Elliott-Halberstam conjecture - Revision history

Vipul at 20:39, 3 May 2010

2010-05-03T20:39:50Z

← Older revision		Revision as of 20:39, 3 May 2010
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	==Statement==		==Statement==

	For <math>n</math> a [[natural number]] and <math>a</math> an integer relatively prime to <math>m</math>, consider the [[fact about::modular prime-counting function]] <math>x \mapsto \pi(x;n,a)</math>, which counts the number of [[prime number]]s less than or equal to <math>x</math> that are congruent to <math>a</math> modulo <math>n</math>. Let <math>\pi(x)</math> be the [[fact about::prime-counting function]] at <math>x</math> and <math>\varphi(n)</math> the [[fact about::Euler phi-function]] of <math>n</math>. Consider the expression:		For <math>n</math> a [[natural number]] and <math>a</math> an integer relatively prime to <math>n</math>, consider the [[fact about::modular prime-counting function]] <math>x \mapsto \pi(x;n,a)</math>, which counts the number of [[prime number]]s less than or equal to <math>x</math> that are congruent to <math>a</math> modulo <math>n</math>. Let <math>\pi(x)</math> be the [[fact about::prime-counting function]] at <math>x</math> and <math>\varphi(n)</math> the [[fact about::Euler phi-function]] of <math>n</math>. Consider the expression:

	<math>E(x;n) := \max_{(a,n) = 1} \left\| \pi(x;n,a) - \frac{\pi(x)}{\varphi(n)} \right\|</math>		<math>E(x;n) := \max_{(a,n) = 1} \left\| \pi(x;n,a) - \frac{\pi(x)}{\varphi(n)} \right\|</math>

Vipul: /* Relation with other facts/conjectures */

2010-02-09T03:10:49Z

Relation with other facts/conjectures

← Older revision		Revision as of 03:10, 9 February 2010
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	! Name !! Statement !! Status		! Name !! Statement !! Status
	\|-		\|-
	\| [[Stronger than::Bombieri-~~Vingradov~~ theorem]] \|\| Elliott-Halberstam conjecture holds for <math>\theta < 1/2</math> \|\| Proved assuming [[generalized Riemann hypothesis]]		\| [[Stronger than::Bombieri-Vinogradov theorem]] \|\| Elliott-Halberstam conjecture holds for <math>\theta < 1/2</math> \|\| Proved assuming [[generalized Riemann hypothesis]]
	\|}		\|}

Vipul: Created page with '==Statement== For $n$ a natural number and $a$ an integer relatively prime to $m$ , consider the [[fact about::modular prime-counting functio…'

2010-02-09T03:09:59Z

Created page with '==Statement== For <math>n</math> a natural number and <math>a</math> an integer relatively prime to <math>m</math>, consider the [[fact about::modular prime-counting functio…'

New page

==Statement==

For <math>n</math> a [[natural number]] and <math>a</math> an integer relatively prime to <math>m</math>, consider the [[fact about::modular prime-counting function]] <math>x \mapsto \pi(x;n,a)</math>, which counts the number of [[prime number]]s less than or equal to <math>x</math> that are congruent to <math>a</math> modulo <math>n</math>. Let <math>\pi(x)</math> be the [[fact about::prime-counting function]] at <math>x</math> and <math>\varphi(n)</math> the [[fact about::Euler phi-function]] of <math>n</math>. Consider the expression:

<math>E(x;n) := \max_{(a,n) = 1} \left| \pi(x;n,a) - \frac{\pi(x)}{\varphi(n)} \right|</math>

The intuition here is that the primes should be roulghy equally distributed between the various congruence classes modulo <math>n</math>, so the ''expected'' number of primes in each congruence class is <math>\pi(x)/\varphi(n)</math>. We are interested in the largest of the deviations from this expected value.

Then, the Elliott-Halberstam conjecture states that for every <math>\theta < 1</math> and <math>A > 0</math> there exists a constant <math>C > 0</math> such that

<math>\sum_{1 \le n \le x^{\theta}} E(x;n) \le \frac{Cx}{\log^A x}</math>

==Relation with other facts/conjectures==

{| class="sortable" border="1"
! Name !! Statement !! Status
|-
| [[Stronger than::Bombieri-Vingradov theorem]] || Elliott-Halberstam conjecture holds for <math>\theta < 1/2</math> || Proved assuming [[generalized Riemann hypothesis]]
|}

Elliott-Halberstam conjecture - Revision history

Vipul at 20:39, 3 May 2010

Vipul: /* Relation with other facts/conjectures */

Vipul: Created page with '==Statement== For n a natural number and a an integer relatively prime to m, consider the [[fact about::modular prime-counting functio…'

Vipul: Created page with '==Statement== For $n$ a natural number and $a$ an integer relatively prime to $m$ , consider the [[fact about::modular prime-counting functio…'