Explicit $L^2$ bounds for the Riemann $\zeta $ function

Daniele Dona; Harald A. Helfgott; Sebastian Zuniga Alterman

doi:10.5802/jtnb.1194

Explicit

L^{2}

bounds for the Riemann

ζ

function

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

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 1, pp. 91-133.

Résumé
Abstract

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 $0 < σ \leq 1$ . 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 $0 \leq σ \leq 1$ .

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 $0 < σ \leq 1$ 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 $0 \leq σ \leq 1$ .

Reçu le : 2020-08-15
Révisé le : 2021-09-24
Accepté le : 2021-10-29
Publié le : 2022-07-07

DOI : 10.5802/jtnb.1194

Classification : 11M06
Mots-clés : Riemann zeta function,

L^{2}

norm, mean square bounds, explicit bounds, mean value theorem.

Affiliations des auteurs :

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@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, Tome 34 (2022) no. 1, pp. 91-133. doi : 10.5802/jtnb.1194. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1194/

Bibliographie
Cité par

[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 $| ζ (\frac{1}{2} + i t) |$ , 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 $ζ (1 / 2 + i t)$ , 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 $| ζ (\frac{1}{2} + i t) |$ , 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

Aleksander Simonič; Valeriia V. Starichkova Atkinson's formula for the mean square of $ζ (s)$ with an explicit error term, Journal of Number Theory, Volume 244 (2023), pp. 111-168 | DOI:10.1016/j.jnt.2022.09.010 | Zbl:1509.11081
Ramachandran Balasubramanian; Olivier Ramaré; Priyamvad Srivastav Explicit bounds for products of primes in AP, Mathematics of Computation, Volume 92 (2023) no. 343, pp. 2381-2411 | DOI:10.1090/mcom/3853 | Zbl:1535.11129
Daniele Dona; Sebastian Zuniga Alterman On the Atkinson formula for the $ζ$ function, Journal of Mathematical Analysis and Applications, Volume 516 (2022) no. 2, p. 34 (Id/No 126478) | DOI:10.1016/j.jmaa.2022.126478 | Zbl:1512.11062

Cité par 3 documents. Sources : zbMATH