Bombieri–Vinogradov Theorem
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of
analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...
, obtained in the mid-1960s, concerning the distribution of primes in
arithmetic progression An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...
s, averaged over a range of moduli. The first result of this kind was obtained by Mark Barban in 1961 and the Bombieri–Vinogradov theorem is a refinement of Barban's result. The Bombieri–Vinogradov theorem is named after
Enrico Bombieri Enrico Bombieri (born 26 November 1940) is an Italian mathematician, known for his work in analytic number theory, Diophantine geometry, complex analysis, and group theory. Bombieri is currently professor emeritus in the School of Mathematics ...
and A. I. Vinogradov, who published on a related topic, the density hypothesis, in 1965. This result is a major application of the large sieve method, which developed rapidly in the early 1960s, from its beginnings in work of
Yuri Linnik Yuri Vladimirovich Linnik (; January 8, 1915 – June 30, 1972) was a Soviet mathematician active in number theory, probability theory and mathematical statistics. Biography Linnik was born in Bila Tserkva, in present-day Ukraine. He went to ...
two decades earlier. Besides Bombieri,
Klaus Roth Klaus Friedrich Roth (29 October 1925 – 10 November 2015) was a German-born British mathematician who won the Fields Medal for proving Roth's theorem on the Diophantine approximation of algebraic numbers. He was also a winner of the De ...
was working in this area. In the late 1960s and early 1970s, many of the key ingredients and estimates were simplified by
Patrick X. Gallagher Patrick Ximenes Gallagher (January 2, 1935 – March 30, 2019) was an American mathematician who pioneered large sieve, large sieve theory and invented the larger sieve. Biography Early life Patrick Ximenes Gallagher was born on January 2, 1935, ...
.


Statement of the Bombieri–Vinogradov theorem

Let x and Q be any two positive
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s with :x^\log^x \leq Q \leq x^. Then :\sum_\max_\max_\left, \psi(y;q,a)-\=O\left(x^Q(\log x)^5\right)\!. Here \varphi(q) is the
Euler totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In oth ...
, which is the number of summands for the modulus ''q'', and :\psi(x;q,a)=\sum_\Lambda(n), where \Lambda denotes the
von Mangoldt function In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive. Definition The von Mang ...
. A verbal description of this result is that it addresses the error term in the prime number theorem for arithmetic progressions, averaged over the moduli ''q'' up to ''Q''. For a certain range of ''Q'', which are around \sqrt x if we neglect logarithmic factors, the error averaged is nearly as small as \sqrt x. This is not obvious, and without the averaging is about of the strength of the
Generalized Riemann Hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whi ...
(GRH).


See also

* Elliott–Halberstam conjecture (a generalization of Bombieri–Vinogradov) *
Vinogradov's theorem In number theory, Vinogradov's theorem is a result which implies that any sufficiently large odd integer can be written as a sum of three prime numbers. It is a weaker form of Goldbach's weak conjecture, which would imply the existence of such a re ...
(named after
Ivan Matveyevich Vinogradov Ivan Matveevich Vinogradov ( rus, Ива́н Матве́евич Виногра́дов, p=ɪˈvan mɐtˈvʲejɪvʲɪtɕ vʲɪnɐˈɡradəf, a=Ru-Ivan_Matveyevich_Vinogradov.ogg; 14 September 1891 – 20 March 1983) was a Soviet mathematician ...
) *
Siegel–Walfisz theorem In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel to primes in arithmetic progressions. It is a refinement both of the prime number theorem and of Dirichlet ...


Notes


External links

*
''The Bombieri-Vinogradov Theorem''
R.C. Vaughan's Lecture note. {{DEFAULTSORT:Bombieri-Vinogradov theorem Sieve theory Theorems in analytic number theory