View source for Bombieri prime number theorem

For many important problems in the theory of numbers one needs information about the average distribution of prime numbers in arithmetic progressions (cf. also [[Number theory|Number theory]]; [[Prime number|Prime number]]; [[Dirichlet theorem|Dirichlet theorem]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b1107101.png" /> denote the number of primes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b1107102.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b1107103.png" />. One looks for inequalities of the type (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b1107104.png" /> arbitrary but fixed)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b1107105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b1107106.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b1107107.png" /> is the logarithmic integral (cf. also [[Distribution of prime numbers|Distribution of prime numbers]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b1107108.png" /> is the Euler totient function (cf. [[Totient function|Totient function]]).

The first attempt to obtain a  "non-trivial"  estimate of this kind was made by A. Rényi in 1948. He showed that (a1) is true with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b1107109.png" /> for some small positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b11071010.png" />. Due to subsequent refinements of M.B. Barban, Pan Cheng Dong, A.I. Vinogradov and, finally, E. Bombieri it is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b11071011.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b11071012.png" />. Somewhat later, P.X. Gallagher introduced major simplifications in Bombieri's arguments. More recently, R.C. Vaughan developed an ingenious new method which gives a still simpler proof by essentially elementary means; a general reference is [[#References|[a2]]].

The critical exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b11071013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b11071014.png" /> can conjecturally be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b11071015.png" /> (the Halberstam conjecture). Under certain restrictive conditions, Fouvry–Iwaniec and Bombieri–Friedlander–Iwaniec have given refinements to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b11071016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b11071017.png" />; see [[#References|[a1]]].

The main advantage of Bombieri's theorem becomes clear by noting that the classical Page–Siegel–Walfisz prime number theorem (cf. [[Page theorem|Page theorem]]) only leads to the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b11071018.png" /> for the moduli <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b11071019.png" /> in (a1). Moreover, Bombieri's bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110710/b11071020.png" /> is as good, apart from the logarithmic factor, as one can obtain on the assumption of the generalized Riemann hypothesis (cf. [[Riemann hypotheses|Riemann hypotheses]]). This makes it often possible to circumvent the use of the extended Riemann hypothesis, which has far-reaching implications in number theory; for example, it gives approaches to such important results as the Titchmarsh divisor problem, the Hardy–Littlewood formula for the number of representations of an integer as a sum of a prime and two squares or Chen's celebrated theorem that every sufficiently large even integer is the sum of a prime and an almost-prime having at most two prime factors; a general reference is [[#References|[a3]]].

Bombieri's result has also been generalized to algebraic number fields, by a number of scientists. There are various ways in which this can be done, but the principle underlying the treatment is always that of the [[Large sieve|large sieve]]. In view of applications Hinz's multi-dimensional version of Bombieri's theorem is of some interest, see [[#References|[a4]]]. This leads, among other things, to an analogue of Chen's theorem in totally real fields, see [[#References|[a5]]].

See also [[Selberg sieve|Selberg sieve]]: [[Density theorems|Density theorems]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Bombieri,   J. Friedlander,   H. Iwaniec,   "Primes in arithmetic progressions"  ''Acta Math.'' , '''156'''  (1986)  pp. 203–251</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Davenport,   "Multiplicative number theory" , Berlin  (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Halberstam,   H.-E. Richert,   "Sieve methods" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Hinz,   "A generalization of Bombieri's prime number theorem to algebraic number fields"  ''Acta Arith.'' , '''LI'''  (1988)  pp. 173–193</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Hinz,   "Chen's theorem in totally real algebraic number fields"  ''Acta Arith.'' , '''LVIII'''  (1991)  pp. 335–361</TD></TR></table>