of q), it is an integer. For a fixed value of n it is multiplicative in q: If q and r are coprime, then ψ(n) - Dedekind psi function The Dedekind psi function, used in the theory of modular functions, is defined by the formula Completely multiplicative functions λ(n) – Liouville function λ(n), the Liouville function, is defined by χ(n) – characters All Dirichlet characters χ(n) are completely multiplicative. Two characters have special notations: The principal character (mod n) is denoted by χ0(a) (or χ1(a)). It is defined as The quadratic character (mod n) is denoted by the Jacobi symbol for odd n (it is not defined for even n): In this formula is the Legendre symbol, defined for all integers a and all odd primes p by Following the normal convention for the empty product, Additive functions ω(n) – distinct prime divisors ω(n), defined above as the number of distinct primes dividing n, is additive (see Prime omega function). Completely additive functions Ω(n) – prime divisors Ω(n), defined above as the number of prime factors of n counted with multiplicities, is completely additive (see Prime omega function). νp(n) – p-adic valuation of an integer n For a fixed prime p, νp(n), defined above as the exponent of the largest power of p dividing n, is completely additive. Logarithmic derivative , where is the arithmetic derivative. Neither multiplicative nor additive (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. (x), the prime-counting function, is the number of primes not exceeding x. It is the summation function of the characteristic function of the prime numbers. A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes, ... 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. θ(x) and ψ(x), the Chebyshev functions, are defined as sums of the natural logarithms of the primes not exceeding x. The Chebyshev function ψ(x) is the summation function of the von Mangoldt function just below. Λ(n) – von Mangoldt function