Minor arcs for Goldbach's problem

Abstract

The ternary Goldbach conjecture states that every odd number n>=7 is the sum of three primes. The estimation of sums of the form \sum_{p\leq x} e(αp), α= a/q + O(1/q^2), has been a central part of the main approach to the conjecture since (Vinogradov, 1937). Previous work required q or x to be too large to make a proof of the conjecture for all n feasible. The present paper gives new bounds on minor arcs and the tails of major arcs. This is part of the author's proof of the ternary Goldbach conjecture. The new bounds are due to several qualitative improvements. In particular, this paper presents a general method for reducing the cost of Vaughan's identity, as well as a way to exploit the tails of minor arcs in the context of the large sieve.

79 pages; third version. (A couple of explanatory paragraphs have been added.)

References in corpus (4)

• The ternary Goldbach conjecture is true
• A Density Version of the Vinogradov Three Primes Theorem
• Numerical Computations Concerning the GRH
• Every odd number greater than 1 is the sum of at most five primes

Cited by in corpus (15)

• The ternary Goldbach conjecture is true
• A Density Version of the Vinogradov Three Primes Theorem
• Short effective intervals containing primes
• A computational history of prime numbers and Riemann zeros
• Numerical Verification of the Ternary Goldbach Conjecture up to 8.875e30
• Updating the error term in the prime number theorem
• Numerical Computations Concerning the GRH
• Every odd number greater than 1 is the sum of at most five primes
• Prime solutions to polynomial equations in many variables and differing degrees
• The ternary Goldbach problem with primes in positive density sets
• A New Proof of Vinogradov's Three Primes Theorem
• Reasoning about Primes (II)
• Dealing with prime numbers I.: On the Goldbach conjecture
• A Vinogradov-type problem in almost primes
• Finite Prime Distance Graphs and 2-Odd Graphs