Sigma function

In mathematics, the term sigma function can refer to the sum-of-divisors function in number theory or the Weierstrass sigma function in complex analysis.^[1]^[2] In number theory, the sigma function, denoted σ(n), is an arithmetic function that computes the sum of all positive divisors of a positive integer n, formally defined as σ(n) = ∑_{d|n} d, where the sum is over all positive divisors d of n. For example, σ(6) = 1 + 2 + 3 + 6 = 12, since the divisors of 6 are 1, 2, 3, and 6.^[3] The function is multiplicative, meaning that if m and n are coprime positive integers, then σ(mn) = σ(m)σ(n), which allows computation via prime factorization: for n = p_1^{a_1} p_2^{a_2} ⋯ p_k^{a_k}, σ(n) = ∏{i=1}^k (1 + p_i + p_i^2 + ⋯ + p_i^{a_i}) = ∏{i=1}^k \frac{p_i^{a_i+1} - 1}{p_i - 1}. This property facilitates efficient evaluation and has applications in studying perfect numbers, abundant numbers, and deficient numbers, where σ(n) - n = n defines perfect numbers like 6 and 28.^[1] More generally, the sigma function extends to σ_k(n) = ∑_{d|n} d^k for any positive integer k, with σ_0(n) counting the number of divisors (often denoted d(n) or τ(n)) and σ_1(n) recovering the standard sum-of-divisors function. Dirichlet established in 1838 that the average order of the number of divisors is asymptotic to ln n + 2γ - 1, where γ is the Euler-Mascheroni constant, highlighting its role in analytic number theory. Gronwall's theorem further bounds the growth of σ(n)/n, showing lim sup σ(n)/(n ln ln n) = e^γ as n → ∞.^[1]

Sum of Divisors Function

Definition

The sum-of-divisors function, denoted \sigma(n), gives the sum of all positive divisors of the positive integer n:

\sigma(n) = \sum_{d \mid n} d.

It is a multiplicative function: if m and n are coprime, then \sigma(mn) = \sigma(m) \sigma(n). For n = p_1^{a_1} \cdots p_k^{a_k},

\sigma(n) = \prod_{i=1}^k \frac{p_i^{a_i + 1} - 1}{p_i - 1}.

^[1]

Examples

The sigma function, denoted σ(n), computes the sum of all positive divisors of n. For small values of n, these sums illustrate how the function aggregates divisors to produce results that reveal number-theoretic properties, such as abundance or perfection.^[1] The following table lists σ(n) for n from 1 to 20, showcasing patterns like primes yielding σ(p) = p + 1 and highly composite numbers like 12 yielding larger sums relative to n.^[4]

n	Divisors	σ(n)
1	1	1
2	1, 2	3
3	1, 3	4
4	1, 2, 4	7
5	1, 5	6
6	1, 2, 3, 6	12
7	1, 7	8
8	1, 2, 4, 8	15
9	1, 3, 9	13
10	1, 2, 5, 10	18
11	1, 11	12
12	1, 2, 3, 4, 6, 12	28
13	1, 13	14
14	1, 2, 7, 14	24
15	1, 3, 5, 15	24
16	1, 2, 4, 8, 16	31
17	1, 17	18
18	1, 2, 3, 6, 9, 18	39
19	1, 19	20
20	1, 2, 4, 5, 10, 20	42

For n=12, the divisors are 1, 2, 3, 4, 6, and 12, summing to 28, which exceeds 12 and indicates an abundant number.^[1]^[4] A related function is the aliquot sum s(n) = σ(n) - n, which sums the proper divisors excluding n itself; for n=6, s(6) = 12 - 6 = 6, making 6 the smallest perfect number where the aliquot sum equals n.^[5] The sigma function generalizes to σ_k(n), the sum of the k-th powers of the divisors of n. For k=0, σ_0(12) counts the divisors of 12, yielding 6. For k=2, σ_2(12) sums the squares of those divisors (1² + 2² + 3² + 4² + 6² + 12² = 1 + 4 + 9 + 16 + 36 + 144 = 210).^[6]^[7]

