Arithmetic function

Notation

In this article, $\sum _{p}f(p)$ and $\prod _{p}f(p)$ mean that the sum or product is over all prime numbers: $${\displaystyle \sum _{p}f(p)=f(2)+f(3)+f(5)+\cdots }$$ and $${\displaystyle \prod _{p}f(p)=f(2)f(3)f(5)\cdots .}$$ Similarly, $\sum _{p^k}f(p^k)$ and $\prod _{p^k}f(p^k)$ mean that the sum or product is over all prime powers with strictly positive exponent (so k = 0 is not included): $${\displaystyle \sum _{p^k}f(p^k)=\sum _p\sum _{k>0}f(p^k)=f(2)+f(3)+f(4)+f(5)+f(7)+f(8)+f(9)+\cdots .}$$

The notations $\sum _{d\mid n}f(d)$ and $\prod _{d\mid n}f(d)$ mean that the sum or product is over all positive divisors of n, including 1 and n. For example, if n = 12, then $${\displaystyle \prod _{d\mid 12}f(d)=f(1)f(2)f(3)f(4)f(6)f(12).}$$

The notations can be combined: $\sum _{p\mid n}f(p)$ and $\prod _{p\mid n}f(p)$ mean that the sum or product is over all prime divisors of n. For example, if n = 18, then $${\displaystyle \sum _{p\mid 18}f(p)=f(2)+f(3),}$$ and similarly $\sum _{p^k\mid n}f(p^k)$ and $\prod _{p^k\mid n}f(p^k)$ mean that the sum or product is over all prime powers dividing n. For example, if n = 24, then $${\displaystyle \prod _{p^k\mid 24}f(p^k)=f(2)f(3)f(4)f(8).}$$

Ω(n), ω(n), νp(n) – prime power decomposition

The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: ${\displaystyle n=p_1^{a_1}\cdots p_k^{a_k}}$ where p₁ < p₂ < ... < p_k are primes and the a_j are positive integers. (1 is given by the empty product.)

It is often convenient to write this as an infinite product over all the primes, where all but a finite number have a zero exponent. Define the p-adic valuation ν_p(n) to be the exponent of the highest power of the prime p that divides n. That is, if p is one of the p_i then ν_p(n) = a_i, otherwise it is zero. Then $${\displaystyle n=\prod _p{p}^{{\nu }_p(n)}.}$$

In terms of the above the prime omega functions ω and Ω are defined by

ω(n) = k, Ω(n) = a₁ + a₂ + ... + a_k.

To avoid repetition, formulas for the functions listed in this article are, whenever possible, given in terms of n and the corresponding p_i, a_i, ω, and Ω.

Multiplicative functions

σk(n), τ(n), d(n) – divisor sums

σ_k(n) is the sum of the kth powers of the positive divisors of n, including 1 and n, where k is a complex number.

σ₁(n), the sum of the (positive) divisors of n, is usually denoted by σ(n).

Since a positive number to the zero power is one, σ₀(n) is therefore the number of (positive) divisors of n; it is usually denoted by d(n) or τ(n) (for the German Teiler = divisors).

$${\displaystyle {\sigma }_k(n)=\prod _{i=1}^{\omega (n)}\frac{p_i^{(a_i+1)k}-1}{p_i^k-1}=\prod _{i=1}^{\omega (n)}\left(1+p_i^k+p_i^{2k}+\cdots +p_i^{a_{i}k}\right).}$$

Setting k = 0 in the second product gives $${\displaystyle \tau (n)=d(n)=(1+a_1)(1+a_2)\cdots (1+a_{\omega (n)}).}$$

φ(n) – Euler totient function

φ(n), the Euler totient function, is the number of positive integers not greater than n that are coprime to n. $${\displaystyle \phi (n)=n\prod _{p\mid n}\left(1-\frac{1}{p}\right)=n\left(\frac{p_1-1}{p_1}\right)\left(\frac{p_2-1}{p_2}\right)\cdots \left(\frac{p_{\omega (n)}-1}{p_{\omega (n)}}\right).}$$

