Logic, Semantics, Metamathematics. By A. Tarski. Translated by J. H. Woodger. Pp. 471. 60s. 1956. (Clarendon Press)

Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N →∞, for the number of integer points n ∈ ℤ d ∩ K for which the integers ψ 1 (n), ... , ψ t (n) are simultaneously prime. This implies many other well-known conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime. In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms ψ 1 , ... , ψ t are affinely related; this excludes the important "binary" cases such as the twin prime or Goldbach conjectures, but does allow one to count "nondegenerate" configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the inverse Gowers-norm conjecture (GI(s)) and the Mobius and nilsequences conjecture (MN(s)), where s ∈ {1,2,...} is the complexity of the system and measures the extent to which the forms ψ i depend on each other. The case s = 0 is somewhat degenerate, and follows from the prime number theorem in APs. Roughly speaking, the inverse Gowers-norm conjecture GI(s) asserts the Gowers U s+1 -norm of a function f: [N] → [-1, 1] is large if and only if f correlates with an s-step nilsequence, while the Mobius and nilsequences conjecture MN(s) asserts that the Mobius function μ is strongly asymptotically orthogonal to s-step nilsequences of a fixed complexity. These conjectures have long been known to be true for s = 1 (essentially by work of Hardy-Littlewood and Vinogradov), and were established for s = 2 in two papers of the authors. Thus our results in the case of complexity s ≤ 2 are unconditional. In particular we can obtain the expected asymptotics for the number of 4-term progressions p 1 < p 2 < p 3 < p 4 ≤ N of primes, and more generally for any (nondegenerate) problem involving two linear equations in four prime unknowns.

https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n3-p08-p.pdf

Linear equations in primes

On the Waring–Goldbach problem for fourth and sixth powers

The undecidability of the additive theory of prime numbers (with identity) as well as the theory Th(N, +, n → p_n), where pn denotes the (n + 1)-th prime, are open questions. In a first part, we show the undecidability of Th(N, +, n → nf(n)) where f is a good approximation of the enumeration n → p_n/n. In a second part, as a possible approach, we extend the former theory by adding some extra function. In this direction we show the undecidability of the existential part of the theory Th(N, +, n → p_n, n → r_n), where r_n is the remainder of pn divided by n in the euclidian division.

L'indecidabilite de la theorie additive des nombres premiers ainsi que de la theorie Th(N, +, n → p_n), ou p_n designe le (n+1)-ieme premier, sont deux questions ouvertes. Dans une premiere partie, nous montrons l'indecidabilite de Th(N, +, n → nf(n)) ou f est une bonne approximation de la fonction n → p_n/n des nombres premiers. Dans une seconde partie, nous etendons la premiere theorie en lui ajoutant une fonction supplementaire et nous montrons l'indecidabilite de la theorie Th(N, +, n → p_n, n → r_n), ou r_n designe le reste de p_n dans la division euclidienne de p_n par n, et meme de sa seule partie existentielle.

https://hal.archives-ouvertes.fr/hal-00096769/document

On the Additive Theory of Prime Numbers

where α is a real number, k is a positive integer, e(z) = exp(2πiz), and the summation is over prime numbers. This sum was introduced as a tool in analytic number theory by I. M. Vinogradov in the late 1930’s. In 1937, Vinogradov developed an ingenious new method for estimating sums over primes and applied that method to obtain the first unconditional estimate for f(α) with k = 1. That estimate is the main novelty in his celebrated proof [25] that every sufficiently large odd integer is the sum of three primes. In the sharper form given in [27, Chapter 6], Vinogradov’s result states (essentially) that if a and q are integers satisfying

/pdf/on-weyl-sums-over-primes-and-almost-primes-6gzttppabe.pdf

On Weyl sums over primes and almost primes

Decidability of the theory of the natural integers with the cantor pairing function and the successor

On arithmetical first-order theories allowing encoding and decoding of lists

This paper gives a perfect, ideal, discretization of continuous notions. This is a very convenient frame to treat continuous problems or theories with the help of a computer. This is illustrated by the conversion of algorithms using real numbers into algorithms using integers only and the founding of discrete geometry.

Back and forth between continuous and discrete for the working computer scientist

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S002248120001495X

Denis Richard

Papers

On the Additive Theory of Prime Numbers

Decidability of the theory of the natural integers with the cantor pairing function and the successor

On arithmetical first-order theories allowing encoding and decoding of lists

Back and forth between continuous and discrete for the working computer scientist

Undecidable Extensions of Skolem Arithmetic