Exponential sums over primes and the prime twin problem

Abstract

The purpose of this paper is to obtain an effective estimate of the exponential sum n x Λ(n)e a q + β n (where e(α) = e 2πiα , α, β ∈ R, (a, q) = 1 and Λ is the von Mangoldt function) in the range (log x) 1/2+ε q x (log log log x) 1+ε and |β| < 1 q(log log log x) 1+ε . It improves Daboussi's estimate [2, Theorem 1] in the range q (log x) D and x(log x) −D q, D > 0 and is valid in a wider range for β.

Every odd number greater than $1$ is the sum of at most five primes

“…By using subsequent numerical verifications of both the even Goldbach conjecture and the Riemann hypothesis, it is possible to increase this lower threshold somewhat, but there is still a very significant gap between the lower and upper thresholds for which the odd Goldbach conjecture is known. 2 To explain the reason for this, let us first quickly recall how Vinogradov-type theorems are proven. To represent a number x as the sum of three primes, it suffices to obtain a sufficiently non-trivial lower bound for the sum n 1 ,n 2 ,n 3 :n 1 +n 2 +n 3 =x Λ(n 1 )Λ(n 2 )Λ(n 3 ), where Λ is the von Mangoldt function (see Section 2 for definitions).…”

The tail of the singular series for the prime pair and Goldbach problems

“…Bounds of this type are useful in applications related to both the Goldbach and prime pair conjectures. For a recent application, see [1]. The proof of (1.10) is rather complicated and left open the question of whether the result can be improved further or is best possible.…”

The ternary Goldbach problem

“…2a,2b] was almost log-free, but not quite. There were also bounds [Kar93], [But11] that used L-functions, and thus were not really useful in a truly minor-arc regime. )…”