Arithmetic function: Difference between revisions

Revision as of 13:38, 20 April 2002

Revision as of 10:49, 30 September 2002

Line 6:

The articles on additive and multiplicative functions contain several important examples.

Examples of a non-multiplicative functions are:

An example of a non-multiplicative function is ''c''_''4''(''n'') - the number of ways that ''n'' can be expressed as the sum of four squares of nonnegative integers, where we distinguish between different orders of the summands. For example:

An example of a non-multiplicative function is* ''c''_''4''(''n'') - the number of ways that ''n'' can be expressed as the sum of four squares of nonnegative integers, where we distinguish between different orders of the summands. For example:

:1 = 1²+0²+0²+0² = 0²+1²+0²+0² = 0²+0²+1²+0² = 0²+0²+0²+1²,

::1 = 1²+0²+0²+0² = 0²+1²+0²+0² = 0²+0²+1²+0² = 0²+0²+0²+1²,

hence ''c''₄(1)=4 ≠ 1.

:hence ''c''₄(1)=4 ≠ 1.

Another example of a non-multiplicative function is the [[partition function]] ''P''(''n''), the number of representations of ''n'' as a sum of positive integers, where we don't distinguish between different orders of the summands. For instance: ''P''(2 · 5) = ''P''(10) = 42 and ''P''(2)''P''(5) = 2 · 7 = 14 ≠ 42.

Another example of a non-multiplicative function is* the [[partitionPartition function]] ''P''(''n''), - the number of representations of ''n'' as a sum of positive integers, where we don't distinguish between different orders of the summands. For instance: ''P''(2 · 5) = ''P''(10) = 42 and ''P''(2)''P''(5) = 2 · 7 = 14 ≠ 42.

* the [[Prime number theorem|Prime counting function]] π (''n'') - the number of primes less than or equal to a given number ''n''. We have π(1) = 0 ≠ 1, π (2 · 5) = π(10) = 4 and π(2) π(5) = 1 · 3 = 3 ≠ 4.