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Divisor function

In , particularly in , the divisor function, denoted \sigma_k(n), is an that computes the of the k-th powers of the positive divisors of a positive n. For a fixed nonnegative integer k, it is defined as \sigma_k(n) = \sum_{d \mid n} d^k, where the runs over all positive divisors d of n. This function generalizes several important concepts, with \sigma_0(n) (often denoted d(n) or \tau(n)) giving the total number of positive divisors of n, and \sigma_1(n) (commonly just \sigma(n)) yielding the of those divisors. The divisor function is multiplicative, meaning that if a and b are coprime positive integers, then \sigma_k(ab) = \sigma_k(a) \sigma_k(b) for any k. For n with prime factorization n = p_1^{e_1} p_2^{e_2} \cdots p_m^{e_m}, explicit formulas follow: \sigma_0(n) = \prod_{i=1}^m (e_i + 1) and \sigma_1(n) = \prod_{i=1}^m \frac{p_i^{e_i + 1} - 1}{p_i - 1}, with analogous products for higher k. These properties make the function central to the study of and divisor distributions. In , the divisor functions play a through their representations, such as \sum_{n=1}^\infty \sigma_0(n) n^{-s} = \zeta(s)^2 and more generally \sum_{n=1}^\infty \sigma_k(n) n^{-s} = \zeta(s) \zeta(s - k) for \Re(s) > \max(1, k + 1), where \zeta(s) is the . This connection links the functions to deep problems like the distribution of prime numbers and the , as bounds on \sigma_1(n) relate to error terms in prime-counting formulas. Additionally, \sigma_1(n) classifies numbers by abundancy: a number n is perfect if \sigma_1(n) = 2n, abundant if \sigma_1(n) > 2n, and deficient if \sigma_1(n) < 2n, with applications to ancient problems like Euclid-Euler perfect numbers. The maximal order of \sigma_1(n) is asymptotically e^\gamma n \log \log n, where \gamma is the Euler-Mascheroni constant, reflecting the divisor structure for highly composite numbers.

Definition and Notation

Formal Definition

The divides relation in number theory, denoted a \mid b for integers a and b with a \neq 0, holds if there exists an integer k such that b = a k. The positive divisors of a positive integer n are the positive integers d such that d \mid n. The divisor function, often denoted \sigma(n), is an arithmetic function that assigns to each positive integer n the sum of its positive divisors. Formally, \sigma(n) = \sum_{d \mid n} d. Here, the sum is taken over all positive divisors d of n. This distinguishes \sigma(n) from the divisor counting function d(n), also known as \tau(n), which is defined as d(n) = \sum_{d \mid n} 1, counting the number of positive divisors of n rather than summing their values. As a base case, consider n=1: the only positive divisor of $1 is $1 itself, so \sigma(1) = 1.

Generalizations and Variants

The divisor function generalizes naturally to the family of functions \sigma_k(n) for integers k \geq 0, defined as the sum of the kth powers of the positive divisors of n: \sigma_k(n) = \sum_{d \mid n} d^k. This encompasses the original as the special case \sigma_1(n) = \sigma(n), which sums the divisors themselves, while \sigma_0(n) counts the total number of divisors, often denoted d(n). The parameter k allows for a unified treatment of various divisor-related sums in analytic number theory, where positive integer values of k emphasize the magnitudes of divisors (as in \sigma_1(n)), k=0 focuses on cardinality, and extensions to negative exponents like k=-1 yield the sum of reciprocals of divisors, \sigma_{-1}(n) = \sum_{d \mid n} d^{-1} = \sigma(n)/n. Notationally, \sigma(n) is the conventional shorthand for \sigma_1(n), with higher k values sometimes linked to the through Dirichlet series representations. Jordan's totient functions J_k(n) serve as higher-order analogs, extending the counting of coprime elements in a manner complementary to the power sums of \sigma_k(n).

