An average form of Artin's conjecture

The Lang-Trotter Conjecture on Average

For an elliptic curve E over Q and an integer r let πr E(x) be the number of primes p ≤ x of good reduction such that the trace of the Frobenius morphism of E/Fp equals r. We consider the quantity π r E(x) on average over certain sets of elliptic curves. More in particular, we establish the following: If A,B > x1/2+ε and AB > x3/2+ε, then the arithmetic mean of πr E(x) over all elliptic curves ...

Average Twin Prime Conjecture for Elliptic Curves

Let E be an elliptic curve over Q. In 1988, N. Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy–Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s Conjecture is still widely open. In this paper we prove that Koblitz’s Conjecture is true on average...

On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysang’s Conjecture

The purpose of this paper is twofold. First we derive theoretically, using appropriate transformation on x(n), the closed-form solution of the nonlinear difference equation x(n+1) = 1/(±1 + x(n)), n ∈ N_0. The form of solution of this equation, however, was first obtained in [10] but through induction principle. Then, with the solution of the above equation at hand, we prove a case ...

An Unsual form of Measles Meningoencephalitis

Towards an ‘average’ Version of the Birch and Swinnerton-dyer Conjecture

The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s, E). Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a ...