Theorem by Bombieri and Vinogradov

The theorem of Bombieri and Vinogradow is a proven 1965 theorem of the analytical number theory of Enrico Bombieri and Askold Ivanovich Vinogradow (sometimes it is also named after Bombieri).

It makes statements about the error term in the statement made in Dirichlet's prime number theorem about the distribution of prime numbers less than or equal in arithmetic progressions . Averaging is carried out over the modulus of the progressions (moduli with a natural number ). For values close to , the error term is on the order of magnitude except for logarithmic factors . Without the averaging over the moduli, the proposition would be just as powerful as the generalized Riemannian conjecture . ${\ displaystyle x}$ ${\ displaystyle q \ leq Q}$ ${\ displaystyle Q}$ ${\ displaystyle Q}$ ${\ displaystyle x}$ ${\ displaystyle {\ sqrt {x}}}$

The proof is an application of the large sieve , with mean values estimated from Dirichlet characters.

It represents a considerable improvement of Siegel-Walfisz's theorem. The theorem corresponds to the conjecture of Elliott and Halberstam for the case (for the definition of the parameter, see there), which thus generalizes the theorem in a certain way (the full conjecture applies to the case ). ${\ displaystyle 0 <\ theta <{\ tfrac {1} {2}}}$ ${\ displaystyle \ theta}$ ${\ displaystyle 0 <\ theta <1}$

Yōichi Motohashi showed in 1976 that the analog of the Bombieri and Winogradow theorem also applies to arithmetic functions that can be represented as linear combinations of Dirichlet convolution of two sequences of complex numbers with certain additional properties. The original theorem of Bombieri and Winogradow is the special case of the Mangoldt function .

formulation

Be

{\ displaystyle E (x; q) = \ max _ {{\ text {gcd}} (a, q) = 1} \ left | \ psi (x; q, a) - {\ frac {x} {\ varphi (q)}} \ right |}

With

{\ displaystyle \ psi (x; q, a) = \ sum _ {n \ leq x \ atop n \ equiv a {\ bmod {q}}} \ Lambda (n),}

with the Mangoldt function and Euler's Phi function . ${\ displaystyle \ Lambda}$ ${\ displaystyle \ varphi}$

Then, according to Bombieri and Winogradow:

{\ displaystyle {\ frac {1} {Q}} \ sum _ {q \ leq Q} \ max _ {y <x} \ E (y; q) = {\ frac {1} {Q}} \ sum _ {q \ leq Q} \ max _ {y <x} \ max _ {1 \ leq a \ leq q \ atop (a, q) = 1} \ left | \ psi (y; q, a) - { y \ over \ varphi (q)} \ right | = O \ left (x ^ {1/2} (\ log x) ^ {5} \ right).}

for solid with ${\ displaystyle A> 0}$

{\ displaystyle x ^ {1/2} \ log ^ {- A} x \ leq Q \ leq x ^ {1/2}.}

${\ displaystyle O}$ denotes the Landau symbol for comparing the growth of two functions.

literature

Harold Davenport : Multiplicative Number Theory, 2nd Edition, Springer 1980, Chapter 28
AI Vinogradov: The density hypothesis for Dirichlet L-series. Izv. Akad. Nauk SSSR Ser. Mat., Vol. 29, 1965, pp. 903-934, Correction Vol. 30, 1966, pp. 719-720.
E. Bombieri: On the large sieve, Mathematika, Volume 12, 1965, pp. 201-225
E. Bombieri: Le Grand Crible dans la Théorie Analytique des Nombres, 2nd edition, Astérisque, Volume 18, 1987

Web links

Bombieri's theorem, Mathworld
Bombieri Vinogradov theorem , blog of Terence Tao

Individual evidence

^ Motohashi, An induction principle for the generalization of Bombieri's prime number theorem, Proc. Japan Academy, Volume 52, 1976, pp. 273-275, see also Bombieri, Friedlander, Iwaniec, Primes in arithmetic progressions to large moduli, Acta Mathematica, Volume 156, 1986, p. 206
^ John Friedlander, Henryk Iwaniec, Opera di Cribro, American Mathematical Society Colloquium Publ., 2010, p. 168

[1] Motohashi, An induction principle for the generalization of Bombieri's prime number theorem, Proc. Japan Academy, Volume 52, 1976, pp. 273-275, see also Bombieri, Friedlander, Iwaniec, Primes in arithmetic progressions to large moduli, Acta Mathematica, Volume 156, 1986, p. 206

[2] John Friedlander, Henryk Iwaniec, Opera di Cribro, American Mathematical Society Colloquium Publ., 2010, p. 168