Editing Arithmetic function (section)

== Neither multiplicative nor additive ==

=== ''π''(''x''), Π(''x''), ''ϑ''(''x''), ''ψ''(''x'') – prime-counting functions===
These important functions (which are not arithmetic functions) are defined for non-negative real arguments, and are used in the various statements and proofs of the [[prime number theorem]]. They are summation functions (see the main section just below) of arithmetic functions which are neither multiplicative nor additive.

''π''(''x''), the [[prime-counting function]], is the number of primes not exceeding ''x''. It is the summation function of the [[indicator function|characteristic function]] of the prime numbers.
<math display="block">\pi(x) = \sum_{p \le x} 1</math>

A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes, etc. It is the summation function of the arithmetic function which takes the value 1/''k'' on integers which are the ''k''th power of some prime number, and the value 0 on other integers.
<math display="block">\Pi(x) = \sum_{p^k\le x}\frac{1}{k}.</math>

''ϑ''(''x'') and ''ψ''(''x''), the [[Chebyshev function]]s, are defined as sums of the natural logarithms of the primes not exceeding ''x''.
<math display="block">\vartheta(x)=\sum_{p\le x} \log p,</math>
<math display="block"> \psi(x) = \sum_{p^k\le x} \log p.</math>

The second Chebyshev function ''ψ''(''x'') is the summation function of the von Mangoldt function just below.

=== Λ(''n'') – von Mangoldt function ===
'''[[von Mangoldt function|Λ(''n'')]]''', the von Mangoldt function, is 0 unless the argument ''n'' is a prime power {{math|''p''<sup>''k''</sup>}}, in which case it is the natural logarithm of the prime ''p'':
<math display="block">\Lambda(n) = \begin{cases}
\log p &\text{if } n = 2,3,4,5,7,8,9,11,13,16,\ldots=p^k \text{ is a prime power}\\
0&\text{if } n=1,6,10,12,14,15,18,20,21,\dots \;\;\;\;\text{ is not a prime power}.
\end{cases}</math>

=== ''p''(''n'') – partition function ===
'''[[partition function (number theory)|''p''(''n'')]]''', the partition function, is the number of ways of representing ''n'' as a sum of positive integers, where two representations with the same summands in a different order are not counted as being different:
<math display="block">p(n) = \left|\left\{ (a_1, a_2,\dots a_k): 0 < a_1 \le a_2 \le \cdots \le a_k\; \land \;n=a_1+a_2+\cdots +a_k \right\}\right|.</math>

=== ''λ''(''n'') – Carmichael function ===
'''[[Carmichael function|''λ''(''n'')]]''', the Carmichael function, is the smallest positive number such that <math>a^{\lambda(n)}\equiv 1 \pmod{n}</math> &nbsp; for all ''a'' coprime to ''n''. Equivalently, it is the [[least common multiple]] of the orders of the elements of the [[Multiplicative group of integers modulo n|multiplicative group of integers modulo ''n'']].

For powers of odd primes and for 2 and 4, ''λ''(''n'') is equal to the Euler totient function of ''n''; for powers of 2 greater than 4 it is equal to one half of the Euler totient function of ''n'':
<math display="block">\lambda(n) = \begin{cases}
\;\;\phi(n) &\text{if }n = 2,3,4,5,7,9,11,13,17,19,23,25,27,\dots\\
\tfrac 1 2 \phi(n)&\text{if }n=8,16,32,64,\dots
\end{cases}</math>
and for general ''n'' it is the least common multiple of ''λ'' of each of the prime power factors of ''n'':
<math display="block">\lambda(p_1^{a_1}p_2^{a_2} \dots p_{\omega(n)}^{a_{\omega(n)}}) = \operatorname{lcm}[\lambda(p_1^{a_1}),\;\lambda(p_2^{a_2}),\dots,\lambda(p_{\omega(n)}^{a_{\omega(n)}}) ].</math>

=== ''h''(''n'') – class number ===
'''[[Ideal class group|''h''(''n'')]]''', the class number function, is the order of the [[ideal class group]] of an algebraic extension of the rationals with [[discriminant]] ''n''. The notation is ambiguous, as there are in general many extensions with the same discriminant. See [[quadratic field]] and [[cyclotomic field]] for classical examples.

=== ''r''<sub>''k''</sub>(''n'') – sum of ''k'' squares ===
'''[[Sum of squares function|''r''<sub>''k''</sub>(''n'')]]''' is the number of ways ''n'' can be represented as the sum of ''k'' squares, where representations that differ only in the order of the summands or in the signs of the square roots are counted as different.
<math display="block">r_k(n) = \left|\left\{(a_1, a_2,\dots,a_k):n=a_1^2+a_2^2+\cdots+a_k^2\right\}\right|</math>

=== ''D''(''n'') – Arithmetic derivative ===
Using the [[Differential operator#Notations|Heaviside notation]] for the derivative, the [[arithmetic derivative]] ''D''(''n'') is a function such that 
* <math> D(n) = 1</math> if ''n'' prime, and
* <math>D(mn) = m D(n) + D(m) n</math> (the [[product rule]])