Explicit upper bounds for exponential sums over primes

Abstract

Abstract. We give explicit upper bounds for linear trigonometric sums over primes.

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

“…arising from the Plancherel identity and the prime number theorem, one can obtain satisfactory estimates for all minor arcs with q log 8 x (assuming that Q was chosen to be significantly less than x/ log 8 x). To finish the proof of Vinogradov's theorem, one thus needs to obtain good prime number theorems in arithmetic progressions whose modulus q can be as large as log 8 x. While explicitly effective versions of such theorems exist (see e.g.…”

“…and by the constants made explicit (with some slight degradation in the logarithmic exponents) by Daboussi and Rivat [8]. Unfortunately, due to the slow decay of the exp(− 1 2 √ log x) term, this estimate only becomes non-trivial for x 10 184 (see [8, §8]) and is thus not of direct use for smaller values of x.…”

The ternary Goldbach problem

“…Since the major-arc bounds are valid only for q ≤ r = 300000 and |δ| ≤ 4r/q, we cannot afford even a single factor of log x (or any other function tending to ∞ as x → ∞) in front of terms such as x/ q|δ 0 |: a factor like that would make the term larger than the trivial bound x if q|δ 0 | is equal to a constant (r, say) and x is very large. Apparently, there was no such "log-free bound" with explicit constants in the literature, even though such bounds were considered to be in principle feasible, and even though previous work ([Che85], [Dab96], [DR01], [Tao14]) had gradually decreased the number of factors of log x. (In limited ranges for q, there were log-free bounds without explicit constants; see [Dab96], [Ram10].…”

On Bombieri's asymptotic sieve

“…It was presented partially in talks, and we somehow forgot about it. The recent papers [4] and [6] made us believe that it would be a good idea to make this work available. Note in particular that in [6], the sequence f is essentially assumed to be of "dimension" 2, where the dimension comes from (H 1 ): the upper bound we assume is c(Log u/Log v) κ where κ = 1 is (an upper bound of) the dimension.…”