Examples and Computations

Illustrative Examples

To illustrate the divisor function σ(n), which sums the positive divisors of a positive integer n, consider n = 6. The divisors of 6 are 1, 2, 3, and 6. Adding these gives σ(6) = 1 + 2 + 3 + 6 = 12. For a prime number p, the only positive divisors are 1 and p, so σ(p) = 1 + p. For example, with p = 5, σ(5) = 1 + 5 = 6. Now consider a prime power such as p^2, where p is prime. For p = 3, this is n = 9, with divisors 1, 3, and 9, yielding σ(9) = 1 + 3 + 9 = 13. For n = 12, the divisors are 1, 2, 3, 4, 6, and 12. Summing these produces σ(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. A number n is perfect if the sum of its proper divisors equals n, or equivalently, if σ(n) = 2n. The smallest such number is 6, since σ(6) - 6 = 6.

Tables of Values

The divisor function \sigma(n), which sums the positive divisors of n, and the divisor count d(n) provide foundational data for understanding the distribution of divisors. The following table presents values for n = 1 to $20, allowing comparison between the sum and the count of divisors.
nd(n)\sigma(n)
111
223
324
437
526
6412
728
8415
9313
10418
11212
12628
13214
14424
15424
16531
17218
18639
19220
20642
For n up to 100, the table below focuses on perfect numbers (where \sigma(n) = 2n), abundant numbers (where \sigma(n) > 2n), and a selection of deficient numbers (where \sigma(n) < 2n) to illustrate classification based on divisor sums. Perfect numbers up to 100 are 6 and 28; abundant numbers include all even multiples of 12 in this range except those exceeding the threshold subtly, with the full list comprising 21 terms, all even. Deficient numbers dominate, comprising most integers in this range.
nType\sigma(n)\sigma(n) - 2n
1Deficient1-1
2Deficient3-1
3Deficient4-2
4Deficient7-1
5Deficient6-4
6120
7Deficient8-6
12284
18393
20422
246012
28560
307212
369119
409010
429612
4812428
6016848
9019212
9625260
10021717
The generalized divisor function \sigma_k(n) sums the k-th powers of the divisors of n, with \sigma_0(n) = d(n), \sigma_1(n) = \sigma(n), and \sigma_2(n) as the sum of squares. The table below gives values for select n, including primes (where patterns emerge clearly) and squares, to highlight differences across k.
nType\sigma_0(n)\sigma_1(n)\sigma_2(n)
1-111
2Prime235
3Prime2410
4Square3721
5Prime2626
6-41250
7Prime2850
9Square31391
11Prime212122
12-628210
25Square331651
36Square9911911
These tables reveal observable patterns, such as for prime p, \sigma(p) = p + 1 (with d(p) = 2 and \sigma_2(p) = 1 + p^2), and increasing complexity for composite forms like squares, where divisor counts and sums grow nonlinearly. For instance, \sigma(6) = 12 exemplifies a perfect number's balance.

Fundamental Properties

Multiplicativity

In number theory, an arithmetic function f: \mathbb{N} \to \mathbb{C} is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever \gcd(m, n) = 1. This property captures the behavior of functions that respect the prime factorization structure of integers in a compatible way. The generalized divisor function \sigma_k(n) = \sum_{d \mid n} d^k, for fixed k \in \mathbb{R}, is multiplicative. To see this, suppose \gcd(m, n) = 1. The divisors of mn are precisely the products d_1 d_2 where d_1 \mid m and d_2 \mid n, and this representation is unique because m and n are coprime. Thus, \sigma_k(mn) = \sum_{d_1 \mid m} \sum_{d_2 \mid n} (d_1 d_2)^k = \sum_{d_1 \mid m} d_1^k \sum_{d_2 \mid n} d_2^k = \sigma_k(m) \sigma_k(n), since (d_1 d_2)^k = d_1^k d_2^k. Additionally, \sigma_k(1) = 1^k = 1. This establishes the multiplicativity of \sigma_k for any fixed k. A key consequence of this multiplicativity is that \sigma_k(n) can be computed directly from the prime factorization of n. If n = \prod_{i=1}^r p_i^{e_i} where the p_i are distinct primes and e_i \geq 1, then the factors p_i^{e_i} are pairwise coprime, so \sigma_k(n) = \prod_{i=1}^r \sigma_k(p_i^{e_i}). For the standard divisor function \sigma(n) = \sigma_1(n), this yields \sigma(n) = \prod_{p^e \parallel n} (1 + p + p^2 + \cdots + p^e), where the product runs over the distinct prime powers in the factorization of n. This extends naturally to the general case: each local factor is a finite geometric series, \sigma_k(p^e) = 1^k + p^k + p^{2k} + \cdots + (p^e)^k = \frac{1 - p^{k(e+1)}}{1 - p^k}, provided p^k \neq 1 (which holds since p \geq 2 and k is fixed). Thus, \sigma_k(n) = \prod_{p^e \parallel n} \frac{1 - p^{k(e+1)}}{1 - p^k}. This formula facilitates efficient computation and analysis of \sigma_k(n) by reducing it to the prime factors of n.

