In this article we answer a question proposed by Gelfond in 1968. We prove that the sum of digits of prime numbers written in a basis q> 2 is equidistributed in arithmetic progressions (except for some well known degenerate cases). We prove also that the sequence.˛sq.p/ / where p runs through the prime numbers is equidistributed modulo 1 if and only if ˛ 2 � n �.

1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exist infinitely many primes p for which #Ep(Fp) has at most 5 prime factors. We also obtain upper bounds for the number of primes p ≤ x for which #Ep(Fp) is a prime. 1

Abstract. We develop novel techniques using only abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through “congruence ” subgroups. We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, f(c, d) = c2 + d2, but restrict (c, d) to the orbit O = (0, 1)Γ, where Γ is an infinite-index non-elementary finitely-generated subgroup of SL(2, Z) containing unipotent elements. We show that the set of values f(O) contains infinitely many integers having at most R prime factors for any R> 4/(δ−θ), where θ> 1/2 is the spectral gap and δ < 1 is the Hausdorff dimension of the limit set of Γ. If δ> 149/150, then we can take θ = 5/6, giving R = 25. The limit of this method is R = 9 for δ − θ> 4/9. This is the same number of prime factors as attained in Brun’s original attack on the twin prime conjecture. 1.

The Twin Prime Conjecture states that there are infinitely many primes p such that p + 2 is also prime. A refined version of this conjecture is that π2(x), the number of prime twins lying below a level x, satisfies

Probabilistic methods are used to prove that for every E> 0 there exists a sequence A, of squares such that every positive integer is the sum of at most four squares in A, and A,(x) = 0(x=8 + y. Key words and phrases: Sums of squares, additive bases, probabilistic methods in additive number theory. The set A of positive integers is a basis of order h if every positive integer is the sum of at most h elements of A. Lagrange proved in 1770 that the set of squares is a basis of order 4. Let A(x) denote the number of elements of

ABSTRACT. In these lectures we give an overview of the circle method introduced by Hardy and Ramanujan at the beginning of the twentieth century, and developed by Hardy, Littlewood and Vinogradov, among others. We also try and explain the main difficulties in proving Goldbach’s conjecture and we give a sketch of the proof of Vinogradov’s three-prime Theorem. 1. ADDITIVE PROBLEMS In the last few centuries many additive problems have come to the attention of mathematicians: famous examples are Waring’s problem and Goldbach’s conjecture. In general, an additive problem can be expressed in the following form: we are given s ≥ 2 subsets of the set of natural numbers N, not necessarily distinct, which we call A1,..., As. We would like to determine the number of solutions of the equation n = a1 + a2 + ·· · + as (1.1) for a given n ∈ N, with the constraint that a j ∈ A j for j = 1,..., s, or, failing that, we would like to prove that the same equation has at least one solution for “sufficiently large ” n. In fact, we can not expect, in general, that for very small n there will be a solution of equation (1.1). Furthermore, depending on the nature of the sets A j, there may be some arithmetical constraints

These are notes of a series of lectures on sieves, presented during the Special

In this article we give a proof of Serre’s conjecture for the cases of odd conductor and even conductor semistable at 2, and arbitrary weight. Our proof in both cases will not depend on any yet unproved generalization of Kisin’s modularity lifting results to characteristic 2 (moreover, we will not consider at all characteristic 2 representations in any step of our proof). The new key ingredient is the use of Sophie Germain primes to perform an efficient weight reduction (“Sophie Germain’s weight reduction”), which is combined with the methods and results of previous articles on Serre’s conjecture by Khare, Wintenberger, and myself. 1

Let E be an elliptic curve over Q. In 1988, 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 and 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 over a two-parameter family of elliptic curves. One of the key ingredients in the proof is a short average distribution result in the style of Barban-Davenport-Halberstam,

Abstract. We prove theorems of Landau and Hardy-Littlewood type for Goldbach, Chen, Lemoime-Levy and other binary partitions of positive integers. We also pose some new conjectures. 1.