SIMPLE ZEROS OF PRIMITIVE DIRICHLET L-FUNCTIONS AND THE ASYMPTOTIC LARGE SIEVE
نویسندگان
چکیده
منابع مشابه
Real zeros of quadratic Dirichlet L - functions
A small part of the Generalized Riemann Hypothesis asserts that L-functions do not have zeros on the line segment ( 2 , 1]. The question of vanishing at s = 2 often has deep arithmetical significance, and has been investigated extensively. A persuasive view is that L-functions vanish at 2 either for trivial reasons (the sign of the functional equation being negative), or for deep arithmetical r...
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Results in the other direction were obtained by Turán (1974, see []) by the application of his powersum method. We just cite one example: Theorem (Turán): Suppose the existence of constants α ≥ 2, 0 < β ≤ 1 and c(α, β) so that τ > c(α, β) the inequality ∣∣∣∣∣ ∑ N1≤p≤N2 exp(−iτ log p) ∣∣∣∣∣ ≤ N log N τβ holds for all N1, N2 integers with τ α ≤ N ≤ N1 < N2 ≤ 2N ≤ exp(τ). Then ζ(s) 6= 0 on the seg...
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This Research in Teams meeting focused on the finer behaviour of the function π(x; q, a), which denotes the number of prime numbers of the form qn+ a that are less than or equal to x. Dirichlet’s famous theorem on primes in arithmetic progressions asserts that that there are infinitely many primes of the form qn + a when a is a reduced residue modulo q (that is, when a and q are relatively prim...
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Let χ be a real odd Dirichlet character of modulus d, and let L(s, χ) be the associated Dirichlet L-function. As a consequence of the work of Low and Purdy, it is known that if d ≤ 800 000 and d 6= 115 147, 357 819, 636 184, then L(s, χ) has no positive real zeros. By a simple extension of their ideas and the advantage of thirty years of advances in computational power, we are able to prove tha...
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ژورنال
عنوان ژورنال: The Quarterly Journal of Mathematics
سال: 2013
ISSN: 0033-5606,1464-3847
DOI: 10.1093/qmath/hat008