Formulas for Prime Powers

The divisor function \sigma_k(n) restricted to a prime power n = p^e, where p is prime and e \geq 0 is a non-negative integer, sums the k-th powers of its divisors. The divisors of p^e are precisely $1, p, p^2, \dots, p^e, so \sigma_k(p^e) = \sum_{i=0}^e (p^i)^k = \sum_{i=0}^e p^{ki}. This is the partial sum of a geometric series with first term 1 and common ratio r = p^k. The closed-form expression is \sigma_k(p^e) = \frac{1 - p^{k(e+1)}}{1 - p^k}, valid when p^k \neq 1, which holds for integer k \geq 1 since p > 1. For the special case k=1, the sum-of-divisors function \sigma(p^e) simplifies to \sigma(p^e) = \frac{p^{e+1} - 1}{p - 1}, a formula central to many applications in number theory, such as determining perfect numbers. When k=0, \sigma_0(n) counts the number of positive divisors of n, so for prime powers, \sigma_0(p^e) = e + 1. This follows directly from the count of terms in the sum \sum_{i=0}^e p^{0 \cdot i} = \sum_{i=0}^e 1 = e+1.

Identities and Relations

Dirichlet Convolutions

The Dirichlet convolution of two f and g, both defined on the positive integers, is the (f * g) given by (f * g)(n) = \sum_{d \mid n} f(d) \, g\left(\frac{n}{d}\right) for each positive integer n. This operation is associative and commutative, and it endows the set of arithmetic functions with a ring structure under pointwise addition and as multiplication. The divisor function \sigma(n) = \sum_{d \mid n} d can be expressed as the Dirichlet convolution of the constant function u(n) = 1 for all n \geq 1 and the \mathrm{id}(n) = n, so \sigma = u * \mathrm{id}. More generally, the k-th power divisor function \sigma_k(n) = \sum_{d \mid n} d^k is the u * \mathrm{id}^k, where \mathrm{id}^k(n) = n^k. The Möbius function \mu, defined on positive integers with \mu(n) = 0 if n has a squared prime factor, \mu(n) = 1 if n has an even number of distinct prime factors, and \mu(n) = -1 if odd, serves as the convolutional inverse of u, satisfying u * \mu = \varepsilon, where \varepsilon is the unit function with \varepsilon(1) = 1 and \varepsilon(n) = 0 for n > 1. Applying this to the divisor functions yields \mu * \sigma_k = \mathrm{id}^k, or equivalently, n^k = \sum_{d \mid n} \mu(d) \, \sigma_k\left(\frac{n}{d}\right) for each positive integer n and integer k \geq 0; this identity is a direct consequence of Möbius inversion and provides a recursive way to compute powers from divisor sums.

