Le nombre de zéros and la distribution des parties réelles des zéros non-réelles de la dérivée de la fonction zêta de Riemann a été étudiée par B. C. Berndt, N. Levinson, H. L. Montgomery, H. Akatsuka et l’auteure. Berndt, Levinson et Montgomery ont étudié le cas inconditionnel, alors qu’Akatsuka et l’auteure ont donné de meilleures estimations sous l’hypothèse de Riemann. Récemment F. Ge a amélioré l’estimation du nombre de zéros par Akatsuka. Dans cet article nous montrons des résultats similaires relatifs à la dérivée des fonctions de Dirichlet associées aux caractères primitifs de Dirichlet, sous l’hypothèse de Riemann généralisée.
The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by B. C. Berndt, N. Levinson, H. L. Montgomery, H. Akatsuka, and the author. Berndt, Levinson, and Montgomery investigated the unconditional case, while Akatsuka and the author gave sharper estimates under the truth of the Riemann hypothesis. Recently, F. Ge improved the estimate on the number of zeros shown by Akatsuka. In this paper, we prove similar results related to the first derivative of Dirichlet -functions associated with primitive Dirichlet characters under the assumption of the generalized Riemann hypothesis.
Accepté le :
Publié le :
Mots clés : Dirichlet $L$-functions, first derivative, zeros
CC-BY-ND 4.0
@article{JTNB_2017__29_2_471_0,
author = {Ade Irma Suriajaya},
title = {Two estimates on the distribution of zeros of the first derivative of {Dirichlet} $L$-functions under the generalized {Riemann} hypothesis},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {471--502},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {29},
number = {2},
year = {2017},
doi = {10.5802/jtnb.988},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.988/}
}
TY - JOUR AU - Ade Irma Suriajaya TI - Two estimates on the distribution of zeros of the first derivative of Dirichlet $L$-functions under the generalized Riemann hypothesis JO - Journal de théorie des nombres de Bordeaux PY - 2017 SP - 471 EP - 502 VL - 29 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.988/ DO - 10.5802/jtnb.988 LA - en ID - JTNB_2017__29_2_471_0 ER -
%0 Journal Article %A Ade Irma Suriajaya %T Two estimates on the distribution of zeros of the first derivative of Dirichlet $L$-functions under the generalized Riemann hypothesis %J Journal de théorie des nombres de Bordeaux %D 2017 %P 471-502 %V 29 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.988/ %R 10.5802/jtnb.988 %G en %F JTNB_2017__29_2_471_0
Ade Irma Suriajaya. Two estimates on the distribution of zeros of the first derivative of Dirichlet $L$-functions under the generalized Riemann hypothesis. Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 471-502. doi : 10.5802/jtnb.988. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.988/
[1] Conditional estimates for error terms related to the distribution of zeros of , J. Number Theory, Volume 132 (2012) no. 10, pp. 2242-2257 | DOI
[2] Zeros of the first derivative of Dirichlet -functions (2016) (https://arxiv.org/abs/1604.08015)
[3] The number of zeros for , J. Lond. Math. Soc., Volume 2 (1970), pp. 577-580 | DOI
[4] The number of zeros of , Int. Math. Res. Not., Volume 2017 (2017) no. 5, pp. 1578-1588
[5] Zeros of the derivatives of the Riemann zeta-function, Acta Math., Volume 133 (1974), pp. 49-65 | DOI
[6] Errata of Multiplicative Number Theory I: Classical Theory (http://www-personal.umich.edu/~hlm/mnt1err.pdf)
[7] Multiplicative Number Theory I: Classical Theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, 2006, xvii+552 pages
[8] Contributions to the theory of Dirichlet’s -functions, Skr. Norske Vid.-Akad., Oslo I., Volume 1946 (1946) no. 3, pp. 1-62
[9] Geometrisches zur Riemannschen Zetafunktion, Math. Ann., Volume 110 (1934), pp. 514-521 | DOI
[10] On the zeros of the -th derivative of the Riemann zeta function under the Riemann Hypothesis, Funct. Approx. Comment. Math., Volume 53 (2015), pp. 69-95 | DOI
[11] The theory of functions, Oxford University Press, 1939, x+454 pages
[12] The theory of the Riemann zeta-function, Oxford Science Publications, Oxford University Press, 1986, x+412 pages
[13] Zeros of derivatives of Dirichlet -functions, Turk. J. Math., Volume 21 (1997) no. 2, pp. 521-534
[14] On the zeros of near the critical line, Duke Math. J., Volume 110 (2001) no. 3, pp. 55-572 | DOI
Cité par Sources :