), Volume 10931, Springer, 2018, pp. 255-263 | DOI | Zbl

[19] Habiba Kadiri A zero density result for the Riemann zeta function, Acta Arith., Volume 160 (2013) no. 2, pp. 185-200 | DOI | MR | Zbl

[20] Edmund Landau Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, 1909

[21] A. F. Lavrik On the principal term in the divisor problem and the power series of the Riemann zeta-function in a neighborhood of its pole, Proc. Steklov Inst. Math., Volume 142 (1976), pp. 165-173 (in Russian)

[22] R. Sherman Lehman On the distribution of zeros of the Riemann zeta-function, Proc. Lond. Math. Soc., Volume 20 (1970) no. 2, pp. 303-320 | DOI | MR | Zbl

[23] John E. Littlewood On the zeros of the Riemann zeta-function, Cambr. Phil. Soc. Proc., Volume 22 (1924), pp. 295-318 | DOI | Zbl

[24] Kohji Matsumoto The mean square of the Riemann zeta-function in the critical strip, Jap. J. Math., New Ser., Volume 15 (1989) no. 1, pp. 1-13 | DOI | MR | Zbl

[25] Kohji Matsumoto Recent Developments in the Mean Square Theory of the Riemann Zeta and Other Zeta-Functions, Number theory (Trends in Mathematics), Birkhäuser, 2000, pp. 241-286 | DOI | Zbl

[26] Kohji Matsumoto; Tom Meurman The mean square of the Riemann zeta-function in the critical strip III, Acta Arith., Volume 64 (1993) no. 4, pp. 357-382 | DOI | MR | Zbl

[27] Hugh L. Montgomery Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, 227, Springer, 1971 | DOI

[28] Hugh L. Montgomery; Robert C. Vaughan Hilbert’s Inequality, J. Lond. Math. Soc., Volume 8 (1974) no. 2, pp. 73-81 | DOI | MR | Zbl

[29] Hugh L. Montgomery; Robert C. Vaughan Multiplicative number theory: I. Classical theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, 2007

[30] David J. Platt; Timothy S. Trudgian An improved explicit bound on |ζ(1 2+it)|, J. Number Theory, Volume 147 (2015), pp. 842-851 | DOI | MR

[31] Emmanuel Preissmann Sur une inégalité de Montgomery–Vaughan, Enseign. Math., Volume 30 (1984), pp. 95-113 | MR | Zbl

[32] Olivier Ramaré An explicit density estimate for Dirichlet L-series, Math. Comput., Volume 85 (2016) no. 297, pp. 325-356 | DOI | MR | Zbl

[33] Olivier Ramaré; P. Akhilesh Explicit averages of non-negative multiplicative functions: going beyond the main term, Colloq. Math., Volume 147 (2017) no. 2, pp. 275-313 | DOI | MR | Zbl

[34] Reinhold Remmert Classical Topics in Complex Function Theory, Graduate Texts in Mathematics, 172, Springer, 1998 | DOI

[35] Aleksander Simonič Explicit zero density estimate for the Riemann zeta-function near the critical line, J. Math. Anal. Appl., Volume 491 (2020) no. 1, 124303, 40 pages | MR | Zbl

[36] Edward C. Titchmarsh On van der Corput’s method and the zeta-function of Riemann. V, Q. J. Math, Volume 5 (1934), pp. 195-210 | DOI | Zbl

[37] Edward C. Titchmarsh The theory of the Riemann Zeta-function, Oxford Science Publications, Clarendon Press, 1986

Cited by Sources: