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Multiplicative function

In number theory, a multiplicative function is an arithmetic function f: \mathbb{N} \to \mathbb{C} such that f(1) = 1 and f(mn) = f(m)f(n) whenever \gcd(m, n) = 1. This property distinguishes multiplicative functions from completely multiplicative functions, which satisfy f(mn) = f(m)f(n) for all positive integers m and n, without the coprimality condition. A fundamental property of multiplicative functions is that they are completely determined by their values on prime powers p^k, owing to the unique prime of integers. For any positive integer n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r}, f(n) = f(p_1^{e_1}) f(p_2^{e_2}) \cdots f(p_r^{e_r}). The Dirichlet convolution of two multiplicative functions is also multiplicative, forming an under this operation with the unit function \varepsilon(n) (where \varepsilon(1) = 1 and \varepsilon(n) = 0 otherwise) as the . Prominent examples include the Euler totient function \varphi(n), which counts the integers up to n coprime to n and is multiplicative but not completely multiplicative; the Möbius function \mu(n), defined as \mu(n) = 1 if n is a square-free positive with an even number of prime factors, \mu(n) = -1 if odd, and $0 otherwise, which is also multiplicative; the divisor function \tau(n) or d(n), counting the number of positive divisors of n; and the sum-of-divisors function \sigma(n), summing the positive divisors of n. All of these are multiplicative, with explicit formulas on prime powers: for instance, \tau(p^k) = k+1 and \sigma(p^k) = 1 + p + \cdots + p^k = \frac{p^{k+1} - 1}{p-1}. Multiplicative functions are central to , enabling the Euler product representation of , such as the \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_p (1 - p^{-s})^{-1}, and facilitating tools like Möbius inversion, which states that if F(n) = \sum_{d|n} f(d), then f(n) = \sum_{d|n} \mu(d) F(n/d), preserving multiplicativity. They also arise in applications like perfect numbers, where even perfect numbers satisfy \sigma(n) = 2n.

Definition and Fundamentals

Definition

In , an is a f: \mathbb{N} \to \mathbb{C} defined on the positive integers. Such a f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever \gcd(m, n) = 1. This condition implies that the values of f on arbitrary positive integers are determined solely by its values on prime powers, via the : if n = \prod_{i} p_i^{a_i} for distinct primes p_i and positive integers a_i, then f(n) = \prod_{i} f(p_i^{a_i}). This notion of multiplicativity is distinct from that of a completely multiplicative function, which satisfies f(mn) = f(m)f(n) for all positive integers m and n, without the coprimality restriction.

Characterization

A f is completely determined by its values on prime powers p^k for primes p and integers k \geq 0, with the normalization f(1) = 1. Specifically, if n = \prod_p p^{a_p} is the of n > 1, then f(n) = \prod_p f(p^{a_p}), where the product runs over the primes dividing n. This follows from the unique factorization theorem and the multiplicativity condition f(mn) = f(m)f(n) whenever \gcd(m,n) = 1. An equivalent characterization arises from the Euler product representation of the associated . For a multiplicative function f, the D(f, s) = \sum_{n=1}^\infty f(n) n^{-s} admits an Euler product expansion D(f, s) = \prod_p \left( \sum_{k=0}^\infty \frac{f(p^k)}{p^{ks}} \right), valid in the half-plane \operatorname{Re}(s) > \sigma where the series converges absolutely, for some \sigma depending on f. This product converges absolutely under the same conditions as the series, and the equality holds because every n factors uniquely into prime powers, allowing the double sum over primes and exponents to rearrange into the infinite product. Conversely, if an arithmetic function's factors into such a product over primes with local factors depending only on prime powers, then the function is multiplicative. Multiplicativity is also preserved under Dirichlet convolution with the constant function \mathbf{1}(n) = 1 for all n. If g(n) = \sum_{d \mid n} f(d) = (f * \mathbf{1})(n), then f is multiplicative if and only if g is multiplicative. The forward direction holds because the convolution of two multiplicative functions (here f and \mathbf{1}) is multiplicative: for coprime m, n, the divisors of mn are products of divisors of m and n, so g(mn) = g(m)g(n). The converse follows by Möbius inversion: f(n) = \sum_{d \mid n} \mu(d) g(n/d) = (\mu * g)(n), where \mu is the , which is multiplicative; thus, the convolution \mu * g is multiplicative whenever g is. To verify whether a given f is multiplicative, one must check the defining property f(mn) = f(m)f(n) for all pairs of coprime positive integers m, n. In practice, for computational or theoretical purposes, this involves evaluating f on products of coprime arguments (such as distinct s) and confirming the product equality holds; since multiplicativity implies determination by prime power values, consistency across such pairs suffices to establish the property.

