Everything about Divisor function
Arithmetic function related to the divisors of an integer
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. 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. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Divisor functionA 5-minute orientation
- Skim a recent paper: Identities associated to a generalized divisor function and modified Bessel functionInfluential work since 2022
- Pick up a classic book: Asymptotic prime divisorsThe most-reprinted book on the subject
- Read the landmark paper: The divisor function $d_3(n)$ in arithmetic progressionsCited 52 times
- Go sideways: Ramanujan's sumA closely related topic
- Go sideways: Colossally abundant numberA closely related topic
📖 Definitions
Noun A function yielding the sum of the divisors of an integer. It is denoted by the Greek letter σ (sigma).
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
The divisor function $d_3(n)$ in arithmetic progressions
The divisor function at consecutive integers
Riemann’s zeta-function and the divisor problem
The divisor function over arithmetic progressions
Ambiguous divisor classes in function fields
Weighted divisor sums and Bessel function series, II
The divisor function on residue classes I
Weighted Divisor Sums and Bessel Function Series
🆕 Influential since 2022
Identities associated to a generalized divisor function and modified Bessel function
On the divisor function over Piatetski-Shapiro sequences
Bressoud–Subbarao Type Weighted Partition Identities for a Generalized Divisor Function
On some geometric properties of sequence spaces of generalized arithmetic divisor sum function
On Eisenstein series, color partition and divisor function
📚 Books worth reading
Asymptotic prime divisorsStephen McAdam · 1983
Quasianalytic monogenic solutions of a cohomological equationS. Marmi, D. Sauzin · 2003- 📕Quasianalytic monogenic solutions of a cohomological equationS. Marmi · 2003
Analytic number theoryJ. M. de Koninck · 2012- 📕Dynamical systems (1984-2012)John Willard Milnor · 2014
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Ramanujan's sum In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula cq(n)=∑1≤a≤q(a…
- 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…
- Multiplicative function In number theory, a multiplicative function is an arithmetic function f of a positive integer n with the property that f(1)=1 and f(ab)=f(a)f(b) whenever…
- Unitary divisor In mathematics, a natural number a is a unitary divisor (or Hall divisor) of a number b if a is a divisor of b and if a and b / a are coprime, having no…
- Arithmetic function This article uses technical mathematical notation for logarithms. All instances of log(x) without a subscript base should be interpreted as a natural…
- Euler's totient function This article uses technical mathematical notation for logarithms. All instances of log(x) without a subscript base should be interpreted as a natural…
- Superabundant number In mathematics, a superabundant number is a kind of natural number. A natural number n is called superabundant precisely when, for all m < n: σ(m)m<σ(n)n…
- Riemann zeta function In mathematics, the Riemann zeta function, or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a function of a complex…