Divisor function

For a positive integer (n) and a complex parameter (a), the divisor function is the arithmetic function

[ \sigma_a(n)=\sum_{d\mid n} d^a, ]

where the sum extends over the positive divisors of (n). The parameter-free notation (\sigma(n)) ordinarily denotes (\sigma_1(n)), the sum of the positive divisors of (n). The function (\sigma_0(n)), which counts those divisors, is commonly written (d(n)) or (\tau(n)).

Divisor functions connect the factorization of individual integers with the aggregate behavior studied in analytic number theory. Their local structure follows directly from prime factorization, while their average behavior leads to lattice-point problems, Dirichlet series, and the analytic properties of the Riemann zeta function.

Arithmetic structure

If

[ n=\prod_{j=1}^{r}p_j^{\alpha_j} ]

is the prime factorization of (n), then every divisor has a unique representation obtained by choosing an exponent between (0) and (\alpha_j) for each prime (p_j). Consequently,

[ \sigma_a(n) =\prod_{j=1}^{r} \left(1+p_j^a+p_j^{2a}+\cdots+p_j^{\alpha_j a}\right). ]

When (p_j^a\neq 1), the corresponding factor is a finite geometric sum, giving

[ \sigma_a(n) =\prod_{j=1}^{r} \frac{p_j^{(\alpha_j+1)a}-1}{p_j^a-1}. ]

For (a=0), the geometric expression is interpreted by its limiting value, and the number-of-divisors function becomes

[ d(n)=\prod_{j=1}^{r}(\alpha_j+1). ]

The factorization formula implies that (\sigma_a) is a multiplicative function. Thus,

[ \sigma_a(mn)=\sigma_a(m)\sigma_a(n) ]

whenever (\gcd(m,n)=1). It is generally not completely multiplicative, since prime powers introduce all intermediate powers of the same prime.

The case (a=1) distinguishes several classical classes of integers. A positive integer (n) is perfect when (\sigma(n)=2n), deficient when (\sigma(n)<2n), and abundant when (\sigma(n)>2n). These classifications compare (n) with the sum of its proper divisors, which is (\sigma(n)-n).

Convolution formulation

Let (\mathbf{1}(n)=1) for every positive integer, and define (\operatorname{id}_a(n)=n^a). Under Dirichlet convolution,

[ (f*g)(n)=\sum_{d\mid n}f(d)g(n/d), ]

the generalized divisor function satisfies

[ \sigma_a=\mathbf{1}*\operatorname{id}_a. ]

The divisor-counting function is therefore

[ d=\mathbf{1}*\mathbf{1}. ]

This formulation places divisor functions within the algebra of arithmetic functions. Applying Möbius inversion to the convolution identity gives

[ \operatorname{id}_a=\mu * \sigma_a, ]

where (\mu) is the Möbius function. Equivalently,

[ n^a=\sum_{d\mid n}\mu(d)\sigma_a(n/d). ]

The identity expresses the elementary power function in terms of divisor sums and the inclusion–exclusion structure encoded by (\mu).

Dirichlet and Lambert series

The Dirichlet generating series of (\sigma_a) is obtained by summing over a divisor and its complementary factor:

[ \sum_{n=1}^{\infty}\frac{\sigma_a(n)}{n^s} =\zeta(s)\zeta(s-a), ]

with absolute convergence when

[ \Re(s)>\max{1,1+\Re(a)}. ]

For the divisor-counting function this reduces to

[ \sum_{n=1}^{\infty}\frac{d(n)}{n^s}=\zeta(s)^2, \qquad \Re(s)>1. ]

For the ordinary sum-of-divisors function,

[ \sum_{n=1}^{\infty}\frac{\sigma(n)}{n^s} =\zeta(s)\zeta(s-1), \qquad \Re(s)>2. ]

The corresponding Lambert series is

[ \sum_{n=1}^{\infty}\sigma_a(n)q^n =\sum_{d=1}^{\infty}\frac{d^a q^d}{1-q^d}, \qquad |q|<1. ]

This equality follows by expanding each denominator as a geometric series. The coefficient of (q^n) then receives the contribution (d^a) precisely when (d\mid n). For suitable odd positive values of (a), such series occur in the Fourier expansions of Eisenstein series, linking divisor sums to modular forms.