Multiplicative Properties

The sum-of-divisors function \sigma(n), which sums the positive divisors of n, is multiplicative, meaning that if m and n are coprime positive integers (i.e., \gcd(m, n) = 1), then \sigma(mn) = \sigma(m) \sigma(n).^[8]^[1] This property holds more generally for the family of divisor functions \sigma_k(n) = \sum_{d \mid n} d^k, where k is a positive integer, so that \sigma_k(mn) = \sigma_k(m) \sigma_k(n) for coprime m and n.^[8]^[1] For a prime power p^a where p is prime and a \geq 1, the formula simplifies to \sigma(p^a) = 1 + p + p^2 + \cdots + p^a = \frac{p^{a+1} - 1}{p - 1}, which is the sum of a finite geometric series.^[8]^[1] This closed form arises directly from the divisors of p^a being exactly the powers p^0, p^1, \dots, p^a. For the generalized case, \sigma_k(p^a) = 1 + p^k + p^{2k} + \cdots + p^{ak} = \frac{p^{(a+1)k} - 1}{p^k - 1}.^[1] Given the unique prime factorization theorem, any positive integer n > 1 can be expressed as n = \prod_{i=1}^r p_i^{a_i} where the p_i are distinct primes and a_i \geq 1. The multiplicativity then yields the general formula \sigma(n) = \prod_{i=1}^r \sigma(p_i^{a_i}) = \prod_{i=1}^r \frac{p_i^{a_i+1} - 1}{p_i - 1}.^[8]^[1] Similarly, for \sigma_k(n), it follows that \sigma_k(n) = \prod_{i=1}^r \frac{p_i^{(a_i+1)k} - 1}{p_i^k - 1}.^[1] This product form enables efficient computation of \sigma(n) once the prime factorization of n is known. The proof of multiplicativity relies on the fundamental theorem of arithmetic, which ensures that the divisors of mn for coprime m and n are precisely the products of divisors of m and divisors of n. Thus, \sigma(mn) = \sum_{d \mid mn} d = \sum_{a \mid m} \sum_{b \mid n} (a b) = \left( \sum_{a \mid m} a \right) \left( \sum_{b \mid n} b \right) = \sigma(m) \sigma(n), with the extension to \sigma_k following analogously by replacing d with d^k.^[8]^[1]

Analytic Formulas

The Dirichlet series representation of the sum-of-divisors function \sigma(n) is given by

\sum_{n=1}^\infty \frac{\sigma(n)}{n^s} = \zeta(s) \zeta(s-1)

for \operatorname{Re}(s) > 2, where \zeta(s) denotes the Riemann zeta function. This identity arises from the Euler product expansions of the zeta functions and the multiplicative property of \sigma(n), allowing summation over divisors in a factored form. A related generating function is the Lambert series

\sum_{n=1}^\infty \sigma(n) q^n = \sum_{k=1}^\infty \frac{q^k}{1 - q^k}

valid for |q| < 1. This expansion expresses \sigma(n) as the coefficient of q^n on the right-hand side, where the inner sum generates multiples of each k, weighted by the divisors. The series connects \sigma(n) to partition-like structures in q-series theory. Euler's pentagonal number theorem, originally for the partition function, yields a recurrence relation for \sigma(n) via its generating function. Specifically,

\sum_{k=-\infty}^\infty (-1)^k \sigma\left(n - \frac{k(3k-1)}{2}\right) = n

for n \geq 1, where the terms \frac{k(3k-1)}{2} are generalized pentagonal numbers (with \sigma(m) = 0 for m < 1).^[9] This relation provides a recursive way to compute \sigma(n) using values at prior arguments shifted by pentagonal indices.^[9] The function \sigma(n) also appears as a coefficient in broader expansions within analytic number theory, such as the Dirichlet series product \zeta(s) \zeta(s-1), which embeds it alongside the zeta function's structure; note that the square \zeta(s)^2 = \sum_{n=1}^\infty d(n) n^{-s} instead generates the divisor count d(n), highlighting the parallel role of \sigma(n) in weighted sums.

