Sieve theory

Sieve theory is a branch of number theory that involves the use of combinatorial methods to count or estimate the size of sets of integers, particularly with respect to divisibility conditions. It is often used to study the distribution of primes and other arithmetic functions. The basic idea is to "sieve" out unwanted elements from a set, such as all multiples of a certain integer, in order to isolate the primes or other numbers of interest.

The Bombieri–Vinogradov theorem is a significant result in analytic number theory, particularly in the study of prime numbers. It provides a statistical estimate for the distribution of prime numbers in arithmetic progressions. More specifically, the theorem states that, under certain conditions, the primes are uniformly distributed among the residues of a given modulus.

Brun's theorem is a result in number theory pertaining to the distribution of prime numbers. Specifically, it relates to the sum of the reciprocals of the prime numbers. The theorem states that the sum of the reciprocals of the twin primes (pairs of primes that differ by two, such as (3, 5) and (11, 13)) converges to a finite value.

The Brun sieve is a mathematical algorithm or method used in number theory, particularly in the context of prime numbers and integer sequences. Named after the mathematician Viggo Brun, it is primarily associated with the sieve method, a classical technique used to filter out numbers that have certain properties—often used to identify prime numbers or to count prime numbers within a given range. The Brun sieve is particularly effective for counting twin primes or other related prime configurations.

The Fundamental Lemma of Sieve Theory is a key result in analytic number theory that underpins the operation of sieve methods, which are techniques used to count or estimate the size of sets of integers that have certain properties, particularly those related to prime numbers. The lemma essentially provides a way to bound the sums that arise in these sieve methods.

The Goldston-Pintz-Yıldırım sieve is a mathematical technique used in number theory, specifically in the field of additive combinatorics and the study of prime numbers. It is a sophisticated sieving method that was developed by mathematicians Daniel Goldston, Jacek Pintz, and Cem Yıldırım in the early 2000s. The primary goal of this sieve is to find and analyze bounded gaps between prime numbers.

The Jurkat–Richert theorem is a result in the field of mathematics, specifically within the context of functional analysis and operator theory. The theorem provides conditions under which certain types of linear operators can be decomposed into simpler components. To be more precise, the Jurkat–Richert theorem typically pertains to the behavior of bounded linear operators on Banach spaces (complete normed vector spaces) and is often discussed in relation to the spectrum of operators and their compactness properties.

The "large sieve" is a powerful tool in analytic number theory used primarily in the study of the distribution of prime numbers and the behavior of arithmetical functions. It is a general method that provides inequalities for the sizes of sets of integers with certain properties, particularly focusing on the distribution of integer sequences modulo various bases.

The Larger Sieve, commonly known in the context of number theory, refers to an advanced mathematical technique used for determining the properties of integers, particularly in relation to prime numbers. It is an extension of the Sieve of Eratosthenes and is particularly useful in analytic number theory and areas dealing with the distribution of prime numbers.

The Legendre sieve is a mathematical algorithm used in number theory for finding prime numbers within a certain range. It is based on the idea of sieving out composite numbers from a list of integers by marking the multiples of each prime number. Here's an overview of how the Legendre sieve works: 1. **Initialization**: You start with a range of integers, such as all integers from \( 2 \) to \( n \), where \( n \) is your upper limit.

In the context of sieve theory, the "parity problem" generally refers to questions about the distribution of prime numbers. More specifically, sieve theory involves methods that can help determine how many integers in a given set meet certain criteria, often in relation to being prime or composite. The parity problem in sieve theory can typically involve exploring the even and odd behavior of prime numbers or their residues modulo some integer. One classic observation related to parity and primes is that all prime numbers except for 2 are odd.

The Selberg sieve is a mathematical tool used in number theory, particularly in the field of prime number theory and in the study of additive number theory. It is named after the mathematician A. Selberg, who introduced it as a method for estimating the number of integers that are free of large prime factors or, more generally, to sieve out integers that are not divisible by a specified set of primes.