Abstract. In this paper we prove two results concerning Vinogradov's three primes theorem with primes that can be called almost twin primes. First, for any m, every sufficiently large odd integer N can be written as a sum of three primes p 1 , p 2 and p 3 such that, for each i ∈ {1, 2, 3}, the interval [p i , p i + H] contains at least m primes, for some H = H(m). Second, every sufficiently large integer N ≡ 3 (mod 6) can be written as a sum of three primes p 1 , p 2 and p 3 such that, for each i ∈ {1, 2, 3}, p i + 2 has at most two prime factors.

Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limitations. For a fixed residue class a we extend such averages out to moduli ≤ x 20 39 −δ .

Let X k denote the number of k-term arithmetic progressions in a random subset of Z/N Z or {1, . . . , N } where every element is included independently with probability p. We determine the asymptotics of log P(X k ≥ (1 + δ)EX k ) (also known as the large deviation rate) where p → 0 with p ≥ N −c k for some constant c k > 0, which answers a question of Chatterjee and Dembo. The proofs rely on the recent nonlinear large deviation principle of Eldan, which improved on earlier results of Chatterjee and Dembo. Our results complement those of Warnke, who used completely different methods to estimate, for the full range of p, the large deviation rate up to a constant factor.

Abstract-We present algorithms for the discrete cosine transform (DCT) and discrete sine transform (DST), of types II and III, that achieve a lower count of real multiplications and additions than previously published algorithms, without sacrificing numerical accuracy. Asymptotically, the operation count is reduced from 2N log 2 N + O(N ) to

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes in A. The constant 5/8 in this statement is the best possible.