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Arithmetic function

An arithmetic function is a function f defined on the positive integers \mathbb{N} that takes values in the complex numbers \mathbb{C}. These functions are fundamental objects in , encoding arithmetic properties of integers such as divisibility and prime factorization. Arithmetic functions play a central role in analytic and multiplicative , facilitating the study of prime distribution, divisor sums, and related phenomena through tools like and . Notable examples include the d(n), which counts the number of positive divisors of n; the sum-of-divisors function \sigma(n), which sums the positive divisors of n; and \phi(n), which counts the integers up to n that are relatively prime to n. Another key example is the \mu(n), defined as (-1)^k if n is a product of k distinct primes (square-free) and 0 otherwise, which is essential for inversion formulas in . A significant class of arithmetic functions consists of the multiplicative ones, where f(mn) = f(m)f(n) whenever m and n are coprime; completely multiplicative functions satisfy this for all m, n. Multiplicative functions are preserved under the (f * g)(n) = \sum_{d \mid n} f(d)g(n/d), which forms an on the set of arithmetic functions and underpins theorems like inversion: if g(n) = \sum_{d \mid n} f(d), then f(n) = \sum_{d \mid n} \mu(d)g(n/d). For instance, \sum_{d \mid n} \phi(d) = n follows from this , highlighting to cyclic groups and .

Fundamentals

Definition

An arithmetic function is a function f: \mathbb{N} \to \mathbb{C}, where \mathbb{N} denotes the set of positive integers and \mathbb{C} the complex numbers, though such functions often take real or non-negative real values in applications within number theory. The term "arithmetic function" was introduced in the early by number theorists, notably and E. M. Wright, to provide a unified framework for studying various functions arising in , such as those related to divisors and primes. Basic examples include the \mathrm{id}(n) = n, which maps each positive integer to itself, and the constant function $1(n) = 1 for all n \in \mathbb{N}. Unlike sequences, which are typically ordered lists indexed by integers, arithmetic functions are defined pointwise on the positive integers, emphasizing their role as mappings that encode arithmetic properties of each input individually rather than relational or sequential behavior.

Notation

Arithmetic functions are typically denoted by lowercase letters such as f(n), where n is a positive integer and f(n) represents the value of the function at n. This notation facilitates the study of properties like multiplicativity and growth rates in number theory. Specific arithmetic functions employ standardized symbols for clarity. For instance, the sum of the k-th powers of the positive divisors of n is denoted by \sigma_k(n), where k is a real or complex number, defined as \sigma_k(n) = \sum_{d \mid n} d^k. , which counts the number of positive integers up to n that are relatively prime to n, is denoted by \phi(n). Asymptotic behaviors of functions are described using notations that capture limiting ratios and bounds. The \sim indicates asymptotic : f(n) \sim g(n) as n \to \infty if \lim_{n \to \infty} f(n)/g(n) = 1. For bounding , is used: f(n) = O(g(n)) if there exists a constant M > 0 such that |f(n)| \leq M |g(n)| for sufficiently large n. The little o notation provides a stricter bound: f(n) = o(g(n)) if f(n)/g(n) \to 0 as n \to \infty. A common application is the bound d(n) = O(n^\epsilon) for any \epsilon > 0, where d(n) is the number of divisors of n, indicating subpolynomial . Generalizations of Euler's totient function use multi-index notation, such as the Jordan totient J_k(n), which counts the number of k-tuples of integers from 1 to n relatively prime to n in each component and is given by J_k(n) = n^k \prod_{p \mid n} (1 - p^{-k}) for positive integer k. This notation extends to higher dimensions while preserving multiplicative properties.

