Chebyshev's bias for modular forms

2025/09/04 by Koyama, Shin-ya, Sheth, Arshay
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.04187

Abstract

We study Chebyshev's bias for the signs of Fourier coefficients of cuspidal newforms on Γ₀(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.

Citations

• Chebyshev's bias for Fermat curves of prime degree
• Chebyshev's Bias for Ramanujan's τ-function via the Deep Riemann Hypothesis
• Chebyshev's Bias against Splitting and Principal Primes in Global Fields

Related

• On the non-vanishing of modular L-values and fourier coefficients of cusp forms
• Fourier coefficients and cuspidality of modular forms: a new approach
• Signs of Fourier coefficients of half-integral weight modular forms
• Large Fourier coefficients of half-integer weight modular forms
• Fourier expansions at cusps