divisorrelated

Divisorrelated is a term used in mathematics to describe objects whose definition, structure, or value is determined by the divisors of integers. In number theory, divisor-related constructs include functions that take an integer n and produce a result based on the set of positive divisors of n. The most familiar examples are the divisor functions: tau(n) (also denoted d(n)), the number of divisors of n, and sigma(n), the sum of the divisors of n. Additional examples arise from Dirichlet convolution of arithmetic functions, where a function is defined as a sum over divisors of n.

Many divisor-related functions are multiplicative: if gcd(m,n)=1, then f(mn)=f(m)f(n). Tau and sigma are classic instances. Divisor-related sums of the form f(n)=sum_{d|n} g(d) are common, where g is another arithmetic function. The Möbius function μ and the concept of inversion also connect to divisor sums, providing tools for recovering underlying functions from their divisor sums.

Divisor-related ideas appear in factorization problems, asymptotic estimates, and modular forms, where the distribution of divisors influences growth rates. They also inform algorithms for integer factorization and primality testing, since knowing divisors or divisor counts can constrain search spaces. In practice, divisor-related objects are studied through their generating functions, such as Dirichlet series, and through their behavior under convolution. Related topics include the divisor lattice, abundance and deficiency (based on sigma(n) relative to n), and the study of highly composite numbers.

See also: divisor function, sigma function, tau function, Dirichlet convolution, Möbius function, multiplicative function.