Prime Power Decomposition

The asserts that every integer greater than 1 can be expressed uniquely as a product of one or more primes, disregarding order. Specifically, for any integer n > 1, there exist distinct prime numbers p_1 < p_2 < \cdots < p_k and positive integers a_1, a_2, \dots, a_k such that n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}./06:_New_Page/6.01:_New_Page) This unique prime power decomposition forms the cornerstone of analytic number theory, enabling the systematic study of arithmetic functions through their behavior on prime powers. This factorization underpins several key arithmetic functions that quantify the prime structure of n. The function \omega(n) counts the number of distinct prime factors of n, so if n = \prod_{i=1}^k p_i^{a_i}, then \omega(n) = k. In contrast, the big omega function \Omega(n) tallies the total number of prime factors counting multiplicity, given by \Omega(n) = \sum_{i=1}^k a_i. For a fixed prime p, the p-adic valuation \nu_p(n) is the exponent of p in this decomposition, so \nu_p(n) = a_i if p = p_i and \nu_p(n) = 0 otherwise. These functions capture essential aspects of the prime factorization: \omega(n) measures diversity among primes, \Omega(n) accounts for overall complexity including repetitions, and \nu_p(n) isolates the contribution of a single prime. For example, for n = 12 = 2^2 \cdot 3^1, we have \omega(12) = 2, \Omega(12) = 3, and \nu_2(12) = 2 while \nu_3(12) = 1 and \nu_5(12) = 0. In number theory, the prime power decomposition serves as the foundation for Dirichlet series and the multiplicativity of arithmetic functions. Dirichlet series, such as \sum_{n=1}^\infty f(n) n^{-s} for a multiplicative function f, often factor into Euler products over primes \prod_p \left( \sum_{j=0}^\infty f(p^j) p^{-js} \right), leveraging the unique factorization to separate contributions from each prime power. This structure simplifies analysis, as evaluating f on general n reduces to its values on prime powers via the decomposition.

Classifications

Multiplicative Functions

In number theory, an arithmetic function f is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever m and n are coprime positive integers. This condition distinguishes multiplicative functions from the stricter class of completely multiplicative functions, which satisfy the equality for all positive integers m and n, regardless of coprimality. A fundamental property of multiplicative functions is that they are completely determined by their values at prime powers p^k, where p is prime and k \geq 1. For any positive integer n with prime factorization n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}, it follows that f(n) = f(p_1^{k_1}) f(p_2^{k_2}) \cdots f(p_r^{k_r}). This arises directly from the unique prime factorization of integers and the multiplicativity condition. Prominent examples of multiplicative functions include the divisor sum function \sigma_k(n) = \sum_{d \mid n} d^k, which sums the k-th powers of the positive divisors of n; the divisor function \tau(n) or d(n) = \sum_{d \mid n} 1, which counts the number of positive divisors of n; the Euler totient function \varphi(n) = n \prod_{p \mid n} \left(1 - \frac{1}{p}\right), which counts the number of integers up to n coprime to n; and the Jordan totient function J_k(n) = n^k \prod_{p \mid n} \left(1 - p^{-k}\right), a generalization of \varphi(n) that appears in lattice point counting problems. The Dirichlet series associated with a multiplicative function f, given by \sum_{n=1}^\infty \frac{f(n)}{n^s}, admits an Euler product representation \prod_p \left( \sum_{k=0}^\infty \frac{f(p^k)}{p^{ks}} \right) for \operatorname{Re}(s) sufficiently large, where the product runs over all primes p. This factorization leverages the multiplicativity to decompose the series into local factors at each prime. Multiplicative functions play a central role in applications such as the inclusion-exclusion principle for counting divisors and the Möbius inversion formula, which inverts convolutions involving the constant function 1 to recover original functions from their summatory versions. For instance, Möbius inversion states that if g(n) = \sum_{d \mid n} f(d), then f(n) = \sum_{d \mid n} \mu(d) g(n/d), where \mu is the , enabling the recovery of multiplicative structures in divisor problems.