Growth and Distribution

The growth of the sigma function \sigma(n) is highly irregular, reflecting the varying number of divisors among integers. It satisfies the bound \sigma(n) = O(n \log \log n). A sharper characterization of its maximal order is provided by Gronwall's theorem, which asserts that

\limsup_{n \to \infty} \frac{\sigma(n)}{n \log \log n} = e^\gamma,

where \gamma \approx 0.57721 is the Euler-Mascheroni constant. The average order of \sigma(n) is determined by the partial summation formula

\sum_{n \le x} \sigma(n) = \frac{\pi^2}{12} x^2 + O(x \log x)

as x \to \infty, implying

\frac{1}{x} \sum_{n \le x} \sigma(n) \sim \frac{\pi^2}{12} x.

This result, originally due to Wigert, follows from the Euler product representation of the associated Dirichlet series \zeta(s) \zeta(s-1) and standard Tauberian theorems.^[1] Numbers are classified based on the abundance \sigma(n) - n relative to n: perfect numbers satisfy \sigma(n) = 2n (so \sigma(n) - n = n); abundant numbers have \sigma(n) > 2n (\sigma(n) - n > n); and deficient numbers have \sigma(n) < 2n (\sigma(n) - n < n). The smallest perfect number is 6, with \sigma(6) = 12; the next is 28, with \sigma(28) = 56. An example of an abundant number is 12, with \sigma(12) = 28 > 24. Most natural numbers are deficient, though the precise densities of these classes remain open problems.^[10] The value \sigma(n) is odd if and only if n is a perfect square or twice a perfect square. This follows from the multiplicative property of \sigma: for an odd prime power p^k, \sigma(p^k) = 1 + p + \cdots + p^k is odd precisely when k is even; for the prime 2, \sigma(2^k) is odd only for k = 0 or $1. Thus, the odd part of n must be a square, and the power of 2 (if present) at most 1.^[1] All known perfect numbers are even and of the form $2^{p-1}(2^p - 1), where $2^p - 1 is a Mersenne prime (a result due to Euler). There are 52 known perfect numbers as of 2024, corresponding to known Mersenne primes, but it is unknown whether there are infinitely many or if any odd perfect numbers exist (none are known below $10^{2200} as of 2025).^[10]^[11]^[12] Consequently, the asymptotic density of perfect numbers is unknown, though their count up to x is at most O(\log x).^[10]

Weierstrass Sigma Function

Definition

The Weierstrass sigma function \sigma(z; \Lambda) is a special entire function defined for a lattice \Lambda in the complex plane, generated by fundamental periods $2\omega_1 and $2\omega_2. It is given by the infinite product formula

\sigma(z; \Lambda) = z \prod_{w \in \Lambda \setminus \{0\}} \left(1 - \frac{z}{w}\right) \exp\left( \frac{z}{w} + \frac{1}{2} \left( \frac{z}{w} \right)^2 \right),

where the product converges uniformly on compact subsets of \mathbb{C} excluding the lattice points.^[13]^[14] This function is holomorphic everywhere in the complex plane, possessing a simple zero at z = 0 and no other zeros within any fundamental parallelogram of the lattice, with additional simple zeros precisely at the nonzero lattice points.^[13] It satisfies the normalization condition \sigma(-z; \Lambda) = -\sigma(z; \Lambda), making it an odd function.^[13]^[2] Karl Weierstrass introduced the sigma function in 1856 as a means to quasi-periodize the infinite product representation of the sine function, extending its periodic properties to the doubly periodic setting of elliptic functions.^[15] The sigma function forms a foundational component in the theory of Weierstrass elliptic functions, providing an entire counterpart to the meromorphic Weierstrass \wp-function.^[13]

