منابع مشابه
Large Character Sums
where χ is a non-principal Dirichlet character χ (mod q). It is easy to show that such character sums are always ≤ q in absolute value, while G. Pólya and I.M. Vinogradov (see [3]) improved this to≤ √q log q around 1919, and H.L. Montgomery and R.C. Vaughan [13] to √q log log q in 1977, assuming the Generalized Riemann Hypothesis (GRH). Up to the constant this is “best possible” since R.E.A.C. ...
متن کاملThe Resonance Method for Large Character Sums
We consider the size of large character sums, proving new lower bounds for ∆(N, q) = supχ6=χ0 mod q | ∑ n<N χ(n)| in almost all ranges of N . The proofs use the resonance method and saddle point analysis.
متن کاملLarge Character Sums: Pretentious Characters and the Pólya-vinogradov Theorem
There has been no subsequent improvement in this inequality other than in the implicit constant. Moreover it is believed that (1.1) will be difficult to improve since it is possible (though highly unlikely) that there is an infinite sequence of primes q ≡ 3 (mod 4) for which (pq ) = 1 for all p < q , in which case M(( · q )) √ q log q. The unlikely possibility described above involves a quadrat...
متن کامل2 3 M ar 1 99 9 LARGE CHARACTER SUMS
where χ is a non-principal Dirichlet character χ (mod q). It is easy to show that such characters sums are always ≤ q in absolute value, while G. Pólya and I.M. Vinogradov [3] improved this to ≤ √q log q around 1919, and H.L. Montgomery and R.C. Vaughan [13] to ≪ √q log log q assuming the Generalized Riemann Hypothesis (GRH), in 1977. Up to the constant this is “best possible” since R.E.A.C. Pa...
متن کاملLarge Character Sums: Burgess’s Theorem and Zeros of L-functions
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, ...
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ژورنال
عنوان ژورنال: Journal of the American Mathematical Society
سال: 2000
ISSN: 0894-0347,1088-6834
DOI: 10.1090/s0894-0347-00-00357-x