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 is stricter than mere multiplicativity, as it holds without requiring \gcd(m,n)=1. Consequently, the values on prime powers are determined by powers of the value at primes: f(p^k) = f(p)^k for any prime p and positive integer k./04%3A_Number_Theoretic_Functions/4.01%3A_Multiplicative_Functions) Prominent examples include the Liouville function \lambda(n) = (-1)^{\Omega(n)}, where \Omega(n) counts the total number of prime factors of n with multiplicity; this is completely multiplicative because \Omega(mn) = \Omega(m) + \Omega(n) for all m,n. Another key class consists of Dirichlet characters \chi modulo q, which are completely multiplicative homomorphisms from the multiplicative group of integers modulo q extended to all integers (vanishing when not coprime to q), satisfying \chi(mn) = \chi(m)\chi(n) universally. The Ramanujan sum c_q(n), defined for fixed modulus q as c_q(n) = \sum_{\substack{1 \leq k \leq q \\ \gcd(k,q)=1}} \exp(2\pi i n k / q), can be expressed using all Dirichlet characters \chi modulo q as c_q(n) = \sum_{\chi \pmod{q}} \overline{\chi}(n) \tau(\chi), where \tau(\chi) = \sum_{k=1}^q \chi(k) \exp(2\pi i k / q) is the Gauss sum; this leverages the complete multiplicativity of these characters to exhibit multiplicative behavior in n. These functions admit simpler Euler product representations for their associated Dirichlet series due to the power structure at primes. For a completely multiplicative f, the series \sum_{n=1}^\infty f(n) n^{-s} factors as \prod_p (1 - f(p) p^{-s})^{-1} for \Re(s) > 1 under suitable convergence conditions. Dirichlet characters, in particular, underpin Dirichlet L-functions L(s, \chi) = \sum_{n=1}^\infty \chi(n) n^{-s} = \prod_p (1 - \chi(p) p^{-s})^{-1}, which play a central role in , such as proving Dirichlet's theorem on primes in arithmetic progressions. In contrast, the Möbius function \mu is multiplicative but not completely multiplicative, as \mu(4) = 0 \neq \mu(2)^2 = 1.

Additive Functions

In , an additive arithmetic function f is defined on the positive integers such that f(1) = 0 and f(mn) = f(m) + f(n) whenever \gcd(m, n) = 1. This property distinguishes additive functions from multiplicative ones, as it preserves rather than multiplication under coprimality. A key property of additive functions is that their values are fully determined by their behavior on prime powers. Specifically, if n = \prod_{i=1}^r p_i^{k_i} is the prime of n, then f(n) = \sum_{i=1}^r f(p_i^{k_i}), where the is over the distinct prime powers dividing n. This decomposition allows additive functions to be constructed by specifying arbitrary values f(p^k) for each prime p and exponent k \geq 1, reflecting the additive structure across coprime factors. Such functions often relate to the prime factorization of integers, where the natural logarithm \log n = \sum_{p^k \| n} \log(p^k) exemplifies an additive form, approximately capturing the cumulative contribution of prime factors \sum_{p \mid n} \log p for square-free n. A prominent example is the function \omega(n), which counts the number of distinct prime factors of n. For a prime power p^k, \omega(p^k) = 1 if k \geq 1, so \omega(n) = \sum_{p \mid n} 1 over the distinct primes dividing n; for instance, \omega(12) = \omega(2^2 \cdot 3) = 2. This makes \omega(n) strictly additive, as the count adds over coprime components. Additive functions are frequently analyzed for their asymptotic behavior, particularly their average orders over integers up to x. For \omega(n), the sum \sum_{n \leq x} \omega(n) \sim x \log \log x as x \to \infty, indicating that the typical value of \omega(n) grows like \log \log n. This logarithmic growth underscores the role of additive functions in probabilistic , where distributions like the describe fluctuations around the mean.

Completely Additive Functions

A completely additive arithmetic function f is an arithmetic function defined on the positive integers such that f(mn) = f(m) + f(n) for all positive integers m and n. This property implies that f(1) = 0 and, for any prime p and positive integer k, f(p^k) = k f(p), allowing the function to be fully determined by its values on primes. Consequently, for the prime factorization n = \prod_p p^{k_p}, f(n) = \sum_p k_p f(p). Prominent examples include the total prime factor counting function \Omega(n), which counts the prime factors of n with multiplicity, so \Omega(n) = \sum_{p^k \| n} k. This satisfies \Omega(mn) = \Omega(m) + \Omega(n) for all m, n, making it completely additive. Another example is the p-adic valuation \nu_p(n), which gives the highest exponent k such that p^k divides n; it is completely additive in the sense that \nu_p(mn) = \nu_p(m) + \nu_p(n) for fixed prime p and all positive integers m, n. The natural logarithm function \log n also exemplifies this class, as \log(mn) = \log m + \log n holds universally, and in terms of prime factors, \log n = \sum_{p^k \| n} k \log p. These functions find applications in , particularly in proofs of the , where \Omega(n) helps analyze the of prime factors and establish asymptotic behaviors for sums involving prime counts. In valuation theory, the p-adic valuations underpin the of p-adic numbers and metrics, enabling the study of congruences and local properties in number fields. Additionally, the completely additive nature of \log n connects to the \Lambda through , as \log n = \sum_{d \mid n} \Lambda(d).

