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- Z Rudnick, K Soundararajan
- Proceedings of the National Academy of Sciences…
- 2005

The moments of central values of families of L-functions have recently attracted much attention and, with the work of Keating and Snaith [(2000) Commun. Math. Phys. 214, 57-89 and 91-110], there are now precise conjectures for their limiting values. We develop a simple method to establish lower bounds of the conjectured order of magnitude for several such… (More)

Throughout this paper d will denote a fundamental discriminant, and χd the associated primitive real character to the modulus |d|. We investigate here the distribution of values of L(1, χd) as d varies over all fundamental discriminants with |d| ≤ x. Our main concern is to compare the distribution of values of L(1, χd) with the distribution of “random Euler… (More)

Here and throughout logj denotes the j-th iterated logarithm, so that log2 = log log, log3 = log log log and so on. Recall that for a real valued function f and a positive function g the symbol f = Ω(g) means that lim supx→∞ |f(x)|/g(x) > 0. We write f = Ω+(g) if lim supx→∞ f(x)/g(x) > 0, and f = Ω−(g) if lim infx→∞ f(x)/g(x) < 0. Lastly f = Ω±(g) means… (More)

Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x + H) − ψ(x), for 0 ≤ x ≤ N , is approximately normal with mean ∼ H and variance ∼ H logN/H , when N ≤ H ≤ N1−δ .

0 |ζ( 1 2 + it)| dt. For positive real numbers k, it is believed that Mk(T ) ∼ CkT (logT ) 2 for a positive constant Ck. A precise value for Ck was conjectured by Keating and Snaith [9] based on considerations from random matrix theory. Subsequently, an alternative approach, based on multiple Dirichlet series and producing the same conjecture, was given by… (More)

- K. SOUNDARARAJAN
- 2008

X |φ(z)|2 dx dy y = 1. Zelditch [19] has shown that as λ → ∞, for a typical Maass form φ the measure μφ := |φ(z)|2 dx dy y approaches the uniform distribution measure 3 π dx dy y . This statement is referred to as “Quantum Ergodicity.” Rudnick and Sarnak [13] have conjectured that an even stronger result holds. Namely, that as λ → ∞, for every Maass form φ… (More)

Improving on a result of J.E. Littlewood, N. Levinson [3] showed that there are arbitrarily large t for which |ζ(1 + it)| ≥ e log2 t + O(1). (Throughout ζ(s) is the Riemann-zeta function, and logj denotes the j-th iterated logarithm, so that log1 n = logn and logj n = log(logj−1 n) for each j ≥ 2.) The best upper bound known is Vinogradov’s |ζ(1 + it)| (log… (More)

In this paper, we study explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specifically, we improve the existing explicit bounds for the least quadratic non-residue and the least prime in an arithmetic progression. We also refine the classical conditional bounds of… (More)

do not seem to obey any simple law.” Following I. Kaplansky, we call a non-negative integer N eligible for a ternary form f(x, y, z) if there are no congruence conditions prohibiting f from representing N. By the classical theory of quadratic forms, it is well known that any given genus of positive definite ternary quadratic forms represents every eligible… (More)

An important problem in number theory asks for asmptotic formulas for the moments of central values of L-functions varying in a family. This problem has been intensively studied in recent years, and thanks to the pioneering work of Keating and Snaith [7], and the subsequent contributions of Conrey, Farmer, Keating, Rubinstein and Snaith [1], and Diaconu,… (More)