Other Key Identities

One notable identity involving the divisor function \sigma(n) is the summation formula \sum_{d \mid n} \sigma(d) = \sum_{a \mid n} a \cdot \tau(n/a), where \tau(k) denotes the number of positive divisors of k. This arises from expanding \sigma(d) = \sum_{e \mid d} e and reindexing the double sum over divisors, counting each possible a by the number of compatible b such that ab \mid n. A key inversion identity, derived via Möbius inversion applied to the relation \sigma = \mathrm{id} * \mathbf{1} (where \mathrm{id}(n) = n and \mathbf{1}(n) = 1), states that \sum_{d \mid n} \sigma(d) \mu(n/d) = n, with \mu the . This expresses n as a signed sum of divisor sums weighted by the parity of prime factors in the complementary divisors. For even positive integers of the form n = 2m with m odd, the multiplicativity of \sigma yields \sigma(n) = \sigma(2) \sigma(m) = (1 + 2) \sigma(m) = 3 \sigma(m). This simplifies computations for numbers with a single factor of 2 and an odd part. In contrast to \sigma(n) = \sum_{d \mid n} d, which sums the divisors themselves, Euler's totient function \phi satisfies \sum_{d \mid n} \phi(d) = n, summing the counts of integers up to each divisor coprime to it. Both identities partition n via its divisors but capture different arithmetic structures.

Analytic Aspects

Generating Functions and Series

The Dirichlet series associated with the divisor function \sigma(n), defined as the sum of the positive divisors of 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 . This representation arises from the multiplicative structure of \sigma(n) and the known Euler product for \zeta(s), allowing the series to be expressed in terms of zeta functions whose analytic properties are well-understood. More generally, for the k-th power divisor function \sigma_k(n) = \sum_{d \mid n} d^k, the corresponding Dirichlet series is \sum_{n=1}^\infty \frac{\sigma_k(n)}{n^s} = \zeta(s) \zeta(s - k) valid for \operatorname{Re}(s) > \max(1, k+1). This formula extends the case for \sigma(n) = \sigma_1(n) and follows from the convolution identity \sigma_k(n) = \sum_{ab = n} a^k, which corresponds to the product of the generating functions \zeta(s) and \sum_{m=1}^\infty m^k / m^s = \zeta(s - k). Due to the multiplicativity of \sigma_k(n), the Dirichlet series admits an Euler product representation: \sum_{n=1}^\infty \frac{\sigma_k(n)}{n^s} = \prod_p \left( \sum_{e=0}^\infty \frac{\sigma_k(p^e)}{p^{e s}} \right), where the product runs over all primes p. The local factor at each prime is \sum_{e=0}^\infty \frac{\sigma_k(p^e)}{p^{e s}} = \sum_{e=0}^\infty \frac{1 + p^k + \cdots + p^{e k}}{p^{e s}} = \frac{1}{(1 - p^{-s})(1 - p^{k - s})}, yielding the full product \prod_p (1 - p^{-s})^{-1} (1 - p^{k - s})^{-1} = \zeta(s) \zeta(s - k). This Euler product form highlights the prime factorization underlying the divisor sums and facilitates and study of the series beyond its region of . The Dirichlet series for \sigma_k(n) also provides a framework for approximating partial sums \sum_{n \leq x} \sigma_k(n) through representations. Specifically, Perron's formula expresses such partial sums as \sum_{n \leq x} \sigma_k(n) = \frac{1}{2\pi i} \int_{c - iT}^{c + iT} \zeta(s) \zeta(s - k) \frac{x^s}{s} \, ds + O\left( \frac{x \log x}{T} \right), for c > \max(1, k+1) and suitable T > 0, allowing approximations by evaluating the via residues or shifting . This approach bridges the to explicit estimates for cumulative divisor sums, leveraging the poles of \zeta(s) at s=1 and s = k+1 (if integer k \geq 0) to capture the main terms. In particular, the average order of \sigma_k(n) follows from this, with \sum_{n \leq x} \sigma_k(n) \sim \frac{\zeta(k+1)}{k+1} x^{k+1} for \Re(k) > 0.

