Goldbach's ternary problem

The ternary Golbach problem is the assertion that every odd integer $N\geq 7$ is the sum of three primes. It was first proved for large enough integers by I. M. Vinogradov in 1937, and the proof is a good starting point for the study of the Hardy-Littlewood circle method and of Vinogradov's method for estimating exponential sums over prime numbers, such as

\[\begin{eqnarray}\sum_{p\leq N}e(\alpha p);\quad e(x):=e^{2\pi i x}.\quad \quad(1)\end{eqnarray}\]

The full ternary problem was solved only in 2013 in a paper by H. Helfgott. In this post, we present the circle method and show how it can be used to resolve the ternary Golbach problem for large enough integers, assuming an estimate for the prime exponential sum $(1)$. This estimate is due to R. C. Vaughan and is based on his refined version of Vinogradov's method. In a later post, we prove this estimate, so the results here serve as an illustration of the capability of Vinogradov's method. We follow Vinogradov's book Method of Trigonometrical Sums in the Theory of Numbers, but replacing the application of Page's theorem by Siegel's theorem allows us to avoid some technicalities and improve the error terms.

Character sums and Pólya-Vinogradov inequality

In this post, we consider character sums, which are one of the central objects in the discrete Fourier analysis of the integers modulo $q$. We will prove the Pólya-Vinogradov estimate, which we will observe to be the best possible estimate up to a logarithmic factor, despite being easy to prove. Character sums arise in many problems in number theory, and we utilize them to prove classical bounds for the least quadratic nonresidue and primitive root modulo $q$. We also show that the generalized Riemann hypothesis reduces the bounds for the least quadratic nonresidue and primitive root considerably.

Selberg's upper bound sieve

In this post, we derive Selberg's upper bound sieve. Selberg's sieve is a combinatorial sieve based  on the simple but immensely useful idea of introducing a large number of parameters into a combinatorial sieve inequality and optimizing them. We apply this sieve for example to prove the Brun-Titchmarsh inequality for primes in short intervals belonging to an arithmetic progression, as well as to prove good upper bounds for the number of representations of an integer as the sum of two primes, or the number of primes $ap+b$ where $p\leq x$ is prime (the bounds are conjectured to be sharp up to a constant factor). In a later post, we  establish a corresponding lower bound sieve. We follow the classical book of Halberstam and Richert.

Fourier transform and its mapping properties

We present some classical results about the Fourier transform in $\mathbb{R}^n$ in this post, and some of them will be applied in later posts to partial differential equations, additive number theory, and other topics. In particular, we consider the validity of the Fourier inversion theorem in various forms, the $L^2$ theory of the Fourier transform and the mapping properties of the Fourier transform in many spaces.