Complex Moments and the Distribution of Values of Over Function Fields With Applications to Class Numbers

Abstract

In this paper we investigate the moments and the distribution of L(1, χ D ), where χ D varies over quadratic characters associated to square-free polynomials D of degree n over F q , as n → ∞. Our first result gives asymptotic formulas for the complex moments of L(1, χ D ) in a large uniform range. Previously, only the first moment has been computed due to work of Andrade and Jung. Using our asymptotic formulas together with the saddle-point method, we show that the distribution function of L(1, χ D ) is very close to that of a corresponding probabilistic model. In particular, we uncover an interesting feature in the distribution of large (and small) values of L(1, χ D ), that is not present in the number field setting. We also obtain Ω-results for the extreme values of L(1, χ D ), which we conjecture to be best possible. Specializing n = 2g + 1 and making use of one case of Artin's class number formula, we obtain similar results for the class number h D associated to F q (T )[ √ D]. Similarly, specializing to n = 2g + 2 we can appeal to the second case of Artin's class number formula and deduce analogous results for h

The Moments and statistical Distribution of Class number of Primes over Function Fields

“…which converges almost surely (see [8], [12] and [15]). Let E[Y ] be the expected value of a random variable Y on Ω that is defined by…”

Moments and distribution of values of $L$-functions over function fields inside the critical strip

“…Tools for proving Proposition 4.2. The proof of Proposition 4.2 depends on two preliminary lemmas, some standard estimates for log cosh(t) and tanh(t) and a generalization of [9,Lemma 4.4].…”

“…Recall that prime in this case means monic and irreducible. Further, set P(n) as the product of all irreducible polynomials P such that deg(P ) ≤ n. The following lemma already appears in [9]; however, we include it here as it is important for the proof of Theorem 1.3. We re-prove it to make use of a key estimate which appears within the proof.…”

“…This paper will discuss the distribution of values for L-functions over function fields, F q (T ), with q ≡ 1 (mod 4). It is a follow-up to the article describing the distribution of values of L(1, χ D ) in [9]. In this setting, the monic elements of A = F q [T ] play the role of the positive integers.…”

Selberg's Central limit theorem for quadratic Dirichlet L-functions over function fields

“…In particular, we prove results for log |L( 12 , χ D )| in the "large degree" aspect, as such we suppress the q in the notation of the set, that is H n,q = H n . For information in the remaining ranges of σ, see work of the second author [17] and [18]. In this setting, Weil [28] has proven the Riemann Hypothesis, so one might hope to provide an unconditional result of this flavour.…”