Basic Properties

Arithmetic Properties

Multiplicative functions exhibit several key arithmetic properties that arise directly from their definition, particularly in how they interact under pointwise , Dirichlet , and over divisors. If f and g are multiplicative arithmetic functions, then their pointwise product h(n) = f(n)g(n) is also multiplicative. Similarly, the Dirichlet (f \ast g)(n) = \sum_{d \mid n} f(d) g(n/d) of two multiplicative functions is multiplicative. These closure properties form the basis for the Dirichlet ring structure, where multiplicative functions act as units. A fundamental consequence is the multiplicativity of divisor sums. For a multiplicative function f, the sum-over-divisors function \sigma_f(n) = \sum_{d \mid n} f(d) is itself multiplicative. If n = \prod_p p^a is the prime factorization of n, then \sigma_f(n) = \prod_p \left( \sum_{k=0}^a f(p^k) \right). This Euler-like product formula highlights how the property propagates through the prime factors independently. The average value of a multiplicative function also displays multiplicative behavior asymptotically. Specifically, the mean (1/x) \sum_{n \leq x} f(n) for large x can be expressed in a form that factors multiplicatively over primes, reflecting the function's structure on prime powers. Regarding compositions, if f is multiplicative and h is an arbitrary , then f \circ h is not necessarily multiplicative. However, in specific cases where f is completely multiplicative and h preserves the coprimality condition—such as h = \sigma, the sum-of-divisors function—then f(\sigma(n)) is multiplicative, since \sigma(mn) = \sigma(m)\sigma(n) for \gcd(m,n)=1.

Analytic Properties

Multiplicative functions exhibit rich analytic structure through their associated , defined as D_f(s) = \sum_{n=1}^\infty \frac{f(n)}{n^s} for \operatorname{Re}(s) > \sigma_a, where \sigma_a is the abscissa of . A fundamental property is the Euler product representation, which arises from the multiplicativity of f. Specifically, for a multiplicative function f, the Dirichlet series factors as D_f(s) = \prod_p \left( \sum_{k=0}^\infty \frac{f(p^k)}{p^{ks}} \right), where the product is over all primes p, and this equality holds in the half-plane of . This representation is particularly useful when |f(n)| \leq 1 for all n, ensuring for \operatorname{Re}(s) > 1, as the partial products remain bounded and the converges uniformly on compact sets in this region. The of for multiplicative functions depends on the growth of f(p^k). If |f(p^k)| \leq p^{k\sigma} for some \sigma < 1 and all primes p and exponents k \geq 1, the local Euler factors \sum_{k=0}^\infty |f(p^k)| p^{-k\sigma} converge for \operatorname{Re}(s) > \sigma, yielding of D_f(s) in a half-plane extending leftward of \operatorname{Re}(s) = 1. This condition allows beyond the critical line in many cases, facilitating the study of . For bounded multiplicative functions with |f(n)| \leq 1, the series converges absolutely for \operatorname{Re}(s) > 1, mirroring the behavior of the , though slower growth at prime powers can improve the region of convergence. For non-vanishing multiplicative functions f whose Euler product takes the geometric form \prod_p (1 - f(p) p^{-s})^{-1} (as occurs when f is completely multiplicative), the of the connects directly to prime distributions via the \Lambda. In this setting, \frac{D_f'(s)}{D_f(s)} = -\sum_{n=1}^\infty \frac{\Lambda(n) f(n)}{n^s}, valid for \operatorname{Re}(s) > 1. This relation encodes arithmetic information about primes weighted by f, generalizing the classical identity -\zeta'(s)/\zeta(s) = \sum_{n=1}^\infty \Lambda(n) n^{-s}. Such derivatives prove instrumental in deriving asymptotic estimates and zero-free regions for D_f(s). The analytic properties of multiplicative functions are intimately linked to the \zeta(s), whose Dirichlet series corresponds to the constant multiplicative function f(n) = 1, yielding D_f(s) = \zeta(s) with absolute convergence for \operatorname{Re}(s) > 1. More generally, powers of \zeta(s) arise from Dirichlet series of other multiplicative functions, such as the k-fold d_k(n), where \sum_{n=1}^\infty d_k(n) n^{-s} = \zeta(s)^k for \operatorname{Re}(s) > 1. This factoring enables the decomposition of complex series into products over primes, highlighting the role of multiplicativity in .