Growth Rate and Asymptotics

The divisor function d(n), which counts the number of positive divisors of the positive integer n, exhibits irregular growth, but its asymptotic behavior can be analyzed using tools from , including and probabilistic models of . The generating function for d(n) is the \zeta(s)^2, whose pole structure informs the average behavior, though detailed growth analysis requires more advanced techniques such as the circle method or estimates on the distribution of prime factors. The average order of d(n) is given by the asymptotic \frac{1}{x} \sum_{n \le x} d(n) \sim \log x as x \to \infty, a classical result due to Dirichlet that follows from the partial of the series or the residue at the of \zeta(s)^2. This implies that d(n) grows like \log n on average, reflecting the typical number of divisors contributed by small prime factors. Representative examples include d(12) = 6 and d([60](/page/60)) = 12, consistent with the for moderately composite numbers. For the normal order, which describes the typical value for almost all n \le x (in the sense that the proportion of exceptions tends to 0 as x \to \infty), and Ramanujan proved that \log d(n) \sim (\log 2) \log \log n. This means that d(n) \sim \exp( (\log 2) \log \log n ) for n, arising from the fact that integers have approximately \log \log n distinct prime factors, each contributing a factor of 2 to d(n) in the square-free case, with higher powers being rare. More precisely, the extends this to a for the number of prime factors, implying the limiting distribution for \log d(n) / \log \log n concentrates around \log 2. An upper bound on the maximal order of d(n) is provided by Wigert's theorem, which states that \limsup_{n \to \infty} \frac{\log d(n)}{\log n / \log \log n} = \log 2, equivalent to d(n) < n^{c / \log \log n} for any c > \log 2 and sufficiently large n. This bound is sharp in the sense of the lim sup and is achieved along sequences of highly composite numbers where the exponents in the prime factorization are decreasing. The precise maximal order of d(n) for n \le x is more refined and given asymptotically by \log d(n) \sim \frac{\log n \cdot \log \log \log n}{\log \log n}, with lower bounds matching this form up to lower-order terms, as established by Ford using advanced estimates on the distribution of smooth numbers and the geometry of numbers. This growth is realized for superior highly composite numbers, where the prime exponents are chosen to maximize the divisor count relative to the size. For the sum-of-divisors function \sigma_1(n), the maximal order is given by Gronwall's theorem: \limsup_{n \to \infty} \frac{\sigma_1(n)}{n \log \log n} = e^\gamma, where \gamma \approx 0.57721 is the Euler-Mascheroni constant.

Historical Context and Applications

Historical Development

The study of the divisor function traces its origins to ancient Greek mathematics, where it emerged in the context of perfect numbers—positive integers equal to the sum of their proper divisors. Euclid, in his Elements (circa 300 BCE), provided the first systematic treatment by describing a construction for even perfect numbers of the form $2^{p-1}(2^p - 1), where $2^p - 1 is prime, implicitly relying on the summation of divisors to verify perfection. This marked an early milestone in recognizing divisor sums as a tool for classifying numbers based on their aliquot parts. Nicomachus of Gerasa further elaborated on this around 100 CE in his Introduction to Arithmetic, categorizing numbers as perfect, deficient, or abundant according to whether the sum of proper divisors equals, is less than, or exceeds the number itself, though without new formulas. In the medieval Islamic world, interest in divisor sums intensified through explorations of amicable and perfect numbers. Thābit ibn Qurra (circa 836–901 CE), an Arab scholar in Baghdad, advanced the theory by developing a rule for generating pairs of amicable numbers—pairs where each is the sum of the proper divisors of the other—such as (220, 284), building on earlier Greek ideas and introducing algebraic criteria involving Mersenne-like primes. This work highlighted the divisor function's role in relational number properties, influencing subsequent European mathematicians during the Renaissance. The 18th and 19th centuries saw significant formalization of the divisor function in European number theory. Leonhard Euler, in correspondence and papers from the 1730s–1740s, proved that all even perfect numbers conform to Euclid's form, using the sum-of-divisors function \sigma(n) to confirm the equality of aliquot sums, and extended its application to abundant and deficient numbers. Peter Gustav Lejeune Dirichlet, in his seminal 1837 memoir, introduced the concept of Dirichlet convolution for arithmetic functions, including the divisor function, as part of his proof that primes are infinite in arithmetic progressions; this operation, defined as (f * g)(n) = \sum_{d|n} f(d) g(n/d), provided a foundational framework for multiplicative properties and generating functions in analytic number theory. Early 20th-century developments emphasized congruences and asymptotic behavior. , in unpublished notebooks and papers from the 1910s, discovered numerous identities and congruences involving the divisor function d(n), revealing patterns that spurred further research into its distribution. and J.E. Littlewood, in works from 1915–1922, analyzed the Dirichlet divisor problem—the asymptotic growth of \sum_{n \leq x} d(n) \sim x \log x + (2\gamma - 1)x + \Delta(x)—establishing lower bounds like \Delta(x) = \Omega(\sqrt{x} \log x) and improving error term estimates, which quantified the function's oscillatory behavior. Post-1950 advancements focused on computational bounds and generalizations of the divisor problem. In the 1950s, I.M. Vinogradov and others refined exponent bounds for \Delta(x) \ll x^\alpha, with subsequent improvements by J. van der Corput and I. Kátai achieving \alpha < 1/3. Martin Huxley's 2003 result established \alpha \leq 131/416 \approx 0.315, the strongest unconditional bound as of 2003, using advanced spectral methods and Weyl sums, while generalizations extended to higher-order divisor functions \sigma_k(n) and arithmetic progressions.