Other Notable Functions

Prime-Counting Functions

Prime-counting functions are arithmetic functions that quantify the distribution of prime numbers up to a given x, playing a central role in for studying prime density and related asymptotic behaviors. These functions often involve sums or products over primes and their powers, providing cumulative measures essential for theorems on prime gaps and densities. The , denoted \pi(x), counts the number of prime numbers less than or equal to x. For example, \pi(10) = 4 since there are four primes (2, , 5, ) up to 10. The function, denoted \Pi(x), is the product of all primes less than or equal to x, analogous to the but restricted to primes; for instance, \Pi(10) = 2 \times 3 \times 5 \times 7 = 210. The first , \vartheta(x), is defined as the sum of the natural logarithms of all primes up to x: \vartheta(x) = \sum_{p \leq x} \log p, where the sum is over primes p. This weighted sum captures the logarithmic contribution of primes to the growth of arithmetic structures. The second Chebyshev function, \psi(x), extends this by summing over prime powers: \psi(x) = \sum_{p^k \leq x} \log p, summing \log p for each prime power p^k \leq x, with k \geq 1. This function is particularly useful in proofs involving the due to its relation to the . A key asymptotic result for these functions arises from the , which states that \pi(x) \sim \frac{x}{\log x} as x \to \infty, meaning the ratio \frac{\pi(x) \log x}{x} approaches 1; this implies primes are distributed with density approximately \frac{1}{\log x}. Equivalent forms hold for the Chebyshev functions, such as \vartheta(x) \sim x and \psi(x) \sim x. The , \Lambda(n), is an arithmetic function defined as \Lambda(n) = \log p if n = p^k for some prime p and integer k \geq 1, and \Lambda(n) = 0 otherwise. It isolates prime power contributions in sums, and notably, \psi(x) = \sum_{n \leq x} \Lambda(n), linking the second directly to this indicator-like behavior.

Partition and Related Functions

The partition function p(n) counts the number of ways to express the positive n as a sum of positive integers, disregarding order. For example, p(4) = 5 since 4 can be partitioned as $4, $3+1, $2+2, $2+1+1, and $1+1+1+1. This arises in combinatorial number theory and has the \sum_{n=0}^{\infty} p(n) x^n = \prod_{k=1}^{\infty} \frac{1}{1 - x^k}, where p(0) = 1 by convention. The asymptotic growth of p(n) is given by Hardy and Ramanujan's formula p(n) \sim \frac{1}{4n\sqrt{3}} \exp\left( \pi \sqrt{\frac{2n}{3}} \right) as n \to \infty, reflecting its rapid increase. Ramanujan's tau function \tau(n) is defined as the n-th Fourier coefficient of the discriminant modular form \Delta(z) = q \prod_{n=1}^{\infty} (1 - q^n)^{24}, where q = e^{2\pi i z} and z is in the upper half-plane. Thus, \Delta(z) = \sum_{n=1}^{\infty} \tau(n) q^n. This cusp form of weight 12 on the full modular group \mathrm{SL}_2(\mathbb{Z}) exhibits remarkable congruences, such as \tau(n) \equiv \sigma_{11}(n) \pmod{691} for all positive integers n, and \tau(n) \equiv 0 \pmod{23} whenever $23 \mid n. The function \tau(n) is multiplicative but not completely so, and its values connect analytic number theory with modular forms. The class number h(d) of the imaginary quadratic field \mathbb{Q}(\sqrt{d}), where d < 0 is a fundamental discriminant (an integer congruent to 0 or 1 modulo 4 such that no square divides d appropriately for the ring of integers), measures the size of the ideal class group in the ring of integers of the field. For imaginary quadratic fields (d < 0), h(d) equals the number of reduced positive definite binary quadratic forms of discriminant d. Gauss conjectured that h(-d) \to \infty as d \to \infty, a result proven by Landau in 1903, highlighting the scarcity of fields with small class numbers. There are exactly nine imaginary quadratic fields with h(d) = 1, corresponding to discriminants d = -3, -4, -7, -8, -11, -19, -43, -67, -163. The sum-of-squares function r_k(n) denotes the number of ways to write n as a sum of k squares of integers, allowing zeros and distinguishing order and signs; for instance, r_2(5) = 8 from representations like (\pm 1)^2 + (\pm 2)^2 and permutations. Jacobi derived explicit formulas for small k, such as r_2(n) = 4(d_1(n) - d_3(n)), where d_i(n) counts divisors of n congruent to i modulo 4. Lagrange's four-square theorem states that r_4(n) > 0 for all n \geq 1, with the formula r_4(n) = 8 \sum_{d|n, 4\nmid d} d if n is odd, and a variant for even n. These functions link additive partitions to quadratic forms and sieve methods in .

