An arithmetic function a is
• completely additive if a(mn) = a(m) + a(n) for all natural numbers m and n;
• completely multiplicative if a(mn) = a(m)a(n) for all natural numbers m and n; 查看更多內容

σk(n), τ(n), d(n) – divisor sums
σk(n) is the sum of the kth powers of the positive divisors of n, including 1 and n, where k is a complex number.
σ1(n), the sum of the (positive) divisors of n, is usually denoted by σ(n).
Since a positive … 查看更多內容

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 … 查看更多內容

The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p1 < p2 < ... < pk are primes and the aj are positive integers. (1 is given by the empty product.)
It is often … 查看更多內容

λ(n) – Liouville function
λ(n), the Liouville function, is defined by
χ(n) – characters
All Dirichlet characters χ(n) are completely multiplicative. … 查看更多內容

ω(n) – distinct prime divisors
ω(n), defined above as the number of distinct primes dividing n, is additive (see Prime omega function). 查看更多內容

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 查看更多內容