Given a graph G we study the chromatic number of a random graph G1/2 obtained from G by removing each edge of G independently with probability 1/2. Studying χ(G1/2) has been suggested by Bukh [Buk], who asked whether E[χ(G1/2)] > Ω(χ(G)/ log(χ(G))) holds for all graphs G. In this paper we prove several results related this problem. Denoting the chromatic number of G by k = χ(G) we prove the fol...

Every Hecke character of K satisfying (1.1) and (1.2) is actually a quadratic twist of a canonical Hecke character (see Section 2 for a precise description of these characters and which fields have them). Let L(s, χ) denote the Hecke L-function of χ, and Λ(s, χ) its completion; Λ(s, χ) satisfies the functional equation Λ(s, χ) = W (χ)Λ(2 − s, χ), where W (χ) = ±1 is the root number. If χ is a c...

Every Hecke character of K satisfying (1.1) and (1.2) is actually a quadratic twist of a canonical Hecke character (see Section 2 for a precise description of these characters and which fields have them). Let L(s, χ) denote the Hecke L-function of χ, and Λ(s, χ) its completion; Λ(s, χ) satisfies the functional equation Λ(s, χ) = W (χ)Λ(2 − s, χ), where W (χ) = ±1 is the root number. If χ is a c...

Let q ≥ 2 and N ≥ 1 be integers. W. Zhang recently proved that for any fixed ε > 0 and q ≤ N ≤ q1/2−ε , ∑ χ 6=χ0 ∣∣∣∣ N ∑ n=1 χ(n) ∣∣∣∣2|L(1, χ)|2 = (1+ o(1))αq q N , where the sum is taken over all nonprincipal characters χ modulo q, L(1, χ) denotes the L-functions corresponding to χ , and αq = qo(1) is some explicit function of q. Here we improve this result and show that the same asymptotic ...

Studies have revealed that elevated homocysteine levels can cause damage to motor neurons through multiple neurotoxic mechanisms, thus leading to the pathogenesis of amyotrophic lateral sclerosis (ALS). One way by which homocysteine levels are increased in the body is the consequence of methylenetetrahydrofolate reductase (MTHFR) gene polymorphisms. Therefore, to address this question, we studi...

Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/mfms/notes 2015-16/06d functional equation.pdf] 1. L(s, χ) for even χ 2. L(s, χ) for odd χ We prove the analytic continuation and function equation of Dirichlet L-functions L(s, χ) imitating the argument Riemann used for proving the analytic continuation of ζ(s) and its funct...