Limiting Distributions of the Classical Error Terms of Prime Number Theory

Abstract

ABSTRACT. Let φ : [0, ∞) → R and let y 0 be a non-negative constant. Let (λ n ) n∈N be a nondecreasing sequence of positive numbers which tends to infinity, let (r n ) n∈N be a complex sequence, and c a real number. Assume that φ is square-integrable on [0, y 0 ] and for y ≥ y 0 , φ can be expressed as φ(y) = c + ℜ λn≤X r n e iλny + E(y, X),We prove that, under certain assumptions on the exponents λ n and the coefficients r n , φ(y) is a B 2 -almost periodic function and thus possesses a limiting distribution. Also if {λ n } n∈N is linearly independent over Q, we explicitly calculate the Fourier transform of the limiting distribution measure. Moreover, we prove general versions of the above results for vector-valued functions. Finally, we illustrate some applications of our general theorems by applying them to several classical error terms which occur in prime number theory. Examples include the error term in the prime number theorem for an automorphic L-function, weighted sums of the Möbius function, weighted sums of the Liouville function, the sum of the Möbius function in an arithmetic progression, and the error term in Chebotarev's density theorem.

A conditional determination of the average rank of elliptic curves

“…This follows from the widely believed Katz-Sarnak density conjecture, which asserts that this family has orthogonal symmetry. The family of all Weierstrass curves (1) is also believed to have orthogonal symmetry, and hence it is believed that the average rank of all elliptic curves ordered by height should also be 1 2 . In the family of quadratic twists of a fixed elliptic curve, Goldfeld [12] showed under the Riemann Hypothesis for elliptic curve L-functions, which we will denote by ECRH (see below), that the average analytic rank is at most 3.25.…”

“…For technical reasons, we will mostly consider the values of d that are coprime with N E . Our first main result is a conditional proof that the average analytic rank of E d is exactly 1 2 . Here and throughout, * d will denote a sum over square-free integers d and N (D) will denote the number of square-free integers 0 < |d| D with (d, N E ) = 1.…”

“…Theorem 1.7. Assume ECRH + and Hypothesis M(δ, η) for some 0 < δ < 1 and 0 < η < 1 2 , for some non-negative Schwartz weight w with w(0) > 0. Then, for any fixed > 0, we have…”

“…In particular, the proportion of elliptic curves E d with 0 < |d| D square-free for which the Birch and Swinnerton-Dyer Conjecture does not hold is D − min(δ/4,η)+o (1) . Remark 1.9.…”

“…A precise prediction of the conductor, root number, and Gamma Factors of L(Sym k E, s) is given in [6]. The Riemann Hypothesis for L(Sym k E, s + k/2) = L(s, Sym k f E ) states that all non-trivial zeros of this function have real part 1 2 . The automorphy of L(Sym k E, s) is currently known for 1 k 4 by the work of Gelbart and Jacquet [11], Kim and Shahidi [24,25], and Kim [23].…”