Examples

Classical Examples

One of the most prominent examples of a multiplicative function is the \mu(n), defined on the positive integers. For a positive integer n with prime factorization n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}, \mu(n) = 0 if any k_i \geq 2 (i.e., if n has a squared prime factor), \mu(n) = 1 if n = 1, and \mu(n) = (-1)^r if n is square-free with exactly r distinct prime factors. Specifically, \mu(p) = -1 for a prime p, and \mu(p^k) = 0 for k \geq 2. Another classical example is \phi(n), which counts the number of positive integers up to n that are relatively prime to n. For a p^k, \phi(p^k) = p^k - p^{k-1}. This function arises naturally in the study of cyclic groups and . The , often denoted d(n) or \tau(n), gives the number of positive divisors of n. For n = p^k, d(p^k) = k + 1. It provides a simple measure of the arithmetic structure of n based on its prime factors. The sum-of-divisors function \sigma(n) sums the positive divisors of n. For a , \sigma(p^k) = \frac{p^{k+1} - 1}{p - 1}. This function is central to the classification of perfect numbers and abundance in . Finally, the \lambda(n) is defined as \lambda(1) = 1 and \lambda(n) = (-1)^{\Omega(n)} for n > 1, where \Omega(n) counts the total number of prime factors of n counted with multiplicity (i.e., \Omega(p^k) = k). It serves as an indicator of the of the number of prime factors.

Dirichlet Characters

A Dirichlet character modulo q is a completely multiplicative arithmetic function \chi: \mathbb{Z} \to \mathbb{C} that is periodic with period q, satisfies \chi(n) = 0 whenever \gcd(n, q) > 1, and induces a group homomorphism from (\mathbb{Z}/q\mathbb{Z})^* to the multiplicative group of complex numbers of modulus 1. These functions form a key class of multiplicative functions in analytic number theory, leveraging the structure of the unit group modulo q. The principal Dirichlet character \chi_0 modulo q is defined by \chi_0(n) = 1 if \gcd(n, q) = 1 and \chi_0(n) = 0 otherwise; it corresponds to the trivial homomorphism on (\mathbb{Z}/q\mathbb{Z})^*. There are exactly \phi(q) Dirichlet characters modulo q, where \phi is , forming an abelian group under pointwise multiplication that is isomorphic to the Pontryagin dual of (\mathbb{Z}/q\mathbb{Z})^*. A \chi modulo q is primitive if it is not induced from a character modulo d for any proper d of q; the smallest such modulus is called the of \chi. Primitive characters capture the irreducible representations in this context and play a central role in applications like Dirichlet's theorem on primes in arithmetic progressions. The Dirichlet characters modulo q satisfy the relation \sum_{\chi \bmod q} \chi(a) \overline{\chi(b)} = \begin{cases} \phi(q) & \text{if } a \equiv b \pmod{q} \text{ and } \gcd(a, q) = 1, \\ 0 & \text{otherwise}, \end{cases} where the sum is over all \phi(q) characters \chi modulo q and \overline{\chi} denotes the . This relation, derived from the orthogonality of group characters, enables the of indicator functions for residue classes coprime to q. Associated to each Dirichlet character \chi is the L-function L(s, \chi) = \sum_{n=1}^\infty \chi(n) n^{-s}, which for \operatorname{Re}(s) > 1 admits the Euler product representation L(s, \chi) = \prod_p (1 - \chi(p) p^{-s})^{-1}. For the principal character, L(s, \chi_0) relates to the Riemann zeta function by removing factors for primes dividing q. These L-functions extend meromorphically and are nonzero at s=1 for non-principal \chi, underpinning density results in number theory.

