Divisor function in the context of "Arithmetic function"

⭐ In the context of arithmetic functions, the divisor function is considered…

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⭐ Core Definition: Divisor function

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.

A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.

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πŸ‘‰ Divisor function in the context of Arithmetic function

In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes.

An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n.

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Divisor function in the context of Divisor summatory function

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.

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Divisor function in the context of Colossally abundant number

In number theory, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever integer has the highest ratio is a colossally abundant number. It is a stronger restriction than that of a superabundant number, but not strictly stronger than that of an abundant number.

Formally, a number n is said to be colossally abundant if there is an Ξ΅ > 0 such that for all k > 1,

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LINKED FROM (PARENTS)
← Arithmetic function ← Divisor summatory function ← Colossally abundant number
LINKS TO (CHILDREN)
Mathematics → Number theory → Arithmetic function → Divisor → Integer → Riemann zeta function → Eisenstein series → Modular form → Ramanujan → Modular arithmetic → Identity (mathematics) → Ramanujan's sum → Divisor summatory function →