Jk(n) – Jordan totient function

J_k(n), the Jordan totient function, is the number of k-tuples of positive integers all less than or equal to n that form a coprime (k + 1)-tuple together with n. It is a generalization of Euler's totient, φ(n) = J₁(n). $${\displaystyle J_k(n)=n^k\prod _{p\mid n}\left(1-\frac{1}{p^k}\right)=n^k\left(\frac{p_1^k-1}{p_1^k}\right)\left(\frac{p_2^k-1}{p_2^k}\right)\cdots \left(\frac{p_{\omega (n)}^k-1}{p_{\omega (n)}^k}\right).}$$

μ(n) – Möbius function

μ(n), the Möbius function, is important because of the Möbius inversion formula. See § Dirichlet convolution, below. $${\displaystyle \mu (n)=\{\begin{array}{ll}(-1{)}^{\omega (n)}=(-1{)}^{\mathrm{\Omega }(n)}& \text{if }\omega (n)=\mathrm{\Omega }(n)\\ 0& \text{if }\omega (n)\ne \mathrm{\Omega }(n).\end{array}}$$

This implies that μ(1) = 1. (Because Ω(1) = ω(1) = 0.)

τ(n) – Ramanujan tau function

τ(n), the Ramanujan tau function, is defined by its generating function identity: $${\displaystyle \sum _{n\ge 1}\tau (n)q^n=q\prod _{n\ge 1}(1-q^n{)}^{24}.}$$

Although it is hard to say exactly what "arithmetical property of n" it "expresses", (τ(n) is (2π)⁻¹² times the nth Fourier coefficient in the q-expansion of the modular discriminant function) it is included among the arithmetical functions because it is multiplicative and it occurs in identities involving certain σ_k(n) and r_k(n) functions (because these are also coefficients in the expansion of modular forms).

cq(n) – Ramanujan's sum

c_q(n), Ramanujan's sum, is the sum of the nth powers of the primitive qth roots of unity: $${\displaystyle c_q(n)=\sum _{\stackrel{1\le a\le q}{gcd(a,q)=1}}{e}^{2\pi i\frac{a}{q}n}.}$$

Even though it is defined as a sum of complex numbers (irrational for most values of q), it is an integer. For a fixed value of n it is multiplicative in q:

If q and r are coprime, then ${\displaystyle c_q(n)c_r(n)=c_{qr}(n).}$

ψ(n) – Dedekind psi function

The Dedekind psi function, used in the theory of modular functions, is defined by the formula $${\displaystyle \psi (n)=n\prod _{p|n}\left(1+\frac{1}{p}\right).}$$

Completely multiplicative functions

λ(n) – Liouville function

λ(n), the Liouville function, is defined by $${\displaystyle \lambda (n)=(-1{)}^{\mathrm{\Omega }(n)}.}$$

χ(n) – characters

All Dirichlet characters χ(n) are completely multiplicative. Two characters have special notations:

The principal character (mod n) is denoted by χ₀(a) (or χ₁(a)). It is defined as $${\displaystyle {\chi }_0(a)=\{\begin{array}{ll}1& \text{if }gcd(a,n)=1,\\ 0& \text{if }gcd(a,n)\ne 1.\end{array}}$$

The quadratic character (mod n) is denoted by the Jacobi symbol for odd n (it is not defined for even n): $${\displaystyle \left(\frac{a}{n}\right)={\left(\frac{a}{p_1}\right)}^{a_1}{\left(\frac{a}{p_2}\right)}^{a_2}\cdots {\left(\frac{a}{p_{\omega (n)}}\right)}^{a_{\omega (n)}}.}$$

In this formula ${\displaystyle (\frac{a}{p})}$ is the Legendre symbol, defined for all integers a and all odd primes p by $${\displaystyle \left(\frac{a}{p}\right)=\{\begin{array}{ll}0& \text{if }a\equiv 0(\mathrm{mod}p),\\ +1& \text{if }a\not\equiv 0(\mathrm{mod}p)\text{ and for some integer }x,a\equiv x^2(\mathrm{mod}p)\\ -1& \text{if there is no such }x.\end{array}}$$

Following the normal convention for the empty product, ${\displaystyle \left(\frac{a}{1}\right)=1.}$

Additive functions

ω(n) – distinct prime divisors

ω(n), defined above as the number of distinct primes dividing n, is additive (see Prime omega function).

Completely additive functions

Ω(n) – prime divisors

Ω(n), defined above as the number of prime factors of n counted with multiplicities, is completely additive (see Prime omega function).

