The Fourth Moment of Dirichlet L-functions
نویسندگان
چکیده
Here ∑∗ denotes summation over primitive characters χ (mod q), φ(q) denotes the number of primitive characters (mod q), and ω(q) denotes the number of distinct prime factors of q. Note that φ(q) is a multiplicative function given by φ(p) = p − 2 for primes p, and φ(p) = p(1 − 1/p) for k ≥ 2 (see Lemma 1 below). Also note that when q ≡ 2 (mod 4) there are no primitive characters (mod q), and so below we will assume that q 6≡ 2 (mod 4). For q 6≡ 2 (mod 4) it is useful to keep in mind that the main term in (1.1) is ≍ q(φ(q)/q)(log q). Heath-Brown’s result represents a q-analog of Ingham’s fourth moment for ζ(s):
منابع مشابه
The Fourth Moment of Dirichlet L-functions
for certain explicitly computable constants ai. The difficult part of extending Ingham’s result to include the lower-order terms is asymptotically evaluating the off-diagonal terms. The family of all primitive Dirichlet L-functions of modulus q is similar in some ways to the Riemann zeta function in t-aspect, but is more difficult to analyze. In 1981, Heath-Brown obtained an asymptotic formula ...
متن کاملA Note on the Fourth Moment of Dirichlet L-functions
For χ a Dirichlet character (mod q), the moments of L(s, χ) have many applications, for example to the distribution of primes in the arithmetic progressions to modulus q. The asymptotic formula of the fourth power moment in the q-aspect has been obtained by Heath-Brown [1], for q prime, and more recently by Soundararajan [5] for general q. Following Soundararajan’s work, Young [7] pushed the re...
متن کاملStein’s method, Malliavin calculus, Dirichlet forms and the fourth moment theorem
The fourth moment theorem provides error bounds in the central limit theorem for elements of Wiener chaos of any order. It was proved by Nourdin and Peccati [31] using Stein’s method and the Malliavin calculus. It was also proved by Azmoodeh, Campese and Poly [3] using Stein’s method and Dirichlet forms. This paper is an exposition on the connections between Stein’s method and the Malliavin cal...
متن کاملRelative order and type of entire functions represented by Banach valued Dirichlet series in two variables
In this paper, we introduce the idea of relative order and type of entire functions represented by Banach valued Dirichlet series of two complex variables to generalize some earlier results. Proving some preliminary theorems on the relative order, we obtain sum and product theorems and we show that the relative order of an entire function represented by Dirichlet series is the same as that of i...
متن کاملThe First Moment of Quadratic Dirichlet L-functions
We obtain an asymptotic formula for the smoothly weighted first moment of primitive quadratic Dirichlet L-functions at the central point, with an error term that is “square-root” of the main term. Our approach uses a recursive technique that feeds the result back into itself, successively improving the error term.
متن کامل