Dirichlet Convolution and Series

Dirichlet Convolution

The Dirichlet convolution provides a fundamental binary operation on the set of arithmetic functions, enabling the study of their algebraic structure in number theory. For two arithmetic functions f and g, their Dirichlet convolution (f \ast g) is defined by (f \ast g)(n) = \sum_{d \mid n} f(d) \, g\left( \frac{n}{d} \right) for each positive integer n, where the sum runs over all positive divisors d of n. This operation is both commutative, satisfying f \ast g = g \ast f, and associative, satisfying (f \ast g) \ast h = f \ast (g \ast h) for any arithmetic functions f, g, and h. These properties follow directly from the summation over divisors and the symmetry in the arguments. A key feature of Dirichlet convolution is its preservation of multiplicativity. Specifically, if both f and g are multiplicative functions, then their convolution f \ast g is also multiplicative. This closure property arises because, for m and n, the divisors of mn are products of divisors of m and n, allowing the convolution to factor accordingly over prime powers. The for Dirichlet convolution is the unit function \varepsilon, defined by \varepsilon(1) = 1 and \varepsilon(n) = 0 for all n > 1. Every f with f(1) \neq 0 possesses a unique Dirichlet inverse h, satisfying f \ast h = \varepsilon and h \ast f = \varepsilon. If f is multiplicative, then so is its inverse h. The inverse is explicitly given by Möbius inversion: for n \geq 1, h(n) = \sum_{d \mid n} \mu(d) \, f\left( \frac{n}{d} \right), where \mu denotes the . This formula inverts the convolution through the inclusion-exclusion principle inherent to the divisors. Under pointwise addition and , the set of all arithmetic functions forms a with unity \varepsilon. In this ring, the multiplicative functions with value 1 at 1 constitute a submonoid under convolution. This algebraic framework underpins many results in , such as the decomposition of functions via their inverses.