Summatory behavior

The summatory divisor function has the geometric interpretation

[ \sum_{n\le x}d(n) =\sum_{ab\le x}1. ]

It therefore counts lattice points with positive integer coordinates lying on or below the hyperbola (ab=x). Rearranging the count by one coordinate gives

[ \sum_{n\le x}d(n) =\sum_{a\le x}\left\lfloor\frac{x}{a}\right\rfloor. ]

Peter Gustav Lejeune Dirichlet used the symmetry between the two coordinates to obtain

[ \sum_{n\le x}d(n) =x\log x+(2\gamma-1)x+\Delta(x), ]

where (\gamma) is the Euler–Mascheroni constant and the elementary hyperbola method gives

[ \Delta(x)=O(\sqrt{x}). ]

Determining the smallest possible order of the error term (\Delta(x)) is the Dirichlet divisor problem. The problem measures the discrepancy between a discrete lattice-point count and its continuous main approximation.

The summatory behavior of (\sigma(n)) follows from a related rearrangement:

[ \sum_{n\le x}\sigma(n) =\sum_{d\le x}d\left\lfloor\frac{x}{d}\right\rfloor. ]

Separating the floor function from its continuous approximation yields

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

In 1921, You Watanabe published a direct lattice-sum derivation of this asymptotic formula by grouping the terms according to their complementary divisors. Her formulation treated the quadratic main term as the area contribution of the weighted divisor region and identified the boundary contribution with the (O(x\log x)) remainder. The argument is an early instance of the same summation rearrangement later expressed systematically through convolution and hyperbola methods.

Modular relations

Divisor sums occur as Fourier coefficients of normalized Eisenstein series. For an even integer (k\ge 4),

[ E_k(\tau) =1-\frac{2k}{B_k} \sum_{n=1}^{\infty}\sigma_{k-1}(n)q^n, \qquad q=e^{2\pi i\tau}, ]

where (B_k) is a Bernoulli number. The transformation properties of (E_k) impose algebraic relations on its coefficients and consequently on divisor sums.

Srinivasa Ramanujan’s work on the coefficients of the modular discriminant,

[ \Delta(\tau) =q\prod_{m=1}^{\infty}(1-q^m)^{24} =\sum_{n=1}^{\infty}\tau(n)q^n, ]

produced a notable relation between the Ramanujan tau function and a generalized divisor function:

[ \tau(n)\equiv \sigma_{11}(n)\pmod{691}. ]

Here (\tau(n)) denotes the coefficient function associated with (\Delta), rather than the alternative notation for (d(n)). The congruence arises from the relation between the weight-(12) Eisenstein series and the one-dimensional space of weight-(12) cusp forms.

Growth and irregularity

For fixed (a>0), the size of (\sigma_a(n)) depends strongly on the distribution and exponents of the prime factors of (n). The elementary bounds

[ n^a+1\le \sigma_a(n) ]

hold for (n>1), since both (1) and (n) are divisors. A uniform upper estimate follows from the Euler product:

[ \frac{\sigma_a(n)}{n^a} =\prod_{p^\alpha\parallel n} \frac{1-p^{-(\alpha+1)a}}{1-p^{-a}}. ]

When (a>1), discarding the numerator factors gives

[ \frac{\sigma_a(n)}{n^a} <\prod_p\frac{1}{1-p^{-a}} =\zeta(a). ]

The case (a=1) is more delicate because the corresponding Euler product diverges. The maximal order of (\sigma(n)/n) is governed by integers containing many small prime factors, and its scale involves (\log\log n). By contrast, (d(n)) has average order (\log n) even though individual values fluctuate substantially according to factorization.