Quasi-Periodicity

The Weierstrass sigma function \sigma(z; \Lambda) exhibits quasi-periodic behavior with respect to the periods of the lattice \Lambda. Specifically, for the fundamental half-periods \omega_j (j=1,2), it satisfies the functional equations

\sigma(z + 2\omega_j; \Lambda) = -\exp\left(2\eta_j (z + \omega_j)\right) \sigma(z; \Lambda),

where the \eta_j are constants known as the quasi-periods.^[13] These relations indicate that \sigma(z; \Lambda) is not strictly periodic but acquires an exponential multiplier upon translation by the full periods $2\omega_j.^[15] The quasi-periods \eta_j are defined in terms of the Weierstrass zeta function \zeta(z; \Lambda), the logarithmic derivative of the sigma function, via \eta_j = \zeta(\omega_j; \Lambda).^[13] This connection arises because the zeta function itself is quasi-periodic: \zeta(z + 2\omega_j; \Lambda) = \zeta(z; \Lambda) + 2\eta_j.^[13] The quasi-periodicity can be derived from the infinite product representation of \sigma(z; \Lambda), which is an entire function with simple zeros at the lattice points. Taking the natural logarithm of this product yields a series expansion, and differentiating term by term with respect to z produces the zeta function \zeta(z; \Lambda) = \sigma'(z; \Lambda)/\sigma(z; \Lambda). Evaluating the shifted product \sigma(z + 2\omega_j; \Lambda) then reveals the exponential factor through summation over the lattice translates, confirming the functional equation.^[13] The sigma function is uniquely determined, up to a nonzero constant multiple, as the entire function with simple zeros precisely at the points of \Lambda, satisfying the above quasi-periodicity relations, and normalized such that \sigma(0; \Lambda) = 0 and \sigma'(0; \Lambda) = 1.^[15]

Relation to Elliptic Functions

The Weierstrass zeta function is defined as the logarithmic derivative of the sigma function:

\zeta(z; \Lambda) = \frac{\sigma'(z; \Lambda)}{\sigma(z; \Lambda)},

where \Lambda is the period lattice and the prime denotes differentiation with respect to z.^[16] This relation holds for z \notin \Lambda, with \zeta(z; \Lambda) being meromorphic and having simple poles at the lattice points with residue 1.^[17] The series expansion of the zeta function is

\zeta(z; \Lambda) = \frac{1}{z} + \sum_{w \in \Lambda \setminus \{0\}} \left( \frac{1}{z - w} + \frac{1}{w} + \frac{z}{w^2} \right),

which converges absolutely and uniformly on compact sets avoiding the lattice points.^[17] The Weierstrass elliptic \wp-function is then obtained by differentiating the zeta function:

\wp(z; \Lambda) = -\frac{d}{dz} \zeta(z; \Lambda).

Its series expansion follows as

\wp(z; \Lambda) = \frac{1}{z^2} + \sum_{w \in \Lambda \setminus \{0\}} \left( \frac{1}{(z - w)^2} - \frac{1}{w^2} \right),

converging similarly on compact sets excluding lattice points.^[18] This \wp-function is elliptic, doubly periodic with periods given by \Lambda, and satisfies the differential equation

\left( \wp'(z; \Lambda) \right)^2 = 4 \wp(z; \Lambda)^3 - g_2(\Lambda) \wp(z; \Lambda) - g_3(\Lambda),

where the invariants g_2(\Lambda) and g_3(\Lambda) characterize the lattice.^[19] Addition theorems for the \wp-function can be expressed in terms of ratios of sigma functions. For instance,

