Second Chebyshev function - Revision history

Vipul: /* Relation with other functions */

2009-04-30T20:21:04Z

Relation with other functions

← Older revision		Revision as of 20:21, 30 April 2009
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	The exponential of the second Chebyshev function gives the [[lcm of all numbers so far]]. In other words:		The exponential of the second Chebyshev function gives the [[lcm of all numbers so far]]. In other words:

	<math>e^{\psi(x)} = \operatorname{lcm} \{ 1,2, \dots, ~~\gint{~~x} \}</math>.		<math>e^{\psi(x)} = \operatorname{lcm} \{ 1,2, \dots, [x] \}</math>

			where <math>[x]</math> denotes the [[greatest integer function]] of <math>x</math>, i.e., the largest integer less than or equal to <math>x</math>.

Vipul at 20:20, 30 April 2009

2009-04-30T20:20:15Z

← Older revision		Revision as of 20:20, 30 April 2009
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	This summation is taken over all the natural numbers less than or equal to <math>x</math>; however, a positive contribution comes only from prime powers, and the contribution of a prime power <math>p^k</math> is <math>\log p</math>.		This summation is taken over all the natural numbers less than or equal to <math>x</math>; however, a positive contribution comes only from prime powers, and the contribution of a prime power <math>p^k</math> is <math>\log p</math>.

	==Relation with other ~~counting~~ functions==		==Relation with other functions==

	* [[Prime-counting function]]		* [[Prime-counting function]]
	* [[First Chebyshev function]]		* [[First Chebyshev function]]: This simply adds the logarithms of all the primes up to the point.

			===Exponential===

			The exponential of the second Chebyshev function gives the [[lcm of all numbers so far]]. In other words:

			<math>e^{\psi(x)} = \operatorname{lcm} \{ 1,2, \dots, \gint{x} \}</math>.

Vipul at 04:05, 29 April 2009

2009-04-29T04:05:04Z

← Older revision		Revision as of 04:05, 29 April 2009
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	{{~~till-now summation~~\|von Mangoldt function}}		{{summatory function\|von Mangoldt function}}

	==Definition==		==Definition==

Vipul at 03:14, 29 April 2009

2009-04-29T03:14:25Z

← Older revision		Revision as of 03:14, 29 April 2009
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	{{~~counting~~ function}}		{{till-now summation\|von Mangoldt function}}

	==Definition==		==Definition==

Vipul: Created page with '{{counting function}} ==Definition== Let $x$ be a positive real number. The '''second Chebyshev function''' of $x$ , denoted $\psi(x)$ , is defin...'

2009-04-29T02:28:35Z

Created page with '{{counting function}} ==Definition== Let <math>x</math> be a positive real number. The '''second Chebyshev function''' of <math>x</math>, denoted <math>\psi(x)</math>, is defin...'

New page

{{counting function}}

==Definition==

Let <math>x</math> be a positive real number. The '''second Chebyshev function''' of <math>x</math>, denoted <math>\psi(x)</math>, is defined as the following sum:

<math>\psi(x) = \sum_{n \le x} \Lambda(n)</math>.

Here, <math>\Lambda</math> is the [[defining ingredient::von Mangoldt function]].

This summation is taken over all the natural numbers less than or equal to <math>x</math>; however, a positive contribution comes only from prime powers, and the contribution of a prime power <math>p^k</math> is <math>\log p</math>.

==Relation with other counting functions==

* [[Prime-counting function]]
* [[First Chebyshev function]]

Second Chebyshev function - Revision history

Vipul: /* Relation with other functions */

Vipul at 20:20, 30 April 2009

Vipul at 04:05, 29 April 2009

Vipul at 03:14, 29 April 2009

Vipul: Created page with '{{counting function}} ==Definition== Let x be a positive real number. The '''second Chebyshev function''' of x, denoted \psi(x), is defin...'

Vipul: Created page with '{{counting function}} ==Definition== Let $x$ be a positive real number. The '''second Chebyshev function''' of $x$ , denoted $\psi(x)$ , is defin...'