Miscellaneous Functions

The Carmichael function, denoted \lambda(n), is defined as the exponent of the multiplicative group of integers modulo n, meaning it is the smallest positive integer m such that a^m \equiv 1 \pmod{n} for every integer a coprime to n. For n = \prod_{i=1}^r p_i^{k_i} where the p_i are distinct primes, \lambda(n) is the least common multiple of the values \lambda(p_i^{k_i}) over i = 1 to r. Specifically, for an odd prime p and k \geq 1, \lambda(p^k) = \phi(p^k) = p^{k-1}(p-1), where \phi is Euler's totient function; for p=2, \lambda(2) = 1, \lambda(4) = 2, and \lambda(2^k) = 2^{k-2} for k \geq 3. This function always divides \phi(n), providing a tighter bound on the order of elements in (\mathbb{Z}/n\mathbb{Z})^\times compared to \phi(n). The arithmetic derivative, denoted D(n) or n', extends the concept of differentiation to the natural numbers via rules analogous to those in calculus, based on prime factorization. For a prime p, D(p) = 1; for a prime power p^k with k \geq 1, D(p^k) = k p^{k-1}; and for coprime positive integers m and n, D(mn) = m D(n) + n D(m), mirroring the Leibniz product rule. Additionally, D(1) = 0, and the function extends to rationals via the quotient rule \left(\frac{a}{b}\right)' = \frac{a' b - a b'}{b^2}. This construction yields, for n = \prod p_i^{e_i}, the explicit formula D(n) = n \sum_i \frac{e_i}{p_i}. Unlike standard arithmetic functions such as the divisor function, the arithmetic derivative is neither multiplicative nor additive but satisfies a logarithmic additivity property, D(p^k) = p^k \cdot \frac{k}{p}, highlighting its differential character. The Dedekind psi function, denoted \psi(n), is a multiplicative arithmetic function defined by \psi(n) = n \prod_{p \mid n} \left(1 + \frac{1}{p}\right), where the product runs over the distinct prime factors of n, and \psi(1) = 1. Equivalently, it can be expressed using the Möbius function as \psi(n) = \sum_{d \mid n} d \cdot \mu(n/d)^2 or \psi(n) = n \sum_{d \mid n} \frac{\mu(d)^2}{d}. This function relates to the sum-of-divisors function \sigma(n) through the identity \psi(n) = \sigma(n) \prod_{p \mid n} \frac{p}{p+1}, emphasizing its role in bridging divisor sums and prime factor adjustments. Its Dirichlet series is \sum_{n=1}^\infty \frac{\psi(n)}{n^s} = \frac{\zeta(s) \zeta(s-1)}{\zeta(2s)} for \Re(s) > 2, connecting it to the Riemann zeta function.

Operations and Relations

Summatory Functions

In , the summatory function of an arithmetic function f: \mathbb{N} \to \mathbb{C} is defined by F(x) = \sum_{n \leq x} f(n), where the sum runs over all positive integers n not exceeding the real number x \geq 0. This cumulative extends the discrete nature of f to a step function, facilitating the study of its average behavior and asymptotic growth as x \to \infty. Prominent examples illustrate the utility of summatory functions. For the d(n), which enumerates the positive divisors of n, the asymptotic is \sum_{n \leq x} d(n) \sim x \log x, as established by Dirichlet in 1849 via the Euler product for the associated zeta function. For \phi(n), counting integers up to n coprime to n, the summatory satisfies \sum_{n \leq x} \phi(n) = \frac{3x^2}{\pi^2} + O(x \log x), a classical result also due to Dirichlet, reflecting the density of integers coprime to all smaller values. Summatory functions serve as a bridge between arithmetic functions and continuous , particularly through Abel summation, the discrete counterpart to , which relates sums \sum a_n b_n to integrals involving the partial sums of a_n. This technique connects F(x) to the \sum f(n) n^{-s} by enabling approximations via Perron's formula or Tauberian theorems, thus revealing the analytic continuation and residues that govern long-term behavior. For multiplicative arithmetic functions, where f(mn) = f(m) f(n) for coprime m, n, the summatory F(x) typically exhibits growth of the form F(x) \sim c x (\log x)^k for some constant c > 0 and k \geq 0, determined by the order of the at s=1 in the of f. Such estimates arise from the multiplicativity preserving Euler products, allowing decomposition into contributions.