Dirichlet Series and Euler Products

A fundamental property linking to is the multiplicative behavior of their associated . For arithmetic functions f and g, the Dirichlet series of their convolution f \ast g is the product of their individual : D_{f \ast g}(s) = \sum_{n=1}^\infty (f \ast g)(n) n^{-s} = D_f(s) D_g(s), where D_f(s) = \sum_{n=1}^\infty f(n) n^{-s} and similarly for D_g(s). This holds for \Re(s) in the half-plane where both series converge absolutely. When f and g are multiplicative functions, the Dirichlet series D_{f \ast g}(s) admits an Euler product representation that factors correspondingly. Specifically, D_{f \ast g}(s) = \prod_p \left( \sum_{k=0}^\infty (f \ast g)(p^k) p^{-k s} \right) = \left[ \prod_p \sum_{k=0}^\infty f(p^k) p^{-k s} \right] \left[ \prod_p \sum_{k=0}^\infty g(p^k) p^{-k s} \right], valid for \Re(s) > \sigma_a, the abscissa of . This factorization arises because the convolution of two multiplicative functions is multiplicative, allowing the local factors at each prime p to multiply independently. A classical example illustrates this connection: the Dirichlet series for the divisor function d(n), which counts the number of positive divisors of n and arises as the convolution of the constant function $1 with itself (d = 1 \ast 1), is \zeta(s)^2 = \sum_{n=1}^\infty d(n) n^{-s}, where \zeta(s) is the . The Euler product follows as \zeta(s)^2 = \prod_p (1 - p^{-s})^{-2}. To extract partial sums from these series, Perron's formula provides an integral representation: \sum_{n \leq x} (f \ast g)(n) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} D_{f \ast g}(s) \frac{x^s}{s} \, ds, for c > \sigma_a and x > 0 not an integer, with error terms controllable via truncation. This applies directly to convolutions since D_{f \ast g}(s) = D_f(s) D_g(s), facilitating asymptotic estimates for sums like \sum_{n \leq x} d(n) \sim x \log x + (2\gamma - 1)x, where \gamma is the Euler-Mascheroni constant. Regarding , if D_f(s) and D_g(s) admit meromorphic continuations to a common , their product D_{f \ast g}(s) inherits this property, provided the continuations align. For multiplicative f and g, the Euler product form extends the domain of analyticity beyond the convergence half-plane, as seen in the continuation of \zeta(s) via its product \prod_p (1 - p^{-s})^{-1}, enabling similar extensions for convolution series like \zeta(s)^2.

Special Classes

Rational Multiplicative Functions

A rational-valued multiplicative function is an f that is multiplicative—meaning f(mn) = f(m)f(n) whenever \gcd(m,n)=1—and satisfies f(n) \in \mathbb{Q} for all positive integers n. This class includes many classical number-theoretic functions where the values at prime powers f(p^k) are rational numbers, ensuring the property holds across all n via the multiplicative structure. The structure of such functions follows directly from multiplicativity: if n = \prod_p p^{k_p}, then f(n) = \prod_p f(p^{k_p}), where each f(p^{k_p}) \in \mathbb{Q}. Thus, f(n) is a finite product of rational numbers, remaining in \mathbb{Q}. This decomposition highlights how the function is fully determined by its rational values on prime powers, facilitating analysis via Euler products. Representative examples include the normalized Euler totient function f(n) = \phi(n)/n = \prod_{p \mid n} (1 - 1/p), which is multiplicative and rational-valued since each factor $1 - 1/p is rational. Similarly, for a positive integer k, the normalized Jordan totient function J_k(n)/n^k = \prod_{p \mid n} (1 - p^{-k}) is multiplicative and takes rational values, as p^{-k} = 1/p^k \in \mathbb{Q}. In contrast, functions like \sigma(n)/n^s for fixed non-integer s > 1 are multiplicative but generally yield irrational values, though they are rational-valued when s is a positive integer. The collection of rational-valued multiplicative functions is closed under , as the convolution of two such functions remains multiplicative and rational-valued; together with pointwise addition, this structure embeds them within the broader of rational-valued arithmetic functions. Their associated \sum_{n=1}^\infty f(n) n^{-s} admit Euler products \prod_p \left( \sum_{k=0}^\infty f(p^k) p^{-ks} \right), where each local factor has rational coefficients due to the rationality of f(p^k). Certain rational-valued functions specified on prime powers extend uniquely to multiplicative functions on the positive integers; further details on related identities, such as the Busche-Ramanujan identities, appear in the identities section.

