Vipul: Created page with '{{arithmetic function}} ==Definition== ===Hands-on definition=== The '''von Mangoldt function''', denoted $\Lambda$ , is an arithmetic function defined as follow...'

2009-04-22T22:05:15Z

Created page with '{{arithmetic function}} ==Definition== ===Hands-on definition=== The '''von Mangoldt function''', denoted <math>\Lambda</math>, is an arithmetic function defined as follow...'

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{{arithmetic function}}

==Definition==

===Hands-on definition===

The '''von Mangoldt function''', denoted <math>\Lambda</math>, is an [[arithmetic function]] defined as follows:

* <math>\Lambda(1) = 0</math>.
* <math>\Lambda(p^k) = \log p</math>, for <math>p</math> a prime and <math>k</math> a positive integer.
* <math>\Lambda(n) = 0</math> if <math>n</math> has more than one prime divisor.

===Definition in terms of Dirichlet product===

The von Mangoldt function is the unique function <math>\Lambda</math> such that:

<math>\Lambda * U = \log</math>

where <math>\log</math> is the logarithm, <math>U</math> is the [[all ones function]], and <math>*</math> denotes [[Dirichlet product]].

By the [[Mobius inversion formula]], this is equivalent to defining:

<math>\Lambda := \log * \mu</math>,

where <math>\mu</math> is the [[Mobius function]].

Von Mangoldt function - Revision history

Vipul: Created page with '{{arithmetic function}} ==Definition== ===Hands-on definition=== The '''von Mangoldt function''', denoted \Lambda, is an arithmetic function defined as follow...'

Vipul: Created page with '{{arithmetic function}} ==Definition== ===Hands-on definition=== The '''von Mangoldt function''', denoted $\Lambda$ , is an arithmetic function defined as follow...'