Inspired by the analogy between the group of units F × p of the finite field with p elements and the group of points E(Fp) of an elliptic curve E/Fp, E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum p≤x τ (p + a) ∼ Cx. In this paper, we present a comprehensive study of the constants C(E) emerging in the asymptotic study of these elliptic curve divisor sums. Specifically, by analyzing the division fields of an elliptic curve E/Q, we prove upper bounds for the constants C(E) and, in the generic case of a Serre curve, we prove explicit closed formulae for C(E) amenable to concrete computations. Moreover, we compute the moments of the constants C(E) over two-parameter families of elliptic curves E/Q. Our methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics. ,p (E) are uniquely determined positive integers satisfying d 1,p | d 2,p . Determining the asymptotic behaviour of sums over p ≤ x of arithmetic functions evaluated at the elementary divisors d 1,p and d 2,p may be viewed as Titchmarsh divisor problems for elliptic curves. Such problems unravel striking similarities, but also intriguing contrasts, to the original Titchmarsh divisor problem, as illustrated in [AkGh], [AkFe], [Co], [CoMu], [Fe], [FeMu], [FrKu], [FrPo], [Ki], [Ko], [Po], and [Wu].The focus of our paper is on the constants emerging in the following three Titchmarsh divisor problems for elliptic curves. In all the expressions below, the letters p and ℓ denote primes, with p being a prime of good reduction for the given elliptic curve.
Conjecture 1. (Kowalski [Ko, Section 3.2])Let E/Q be an elliptic curve. Then, as x → ∞, 3