Dirichlet Convolution

The Dirichlet convolution of two arithmetic functions f and g is defined by (f * g)(n) = \sum_{d \mid n} f(d) \, g\left(\frac{n}{d}\right) for each positive n. This operation turns the set of all arithmetic functions into a , with defined . The is both commutative and associative: f * g = g * f and (f * g) * h = f * (g * h) for any arithmetic s f, g, h. It admits a unit element \varepsilon, the function given by \varepsilon(1) = 1 and \varepsilon(n) = 0 for n > 1, satisfying f * \varepsilon = \varepsilon * f = f. Moreover, if L_f(s) = \sum_{n=1}^\infty f(n) n^{-s} and L_g(s) = \sum_{n=1}^\infty g(n) n^{-s} are the associated that converge absolutely at some s, then L_{f * g}(s) = L_f(s) L_g(s). Since the convolution forms a ring, invertible elements exist among functions f with f(1) \neq 0; the inverse f^{-1} satisfies f * f^{-1} = \varepsilon. A prominent example is the \mu, which inverts the constant function u(n) = 1 for all n, as u * \mu = \varepsilon. This yields Möbius inversion: if h = f * u, then f = h * \mu. Representative examples illustrate the operation's utility. The divisor function d(n), counting the positive divisors of n, arises as d = u * u. Similarly, \phi satisfies \phi * u = \mathrm{id}, where \mathrm{id}(n) = n, reflecting the identity \sum_{d \mid n} \phi(d) = n. From this, \phi = \mathrm{id} * \mu.

Key Identities and Relations

One of the most fundamental identities in the theory of arithmetic functions is the , which provides a way to recover a from its over divisors. Specifically, if g(n) = \sum_{d \mid n} f(d) for arithmetic s f and g, then f(n) = \sum_{d \mid n} \mu(d) g(n/d), where \mu is the . This identity arises from the property that the of the constant 1 with \mu yields the unit \varepsilon, where \varepsilon(n) = 1 if n = 1 and 0 otherwise. A notable identity links the sum-of-squares function to divisor counts modulo 4. The number of ways to represent n as a sum of two squares, counting order and signs, is given by r_2(n) = 4 \sum_{\substack{d \mid n \\ d \equiv 1 \pmod{4}}} 1 - 4 \sum_{\substack{d \mid n \\ d \equiv 3 \pmod{4}}} 1 = 4(d_1(n) - d_3(n)), where d_i(n) denotes the number of positive of n congruent to i modulo 4. This formula highlights the connection between quadratic forms and the distribution of . Divisor convolutions often simplify through the Möbius function. For instance, the convolution of the divisor function \sigma_0(n) = d(n) (the number of positive divisors of n) with \mu is the unit function: \sigma_0 * \mu = \varepsilon. More generally, the convolution of the identity function \mathrm{id}(n) = n with \mu yields the Euler totient function: \mathrm{id} * \mu = \varphi. This extends to higher powers, where the Jordan totient function J_k(n) satisfies J_k * 1 = \mathrm{id}_k and thus \mathrm{id}_k * \mu = J_k, with J_k(n) = \sum_{d \mid n} \mu(d) (n/d)^k = n^k \prod_{p \mid n} (1 - p^{-k}). In prime-related contexts, the \psi(x) is defined as the summatory function of the : \psi(x) = \sum_{n \leq x} \Lambda(n), where \Lambda(n) = \log p if n = p^k for prime p and positive integer k, and 0 otherwise. This identity connects prime powers to logarithmic sums and plays a key role in the . Class number formulas relate arithmetic functions to values of Dirichlet L-functions. For a fundamental discriminant -d < 0 with d > 0, the class number h(-d) of the imaginary quadratic field \mathbb{Q}(\sqrt{-d}) is given by h(-d) = \frac{w \sqrt{d}}{2\pi} L(1, \chi_{-d}), where w is the number of units (2, 4, or 6), and \chi_{-d} is the non-principal modulo d defined by the Kronecker symbol. Menon's identity provides a gcd sum involving the totient . For any positive n, \sum_{\substack{1 \leq a \leq n \\ \gcd(a,n)=1}} \gcd(a-1, n) = \varphi(n) \, d(n), where the sum is over integers a coprime to n. Among miscellaneous relations, the Ramanujan sum c_q(n) admits an expression via the : c_q(n) = \sum_{d \mid \gcd(n,q)} \mu(q/d) \, d. This follows from Möbius inversion applied to the identity \sum_{d \mid q} c_d(n) = q if q \mid n and 0 otherwise.

