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Divisor summatory function

In number theory, the divisor summatory function is a function that is a sum over the divisor function.

Version
v1 · 2026-09-28 · History
Domain-specific #
9028
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Analytic Number Theory → Mathematics

Core Idea

Divisor summatory function is treated here as the recurring analytic number theory identity summarized by this source-grounded definition: In number theory, the divisor summatory function is a function that is a sum over the divisor 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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The Big Divisor Total

For each number, count how many ways you can share that many cookies fairly: 1 cookie has 1 way, 2 cookies have 2 ways, 3 cookies have 2 ways, 4 cookies have 3 ways. Now add up those counts for all the numbers up to some stopping point. That running total is the divisor summatory function.

Adding Up Divisor Counts

The divisor function counts how many numbers divide a given number evenly; for example, 6 has 4 divisors: 1, 2, 3, and 6. The divisor summatory function adds these counts together for every number from 1 up to some limit. So up to 4 you'd add 1 + 2 + 2 + 3 = 8. Mathematicians can't find a simple exact formula for this total, but they have good estimates of about how big it is as the limit grows, and figuring out how accurate those estimates are is a famous puzzle called a divisor problem.

Summed Divisor Counts

The divisor summatory function, D(x), adds up the divisor function d(n), the number of divisors of n, for all n up to x. There's a neat way to picture it: it counts the pairs of positive whole numbers (a, b) with a × b at most x, which are the grid points lying under a curve called a hyperbola. No exact closed formula for this sum is known, but it can be approximated very well, with an approximation involving x times the logarithm of x and a constant called the Euler–Mascheroni constant, plus an error term. How small that error can be proven to be is one of the 'divisor problems'. If you count lattice points inside a circle instead of under a hyperbola, you get the related Gauss circle problem. The function also comes up when studying how the Riemann zeta function behaves.

 

In analytic number theory, the divisor summatory function is the sum of the divisor function over all positive integers up to x, D(x) = Σ_{n ≤ x} d(n). Geometrically, it counts lattice points in the region under a hyperbola, and this picture drives the standard approach to estimating it. A closed form appears to be beyond available techniques, but asymptotic approximations are known: a main term of order x log x, a secondary term involving the Euler–Mascheroni constant γ, and an error term. Determining the true size of that error term is a central divisor problem, and investigations of the divisor function's behavior are collectively called divisor problems. The analogous lattice-point count inside a circle rather than under a hyperbola is the Gauss circle problem. The divisor summatory function appears frequently in the study of the asymptotic behavior of the Riemann zeta function.

Scope of Application

  • The divisor summatory function is defined as. The divisor function counts the number of ways that the integer n can be written as a product of two integers.

  • The divisor summatory function is defined as. This allows us to provide an alternative expression for D(x), and a simple way to compute it in O(\sqrt{x}) time.

  • The divisor summatory function is defined as. If the hyperbola in this context is replaced by a circle then determining the value of the resulting function is known as the Gauss circle problem.

  • Dirichlet's divisor problem. This estimate can be proven using the Dirichlet hyperbola method, and was first established by Dirichlet in 1849.

  • Dirichlet's divisor problem. Many of the same methods work for this problem and for Gauss's circle problem, another lattice-point counting problem.

Clarity

A clear use of Divisor summatory function names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In number theory, the divisor summatory function is a function that is a sum over the divisor function.

Manages Complexity

Divisor summatory function compresses multiple analytic number theory details into a stable diagnostic relation. The source shows both the central mechanism—thus, for k = 2, D(x) = D 2 (x) counts the number of points on a square lattice bounded on the left by the vertical-axis, on the bottom by the horizontal-axis, and to the upper-right by the hyperbola jk = x.—and the practical consequence—the Dirichlet divisor problem, precisely.

Abstract Reasoning

  1. Type the carrier. Identify the analytic number theory entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In number theory, the divisor summatory function is a function that is a sum over the divisor function.
  3. Check operation and conditions. This allows us to provide an alternative expression for D(x), and a simple way to compute it in O(\sqrt{x}) time.
  4. Demand recognition evidence.

Knowledge Transfer

Within the home domain. Knowledge about Divisor summatory function transfers literally when a new case preserves the same carrier type, relation, and recognition test. The divisor function counts the number of ways that the integer n can be written as a product of two integers. This allows us to provide an alternative expression for D(x), and a simple way to compute it in O(\sqrt{x}) time. Beyond the home domain. No canonical parent is asserted for Divisor summatory function.

Neighborhood in Abstraction Space

Divisor summatory function sits in a moderately populated region (44th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Geometric Figures & Constructions (32 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08