[Le biais de Chebyshev pour les formes modulaires]
We study Chebyshev’s bias for the signs of Fourier coefficients of cuspidal newforms on $\Gamma _0(N)$. Our main result shows that the bias towards either sign is completely determined by the order of vanishing of the $L$-function $L(s, f)$ at the central point of the critical strip. We then give several examples of modular forms where we explicitly compute the order of vanishing of $L(s, f)$ at the central point and as a by-product, verify the super-positivity property, in the sense of Yun–Zhang (2017), for these examples.
Nous étudions le biais de Chebyshev pour les signes des coefficients de Fourier des formes nouvelles cuspidales sur $\Gamma _0(N)$. Notre résultat principal montre que le biais vers l’un ou l’autre signe est entièrement déterminé par l’ordre de disparition de la fonction $L(s, f)$ au point central de la bande critique. Nous donnons ensuite plusieurs exemples de formes modulaires pour lesquelles nous calculons explicitement l’ordre de disparition de $L(s, f)$ au point central et, comme produit dérivé, nous vérifions la propriété de super-positivité, au sens de Yun–Zhang (2017), pour ces exemples.
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Keywords: Chebsyhev’s bias, modular forms, $L$-functions
Mots-clés : Biais de Chebyshev, formes modulaires, fonctions $L$
Shin-ya Koyama 1 ; Arshay Sheth 2 , 3
CC-BY 4.0
Shin-ya Koyama; Arshay Sheth. Chebyshev’s bias for modular forms. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 589-598. doi: 10.5802/crmath.846
@article{CRMATH_2026__364_G3_589_0,
author = {Shin-ya Koyama and Arshay Sheth},
title = {Chebyshev{\textquoteright}s bias for modular forms},
journal = {Comptes Rendus. Math\'ematique},
pages = {589--598},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {364},
doi = {10.5802/crmath.846},
language = {en},
}
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