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For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and () is even; for a square integer, one divisor (namely ) is not paired with a distinct divisor and () is odd. Similarly, the number σ 1 ( n ) {\displaystyle \sigma _{1}(n)} is odd if and only if n is a square or twice a square.
In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function . The various studies of the behaviour of the divisor function are sometimes called divisor problems .
σ k (n) is the divisor function (i.e. the sum of the k-th powers of the divisors of n, including 1 and n). σ 0 (n), the number of divisors of n, is usually written d(n) and σ 1 (n), the sum of the divisors of n, is usually written σ(n). If s > 0,
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function: is a Dirichlet character of modulus (where is a positive integer) if for all integers and : [1] χ ( a b ) = χ ( a ) χ ( b ) ; {\displaystyle \chi (ab)=\chi (a)\chi (b);} that is, χ {\displaystyle \chi } is completely multiplicative .
The restriction of the divisors in the convolution to unitary, bi-unitary or infinitary divisors defines similar commutative operations which share many features with the Dirichlet convolution (existence of a Möbius inversion, persistence of multiplicativity, definitions of totients, Euler-type product formulas over associated primes, etc.).
The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since ...
Particular examples of k-periodic number theoretic functions are the Dirichlet characters = modulo k and the greatest common divisor function () = (,). It is known that every k -periodic arithmetic function has a representation as a finite discrete Fourier series of the form
Number 1 is a unitary divisor of every natural number. The number of unitary divisors of a number n is 2 k, where k is the number of distinct prime factors of n. This is because each integer N > 1 is the product of positive powers p r p of distinct prime numbers p. Thus every unitary divisor of N is the product, over a given subset S of the ...