Theoretical analysis reveals that large Dirichlet coefficient sums repel low-lying L-function zeros from the critical line, indicating stronger bounds on zero distributions.
Let L(s,π )=∑ ₙ₌₁∞λ π(n)n⁻ˢ L ( s , π ) = ∑ n = 1 ∞ λ π ( n ) n - s be an L -function that satisfies a weak form of the generalized Ramanujan conjecture. We prove that large partial sums of λ π(n) λ π ( n ) strongly repel the low-lying zeros of L(s,π ) L ( s , π ) away from the critical line. Our results extend and quantitatively improve preceding work of Granville and Soundararajan.
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Kerr et al. (2026) studied this question.
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