Tabular Data

Values of Selected Functions

This section provides initial values of selected arithmetic functions for positive integers n = 1 to $100, illustrating their behavior through concrete computation. These functions include the \mathrm{id}(n) = n, the d(n), the sum-of-divisors function \sigma(n), \phi(n), the \mu(n), the small omega function \omega(n), the big omega function \Omega(n), and the \lambda(n). Values are computed via the prime factorization of n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}, where p_i are distinct primes and e_i \geq 1: specifically, d(n) = \prod (e_i + 1), \sigma(n) = \prod \frac{p_i^{e_i + 1} - 1}{p_i - 1}, \phi(n) = n \prod (1 - 1/p_i), \mu(n) = (-1)^k if square-free (all e_i = 1) and 0 otherwise, \omega(n) = k, \Omega(n) = \sum e_i, and \lambda(n) = (-1)^{\Omega(n)}. For n=1, all functions take value 0 except \mathrm{id}(1)=1, d(1)=\sigma(1)=\phi(1)=\mu(1)=\lambda(1)=1. The table below enumerates these values, with data drawn from established sequence databases.
n\mathrm{id}(n)d(n)\sigma(n)\phi(n)\mu(n)\omega(n)\Omega(n)\lambda(n)
111111001
22231-111-1
33242-111-1
443720121
55264-111-1
6641221221
77286-111-1
884154013-1
9931360121
101041841221
111121210-111-1
12126284023-1
131321412-111-1
141442461221
151542481221
161653180141
171721816-111-1
18186396023-1
191922018-111-1
20206428023-1
2121432121221
2222436101221
232322422-111-1
242486080241
2525331200121
2626442121221
272744018013-1
282865612023-1
292923028-111-1
30308728-133-1
313123230-111-1
323266316015-1
3333448201221
3434454161221
3535448241221
3636991120241
373723836-111-1
3838460181221
3939456241221
4040890160241
414124240-111-1
424289612-133-1
434324442-111-1
444468420023-1
454567824023-1
4646472221221
474724846-111-1
48481012416025-1
4949357420121
505069320023-1
5151472321221
525269824023-1
535325452-111-1
54548120180241
5555472401221
56568120240241
5757480361221
5858490281221
595926058-111-1
606012168160341
616126260-111-1
6262496301221
6363610436023-1
64647127320161
6565484481221
6666814420-133-1
676726866-111-1
6868612632023-1
6969496441221
7070814424-133-1
717127270-111-1
72721219524025-1
737327472-111-1
74744114361221
7575612440023-1
7676614036023-1
7777496601221
7878816824-133-1
797928078-111-1
80801018632025-1
81815121540141
82824126401221
838328482-111-1
848412224240341
85854108641221
86864132421221
87874120561221
88888180400241
898929088-111-1
909012234240341
91914112721221
9292616844023-1
93934128601221
94944144461221
95954120721221
969612252320261
979729896-111-1
9898617142023-1
9999615660023-1
1001009217400241
These values reveal characteristic patterns: \mu(n) oscillates between -1, 0, and 1, vanishing for non-square-free n and alternating signs based on the parity of \omega(n) for square-free n, as seen in its frequent zeros (e.g., at powers like 4, 8, 9) and sporadic \pm 1 at primes or products of distinct primes. Meanwhile, \sigma(n) exhibits steady growth, accumulating larger sums for highly composite n like 60 (\sigma(60)=168) or 90 (\sigma(90)=234) due to multiple factors, contrasting with minimal values at primes (e.g., \sigma(97)=98).