Publication

Quantitative asymptotics for polynomial patterns in the primes

2026

Abstract

We prove quantitative estimates for averages of the von Mangoldt and Möbius functions along polynomial progressions for a large class of polynomials . The error terms obtained save an arbitrary power of logarithm, matching the classical Siegel–Walfisz error term. These results give the first quantitative bounds for the Tao–Ziegler polynomial patterns in the primes result. The proofs are based on a quantitative generalised von Neumann theorem of Peluse, a recent result of Leng on strong bounds for the Gowers uniformity of the primes, and analysis of a ‘Siegel model’ for the von Mangoldt function along polynomial progressions.

Official source

https://infoscience.epfl.ch/entities/publication/f14b11e9-32b8-4127-a733-02c20ed56f67

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2026

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/f14b11e9-32b8-4127-a733-02c20ed56f67

About this result

Related concepts (32)

Arithmetic function

In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n.

Möbius function

The Möbius function μ(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted μ(x). For any positive integer n, define μ(n) as the sum of the primitive nth roots of unity.

Chebyshev function

In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function θ (x) or θ (x) is given by where denotes the natural logarithm, with the sum extending over all prime numbers p that are less than or equal to x. The second Chebyshev function ψ (x) is defined similarly, with the sum extending over all prime powers not exceeding x where Λ is the von Mangoldt function.

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