Multiplicative Functions over Polynomial Rings

In the context of function fields, multiplicative functions are defined on the ring of polynomials \mathbb{F}_q[X] over a finite field \mathbb{F}_q with q elements, where q is a . Specifically, an f: M \to \mathbb{C}, with M the set of monic polynomials in \mathbb{F}_q[X], is multiplicative if f(1) = 1 and f(FG) = f(F)f(G) whenever F and G are coprime monic polynomials. This definition mirrors the classical notion for the integers but leverages the unique factorization property of \mathbb{F}_q[X], where every nonzero polynomial factors uniquely into a product of monic irreducibles (analogous to primes), up to units in \mathbb{F}_q^\times. The unique irreducible in \mathbb{F}_q[X] ensures that any multiplicative is completely determined by its values on powers of irreducible polynomials. For instance, if F = \prod \pi_i^{k_i} is the irreducible of a monic F, then f(F) = \prod f(\pi_i^{k_i}). This structure facilitates the study of arithmetic properties, much like in the case. A example is the |F| = q^{\deg F}, which is completely multiplicative since \deg(FG) = \deg F + \deg G for any monic polynomials F, G, yielding |FG| = |F| \cdot |G| unconditionally (and thus for coprimes). This serves as the polynomial analog of the n on the positive integers \mathbb{Z}^+. In contrast, the degree \deg(FG) = \deg F + \deg G is additive rather than multiplicative, though it underpins the . Another example is the constant f(F) = 1 for all monic F, which is multiplicative and generates the zeta below. The function over \mathbb{F}_q[X] provides a central analytic tool, defined as \zeta_{\mathbb{F}_q[X]}(T) = \sum_{F \in M} T^{\deg F} = \prod_{\pi \text{ irr. monic}} (1 - T^{\deg \pi})^{-1}, where the sum runs over all monic polynomials and the product over monic irreducibles \pi. This equals \frac{1}{1 - q T} explicitly, converging for |T| < 1/q. For a general multiplicative f, the associated generating function \sum_{F \in M} f(F) T^{\deg F} admits an Euler product \prod_{\pi \text{ irr. monic}} (1 + f(\pi) T^{\deg \pi} + f(\pi^2) T^{2 \deg \pi} + \cdots). Dirichlet series further extend this framework, defined as \sum_{F \in M} f(F) |F|^{-s} = \sum_{F \in M} f(F) q^{-\deg F \cdot s} for \operatorname{Re}(s) > 1 (or appropriately for convergence). For multiplicative f, this series factors into an Euler product over monic irreducibles: \prod_{\pi \text{ irr. monic}} (1 + f(\pi) | \pi |^{-s} + f(\pi^2) | \pi |^{-2s} + \cdots). This polynomial analog parallels the classical \sum f(n) n^{-s} = \prod_p (1 + f(p) p^{-s} + \cdots), enabling analytic techniques like partial and mean value estimates in the function field setting.