Applications in Number Theory

The divisor function \sigma(n) plays a key role in advanced classifications and analytic tools in number theory. Even perfect numbers, which are the only known perfect numbers, take the form n = 2^{p-1}(2^p - 1) where p is prime and $2^p - 1 is a ; in this case, the condition \sigma(2^p - 1) = 2^p ensures \sigma(n) = 2n, as established by Euler using the multiplicativity of \sigma. The Dirichlet series associated with \sigma(n) is \sum_{n=1}^\infty \frac{\sigma(n)}{n^s} = \zeta(s) \zeta(s-1) for \Re(s) > 2, linking the divisor function intimately to the \zeta(s). This Euler product representation arises from the multiplicativity of \sigma(n) and reflects the arithmetic structure of divisors. The analytic continuation of this series, governed by the poles and zeros of \zeta(s), facilitates the asymptotic analysis of partial s \sum_{n \leq x} \sigma(n) \sim \frac{\pi^2}{12} x^2, where the main term stems from the pole of \zeta(s-1) at s=2. These asymptotics draw on the , as the error terms depend on the zero-free region of \zeta(s) near \Re(s)=1, which underpins the theorem's proof via Tauberian methods and informs the distribution of prime factors influencing divisors. In the realm of modular forms, \sigma_k(n)—the sum of the k-th powers of divisors of n—appears as coefficients in the Fourier expansion of E_{k+1}(\tau) = 1 - \frac{2(k+1)}{B_{k+1}} \sum_{n=1}^\infty \sigma_k(n) q^n for even integer weight k+1 \geq 4 and q = e^{2\pi i \tau}. These series are non-holomorphic or holomorphic modular forms of level 1, bridging elementary to automorphic representations. serve as eigenforms under the action of Hecke operators T_m, with eigenvalue \sigma_k(m), highlighting how the divisor function encodes the arithmetic action of these operators on the space of modular forms. This interplay extends to more general settings, such as twisted Eisenstein series, where eigenvalues involve Dirichlet convolutions with characters. Sieve methods provide tools for estimating sums involving the divisor function, particularly in restricted settings like short intervals. For example, by applying combinatorial sieves to control the prime factors of divisors, one can derive non-trivial bounds on \sum_{n \in [x, x+y]} \sigma(n) for y = o(x), revealing the function's local fluctuations. Such estimates, which improve upon trivial bounds, rely on inclusion-exclusion over divisor sets and have applications in understanding the irregularity of \sigma(n) near x, often yielding results like \sigma(n) \ll x (\log x)^{O(1)} on average in short ranges. These techniques, rooted in the general sieve framework, complement analytic approaches by offering elementary yet effective control over divisor distributions.

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