Riemann hypothesis - Revision history

Vipul: /* Related facts and conjectures */

2009-05-07T01:54:31Z

Related facts and conjectures

← Older revision		Revision as of 01:54, 7 May 2009
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	* [[Prime gap corollary to Riemann hypothesis]]		* [[Prime gap corollary to Riemann hypothesis]]
	* [[Lindelof hypothesis]]		* [[Lindelof hypothesis]]

			===Related facts===

			* [[Riemann hypothesis for finite fields]]: This is an analogue of the Riemann hypothesis for finite fields, that has been proved.

	==External links==		==External links==

Vipul at 15:50, 5 May 2009

2009-05-05T15:50:04Z

← Older revision		Revision as of 15:50, 5 May 2009
Line 30:		Line 30:
	* [[Prime gap corollary to Riemann hypothesis]]		* [[Prime gap corollary to Riemann hypothesis]]
	* [[Lindelof hypothesis]]		* [[Lindelof hypothesis]]

			==External links==

			* [[Wikipedia:Riemann hypothesis\|Riemann hypothesis on Wikipedia]]
			* [[Mathworld:RiemannHypothesis\|Riemann hypothesis on Mathworld]]

Vipul at 15:46, 5 May 2009

2009-05-05T15:46:13Z

← Older revision		Revision as of 15:46, 5 May 2009
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	<math>\left\| \pi(x) - \operatorname{li}(x) \right\| \le \frac{1}{8\pi} \sqrt{x} \log x, \qquad x \ge 2657</math>.		<math>\left\| \pi(x) - \operatorname{li}(x) \right\| \le \frac{1}{8\pi} \sqrt{x} \log x, \qquad x \ge 2657</math>.

			==Related facts and conjectures==

			===Stronger conjectures===

			* [[Generalized Riemann hypothesis]]
			* [[Extended Riemann hypothesis]]

			===Weaker facts and conjectures===

			* [[Prime gap corollary to Riemann hypothesis]]
			* [[Lindelof hypothesis]]

Vipul: Created page with '==Statement== ===In terms of zeros of the Riemann zeta-function=== All the nontrivial zeros of the Riemann zeta-function have real part $1/2$ . ===In terms of th...'

2009-05-05T15:42:53Z

Created page with '==Statement== ===In terms of zeros of the Riemann zeta-function=== All the nontrivial zeros of the Riemann zeta-function have real part <math>1/2</math>. ===In terms of th...'

New page

==Statement==

===In terms of zeros of the Riemann zeta-function===

All the nontrivial zeros of the [[Riemann zeta-function]] have real part <math>1/2</math>.

===In terms of the distribution of prime numbers===

For a positive real number <math>x</math>, it states that:

<math>\pi(x) - \operatorname{li}(x) = O(\sqrt{x}\log x)</math>.

Here, <math>\pi(x)</math> denotes the [[fact about::prime-counting function]], i.e., the number of primes less than or equal to <math>x</math>, while <math>\operatorname{li}(x)</math> denotes the [[fact about::logarithmic integral function]]:

<math>\int_0^x \frac{dx}{\log x}</math>.

In fact, more specifically, the following is an equivalent formulation of the Riemann hypothesis:

<math>\left| \pi(x) - \operatorname{li}(x) \right| \le \frac{1}{8\pi} \sqrt{x} \log x, \qquad x \ge 2657</math>.

Riemann hypothesis - Revision history

Vipul: /* Related facts and conjectures */

Vipul at 15:50, 5 May 2009

Vipul at 15:46, 5 May 2009

Vipul: Created page with '==Statement== ===In terms of zeros of the Riemann zeta-function=== All the nontrivial zeros of the Riemann zeta-function have real part 1/2. ===In terms of th...'

Vipul: Created page with '==Statement== ===In terms of zeros of the Riemann zeta-function=== All the nontrivial zeros of the Riemann zeta-function have real part $1/2$ . ===In terms of th...'