Identities and Advanced Results

Busche-Ramanujan Identities

The Busche-Ramanujan identities are functional equations that relate the product of values of a at two arguments to a Dirichlet convolution-like sum over their common divisors. These identities were first explored by Hermann Busche in the late for specific cases of arithmetic functions and later refined by around 1915, with a comprehensive discussion and naming in the work of P. J. McCarthy in 1960. They arise naturally from the structure of for multiplicative functions and provide a tool for expressing multiplicativity in a summed form. Consider a multiplicative function f that can be written as the Dirichlet convolution f = g * h, where g and h are completely multiplicative functions (satisfying g(mn) = g(m)g(n) and h(mn) = h(m)h(n) for all positive integers m, n). Then, for all positive integers m, n, f(mn) = \sum_{a \mid \gcd(m,n)} \mu(a) \, g(a) h(a) \, f\left( \frac{m}{a} \right) f\left( \frac{n}{a} \right), where \mu is the . This form captures Busche's original contribution for functions like the sum-of-divisors function \sigma_k = 1 * \mathrm{id}^k. An equivalent and often more useful variant is the product identity: f(m) f(n) = \sum_{a \mid \gcd(m,n)} g(a) h(a) \, f\left( \frac{mn}{a^2} \right). Ramanujan's refinement emphasizes the case where m and n are coprime, reducing the sum to the single term f(m) f(n) = f(mn) g(1) h(1), which aligns with the defining property of multiplicativity when g(1) = h(1) = 1, but extends to non-coprime cases via the full sum. These identities hold for several classical functions, including the d(n) = 1 * 1 (with g = h = 1) and Ramanujan's tau function \tau(n), where g and h are related to modular forms. A broader class of multiplicative functions admits a Busche-Ramanujan identity if there exists another multiplicative F such that f(m) f(n) = \sum_{d \mid \gcd(m,n)} F(d) \, f\left( \frac{mn}{d^2} \right) for all positive integers m, n. The F is uniquely determined by f when the identity holds, and examples include quadratic functions and certain power sums. The Euler totient \phi(n) = \mathrm{id} * \mu satisfies a restricted form of this identity, holding when m and n do not share a common prime factor to positive powers, as discussed in recent analyses of totients. This restricted identity aids in computations involving the totient, such as \sum_{d \mid n} \phi(d) = n. The proof of these identities relies on Möbius inversion applied to the divisors. Since f = g * h, the Dirichlet series for f factors as \sum f(k)/k^s = (\sum g(k)/k^s)(\sum h(k)/k^s). Expanding the product f(m) f(n) using the multiplicativity of g and h over the prime factors of \gcd(m,n), and inverting via the over the common divisors, yields the summed form. This approach highlights the connection to Euler products for multiplicative functions. Applications include deriving properties of sums, such as those for the totient in partition theory, where Ramanujan's work on highly composite numbers indirectly leverages similar multiplicative structures.

Multivariate Extensions

In number theory, a multivariate multiplicative function is an arithmetic function f: \mathbb{N}^k \to \mathbb{C} satisfying f(1, \dots, 1) = 1 and f(n_1 m_1, \dots, n_k m_k) = f(n_1, \dots, n_k) f(m_1, \dots, m_k) whenever \gcd(n_1 \cdots n_k, m_1 \cdots m_k) = 1. Such functions are completely determined by their values on tuples of prime powers, analogous to the univariate case where multiplicativity holds for coprime arguments. A prominent example is the function on pairs, defined by f(m, n) = \gcd(m, n), which satisfies the multivariate multiplicativity condition because, under the coprimality of the component products, the prime supports are disjoint, preserving the gcd as a product. The function f(m, n) = \operatorname{lcm}(m, n) similarly exhibits this property, as the maximum valuations add separately across disjoint prime sets. In certain contexts, the "inverse" behavior, such as relating gcd and lcm via \gcd(m, n) \cdot \operatorname{lcm}(m, n) = m n, highlights how these functions interact multiplicatively. The associated multivariate Dirichlet series is given by D_f(s_1, \dots, s_k) = \sum_{n_1, \dots, n_k = 1}^\infty \frac{f(n_1, \dots, n_k)}{n_1^{s_1} \cdots n_k^{s_k}}, which factors as an Euler product over primes: D_f(s_1, \dots, s_k) = \prod_p \left( \sum_{i_1 = 0}^\infty \cdots \sum_{i_k = 0}^\infty \frac{f(p^{i_1}, \dots, p^{i_k})}{p^{s_1 i_1 + \cdots + s_k i_k}} \right), due to the multiplicativity of f, with local factors determined by p-adic valuations. This structure mirrors univariate but extends to multiple variables, enabling analysis of convergence and through prime-local behavior. Applications arise in evaluating joint divisor sums, such as \sum_{mn \leq x} f(\gcd(m, n)) for multiplicative f, which admits asymptotic expansions like x (C_f \log x + D_f) + O(x^{(\beta + 1)/2} (\log x)^{\delta + 1}) under growth conditions f(n) \ll n^\beta (\log n)^\delta with \beta < 1, leveraging multiplicativity to reduce to zeta function products. These sums connect to multiple zeta values, where the constant function f \equiv 1 yields series like \zeta(s_1, \dots, s_k) = \sum_{n_1 \geq \cdots \geq n_k \geq 1} (n_1^{-s_1} \cdots n_k^{-s_k}), analyzable via twisted multiplicativity in multiple Dirichlet series frameworks. Multivariate multiplicative functions can be constructed as tensor products of univariate ones: if f_i: \mathbb{N} \to \mathbb{C} are univariate multiplicative for i = 1, \dots, k, then f(n_1, \dots, n_k) = \prod_{i=1}^k f_i(n_i) is multivariate multiplicative, as the global coprimality condition \gcd(n_1 \cdots n_k, m_1 \cdots m_k) = 1 implies pairwise coprimality \gcd(n_i, m_i) = 1 for each i, ensuring the product rule holds. This tensor structure facilitates extensions from single-variable identities, such as the Busche-Ramanujan identities, to multivariable settings.

