Chowla's conjecture on the first Dirichlet prime: Difference between revisions

Latest revision as of 03:35, 9 February 2010

Template:Primes in arithmetic progressions conjecture

Statement

Quick statement

The first Dirichlet prime in any relatively prime congruence class modulo $D$ is $O (D^{1 + ϵ})$ .

In terms of the first Dirichlet prime

For any $ϵ > 0$ , there exists a constant $C$ such that the following holds:

Suppose $a$ and $D$ are relatively prime natural numbers. Then, there exists a prime $p \equiv a (\mod D)$ such that $p < C D^{1 + ϵ}$ .

In terms of the first few Dirichlet primes

For any $ϵ > 0$ and any natural number $k$ , there exists a constant $C$ such that the following holds:

Suppose $a$ and $D$ are relatively prime natural numbers. Then, there exist at least $k$ distinct primes $p \equiv a (\mod D)$ such that $p < C D^{1 + ϵ}$ .

This follows from the version involving the first Dirichlet prime.

Relation with other facts

Stronger conjectures

Heath-Brown's conjecture on the first Dirichlet prime

Weaker facts and conjectures

Chowla's corollary to generalized Riemann hypothesis: Under the generalized Riemann hypothesis, we have the analogous result for $2 + ϵ$ instead of $1 + ϵ$ .
Linnik's theorem: This is an unconditional version where $1 + ϵ$ is replaced by some large constant $L$ . Heath-Brown has shown that $L \leq 5.5$ .

@@ Line 2: / Line 2: @@
 ==Statement==
+===Quick statement===
+The first Dirichlet prime in any relatively prime congruence class modulo <math>D</math> is <math>O(D^{1 + \epsilon})</math>.
+===In terms of the first Dirichlet prime===
 For any <math>\epsilon > 0</math>, there exists a constant <math>C</math> such that the following holds:
-Suppose <math>a</math> and <math>D</math> are relatively prime natural numbers. Then, there exists a prime <math>p \equiv a \pmod D</math> such that <math>p < cD^{1 + \epsilon}</math>.
+Suppose <math>a</math> and <math>D</math> are relatively prime natural numbers. Then, there exists a prime <math>p \equiv a \pmod D</math> such that <math>p < CD^{1 + \epsilon}</math>.
+===In terms of the first few Dirichlet primes===
+For any <math>\epsilon > 0</math> and any natural number <math>k</math>, there exists a constant <math>C</math> such that the following holds:
+Suppose <math>a</math> and <math>D</math> are relatively prime natural numbers. Then, there exist at least <math>k</math> distinct primes <math>p \equiv a \pmod D</math> such that <math>p < CD^{1 + \epsilon}</math>.
+This follows from the version involving the first Dirichlet prime.
 ==Relation with other facts==
@@ Line 15: / Line 29: @@
 ===Weaker facts and conjectures===
-* [[Stronger than::Chowla's corollary to generalized Riemann hypothesis]]: Under the [generalized Riemann hypothesis]], we have the analogous result for <math>2 + \epsilon</math> instead of <math>1 + \epsilon</math>.
+* [[Stronger than::Chowla's corollary to generalized Riemann hypothesis]]: Under the [[generalized Riemann hypothesis]], we have the analogous result for <math>2 + \epsilon</math> instead of <math>1 + \epsilon</math>.
-* [[Stronger than::Linnick's theorem]]: This is an unconditional version where <math>1 + \epsilon</math> is replaced by some large constant <math>L</math>. Heath-Brown have shown that <math>L \le 5.5</math>.
+* [[Stronger than::Linnik's theorem]]: This is an unconditional version where <math>1 + \epsilon</math> is replaced by some large constant <math>L</math>. Heath-Brown has shown that <math>L \le 5.5</math>.