Editing Arithmetic function (section)

=== ''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.