Generalizations

Completely Multiplicative Functions

A completely multiplicative function is an arithmetic function f: \mathbb{N} \to \mathbb{C} satisfying f(mn) = f(m)f(n) for all positive integers m and n. This property implies that f(p^k) = f(p)^k for every prime p and nonnegative integer k, which follows by induction on k. Every completely multiplicative function is multiplicative, meaning it satisfies the multiplicativity condition whenever \gcd(m,n)=1, but the converse does not hold. For instance, \phi is multiplicative yet not completely multiplicative, as \phi(4) = 2 while \phi(2)^2 = 1^2 = 1. A prominent class of completely multiplicative functions consists of the \chi, which are completely multiplicative by definition and extend the notion of characters on the multiplicative group modulo q. The Dirichlet series associated with a completely multiplicative function f takes a particularly simple Euler product form: \sum_{n=1}^\infty \frac{f(n)}{n^s} = \prod_p \left(1 - \frac{f(p)}{p^s}\right)^{-1}, valid in the half-plane of \Re(s) > \sigma_a(f), where \sigma_a(f) is the abscissa of absolute convergence. This factorization arises because the local factors at each prime p simplify to a \sum_{k=0}^\infty f(p)^k p^{-ks} = (1 - f(p)p^{-s})^{-1}. For the divisor sum \sigma_f(n) = \sum_{d \mid n} f(d), the complete multiplicativity of f yields an explicit product formula when n = \prod_p p^{a_p}: \sigma_f(n) = \prod_p \left( \sum_{k=0}^{a_p} f(p)^k \right) = \prod_p \frac{1 - f(p)^{a_p + 1}}{1 - f(p)}, provided f(p) \neq 1 for each prime p dividing n; the sum is a finite at each . This expression highlights how complete multiplicativity preserves product structures across the prime factorization of n.

Strongly Multiplicative Functions

A strongly multiplicative function is a multiplicative function f such that f(p^k) = f(p) for every prime p and integer k \geq 1. This condition means that the value of f on prime powers is constant for exponents at least 1, distinguishing it from the general multiplicative functions (where f(p^k) can vary with k) and from completely multiplicative functions (where f(p^k) = f(p)^k). Such functions are determined by their values on primes, as f(n) = \prod_{p \mid n} f(p) for n > 1, where the product is over the distinct prime factors of n. This follows from multiplicativity and the constancy on prime powers. Representative examples include the function \omega(n), which counts the number of distinct prime factors of n (with \omega(p^k) = 1); and $2^{\omega(n)}, which satisfies $2^{\omega(p^k)} = 2. The associated Dirichlet series \sum_n f(n) n^{-s} admits an Euler product \prod_p \left(1 + f(p) \sum_{k=1}^\infty p^{-k s}\right) = \prod_p \left(1 + f(p) \frac{p^{-s}}{1 - p^{-s}}\right), valid where it converges absolutely. This form reflects the constant behavior on prime powers, differing from the geometric form for completely multiplicative functions.