\frac{\sigma(u + v; \Lambda) \sigma(u - v; \Lambda)}{\sigma^2(u; \Lambda) \sigma^2(v; \Lambda)} = \wp(u; \Lambda) - \wp(v; \Lambda),

which facilitates derivations of formulas like the addition theorem \wp(z_1 + z_2; \Lambda) in terms of \wp(z_1; \Lambda), \wp(z_2; \Lambda), and \wp(z_1 - z_2; \Lambda).^[20] These relations underscore the sigma function's role in generating the algebraic structure of elliptic functions. The invariants are given by lattice sums:

g_2(\Lambda) = 60 \sum_{w \in \Lambda \setminus \{0\}} \frac{1}{w^4}, \quad g_3(\Lambda) = 140 \sum_{w \in \Lambda \setminus \{0\}} \frac{1}{w^6}.

These sums converge conditionally and determine the isomorphism class of the elliptic curve associated with \Lambda.^[19]

Applications

The Weierstrass sigma function played a pivotal role in Karl Weierstrass's unification of elliptic function theory during the 1860s, where he demonstrated that any meromorphic elliptic function could be expressed rationally in terms of the ℘-function and its first derivative, with the sigma function providing the essential entire function for constructing such expressions through quasi-periodic products.^[21] This framework, building on earlier work by Abel and Jacobi, established a rigorous algebraic addition theorem for elliptic functions, distinguishing them from degenerate cases like rational, trigonometric, or exponential functions.^[22] In the context of elliptic integrals, the sigma function inverts these integrals by expressing the argument z as a product analogous to theta functions, facilitating solutions to nonlinear differential equations such as \left( \frac{dy}{dz} \right)^2 = 4y^3 - g_2 y - g_3, where y = \wp(z + z_0) and \sigma(z) encodes the quasi-periodicity.^[23] Specifically, the inverse relation takes the form z = \int_{\wp(z)}^\infty \frac{du}{\sqrt{4u^3 - g_2 u - g_3}}, with \sigma(z) appearing in the explicit construction of the integrand's antiderivative via theta-like infinite products.^[15] The sigma function connects to modular forms through lattice invariants and Eisenstein series, particularly via a modified form \tilde{\sigma}(z) = \sigma(z) \exp(-\gamma_2 z^2 / 2), where \gamma_2 derives from the weight-2 Eisenstein series G_2(\tau), rendering it modular-invariant under SL(2,ℤ) transformations of the period ratio \tau = \omega_2 / \omega_1.^[24] This adjustment aligns the sigma function with quasi-modular forms, as Eisenstein's periodic completion of the related zeta function \zeta(z) = \frac{\sigma'(z)}{\sigma(z)} incorporates G_2 to achieve modular properties.^[24] In degenerate cases, as the modular parameter \tau \to i \infty, the sigma function limits to a trigonometric form: \sigma(z) \sim \frac{2 \omega_1}{\pi} \sin\left( \frac{\pi z}{2 \omega_1} \right) \exp\left( \frac{\eta_1 z^2}{2 \omega_1} \right), where \eta_1 is the quasi-period constant, bridging elliptic functions to ordinary trigonometric ones.^[2] Modern applications include its role in elliptic curve cryptography, where the Weierstrass form of elliptic curves y^2 = x^3 + a x + b is uniformized analytically using \wp(z) and \sigma(z) to parameterize points on the complex torus, aiding in the explicit computation of group operations for protocols like ECDH.^[25] Additionally, the sigma function factors explicitly in terms of Jacobi theta functions as

\sigma(z) = \frac{2 \omega_1}{\pi \theta_1'(0, q)} \exp\left( \frac{\eta_1 z^2}{2 \omega_1} \right) \theta_1 \left( \frac{\pi z}{2 \omega_1}, q \right),

where q = e^{\pi i \tau}, enabling efficient numerical evaluations and connections to theta-based algorithms in integrable systems.^[23]

Sigma function

Sum of Divisors Function

Definition

Examples

Multiplicative Properties

Analytic Formulas

Growth and Distribution

Weierstrass Sigma Function

Definition

Quasi-Periodicity

Relation to Elliptic Functions

Applications

References