On the zeros of Dirichlet $L$-functions. III
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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We show, for any q 3 and distinct reduced residues a, b (mod q), the existence of certain hypothetical sets of zeros of Dirichlet L-functions lying off the critical line implies that π(x; q, a) < π(x; q, b) for a set of real x of asymptotic density 1.
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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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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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Random matrix theory has successfully modeled many systems in physics and mathematics, and often analysis in one area guides development in the other. Hughes and Rudnick computed 1-level density statistics for low-lying zeros of the family of primitive Dirichlet L-functions of fixed prime conductor Q, as Q→∞, and verified the unitary symmetry predicted by random matrix theory. We compute 1and 2...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1976
ISSN: 0002-9947
DOI: 10.1090/s0002-9947-1976-0418410-6