Large Character Sums: Burgess’s Theorem and Zeros of L-functions
نویسندگان
چکیده
We study the conjecture that ∑ n≤x χ(n) = o(x) for any primitive Dirichlet character χ (mod q) with x ≥ q , which is known to be true if the Riemann Hypothesis holds for L(s, χ). We show that it holds under the weaker assumption that ‘100%’ of the zeros of L(s, χ) up to height 14 lie on the critical line. We also establish various other consequences of having large character sums; for example, that if the conjecture holds for χ then it also holds for χ.
منابع مشابه
On the maximum number of consecutive integers on which a character is constant
Let χ be a non-principal Dirichlet character to the prime modulus p. In 1963, Burgess showed that the maximum number of consecutive integers H for which χ remains constant is O ( p1/4 log p ) . This is the best known asymptotic upper bound on this quantity. Recently, McGown proved an explicit version of Burgess’s theorem, namely that H < 7.06p1/4 log p for p ≥ 5 ·1018. Preobrazhenskaya, in the ...
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