Zeros of L-functions and large partial sums of Dirichlet coefficients

Abstract

Abstract Let $$L(s,\pi )=\sum _{n=1}^{\infty }\lambda _{\pi }(n)n^{-s}$$ 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 $$\lambda _{\pi }(n)$$ λ π ( n ) strongly repel the low-lying zeros of $$L(s,\pi )$$ L ( s , π ) away from the critical line. Our results extend and quantitatively improve preceding work of Granville and Soundararajan.