νp(n) – p-adic valuation of an integer n

For a fixed prime p, ν_p(n), defined above as the exponent of the largest power of p dividing n, is completely additive.

Logarithmic derivative

${\displaystyle \mathrm{ld}(n)=\frac{D(n)}{n}=\sum _{\stackrel{p\mid n}{p\text{ prime}}}\frac{v_p(n)}{p}}$, where ${\displaystyle D(n)}$ is the arithmetic derivative.

Neither multiplicative nor additive

π(x), Π(x), ϑ(x), ψ(x) – prime-counting functions

These important functions (which are not arithmetic functions) are defined for non-negative real arguments, and are used in the various statements and proofs of the prime number theorem. They are summation functions (see the main section just below) of arithmetic functions which are neither multiplicative nor additive.

π(x), the prime-counting function, is the number of primes not exceeding x. It is the summation function of the characteristic function of the prime numbers. $${\displaystyle \pi (x)=\sum _{p\le x}1}$$

A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes, etc. It is the summation function of the arithmetic function which takes the value 1/k on integers which are the kth power of some prime number, and the value 0 on other integers. $${\displaystyle \mathrm{\Pi }(x)=\sum _{p^k\le x}\frac{1}{k}.}$$

ϑ(x) and ψ(x), the Chebyshev functions, are defined as sums of the natural logarithms of the primes not exceeding x. $${\displaystyle \vartheta (x)=\sum _{p\le x}\log p,}$$ $${\displaystyle \psi (x)=\sum _{p^k\le x}\log p.}$$

The second Chebyshev function ψ(x) is the summation function of the von Mangoldt function just below.

Λ(n) – von Mangoldt function

Λ(n), the von Mangoldt function, is 0 unless the argument n is a prime power p^k, in which case it is the natural logarithm of the prime p: $${\displaystyle \mathrm{\Lambda }(n)=\{\begin{array}{ll}\log p& \text{if }n=2,3,4,5,7,8,9,11,13,16,\dots =p^k\text{ is a prime power}\\ 0& \text{if }n=1,6,10,12,14,15,18,20,21,\dots \text{ is not a prime power}.\end{array}}$$

p(n) – partition function

p(n), the partition function, is the number of ways of representing n as a sum of positive integers, where two representations with the same summands in a different order are not counted as being different: $${\displaystyle p(n)=\left|\left\{(a_1,a_2,\dots a_k):0<a_1\le a_2\le \cdots \le a_k\wedge n=a_1+a_2+\cdots +a_k\right\}\right|.}$$

λ(n) – Carmichael function

λ(n), the Carmichael function, is the smallest positive number such that ${\displaystyle a^{\lambda (n)}\equiv 1(\mathrm{mod}n)}$ for all a coprime to n. Equivalently, it is the least common multiple of the orders of the elements of the multiplicative group of integers modulo n.

For powers of odd primes and for 2 and 4, λ(n) is equal to the Euler totient function of n; for powers of 2 greater than 4 it is equal to one half of the Euler totient function of n: $${\displaystyle \lambda (n)=\{\begin{array}{ll}\varphi (n)& \text{if }n=2,3,4,5,7,9,11,13,17,19,23,25,27,\dots \\ \frac{1}{2}\varphi (n)& \text{if }n=8,16,32,64,\dots \end{array}}$$ and for general n it is the least common multiple of λ of each of the prime power factors of n: $${\displaystyle \lambda (p_1^{a_1}{p}_2^{a_2}\dots p_{\omega (n)}^{a_{\omega (n)}})=\mathrm{lcm}[\lambda (p_1^{a_1}),\lambda (p_2^{a_2}),\dots ,\lambda (p_{\omega (n)}^{a_{\omega (n)}})].}$$

h(n) – class number

h(n), the class number function, is the order of the ideal class group of an algebraic extension of the rationals with discriminant n. The notation is ambiguous, as there are in general many extensions with the same discriminant. See quadratic field and cyclotomic field for classical examples.

rk(n) – sum of k squares

r_k(n) is the number of ways n can be represented as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the square roots are counted as different. $${\displaystyle r_k(n)=\left|\left\{(a_1,a_2,\dots ,a_k):n=a_1^2+a_2^2+\cdots +a_k^2\right\}\right|}$$

D(n) – Arithmetic derivative

Using the Heaviside notation for the derivative, the arithmetic derivative D(n) is a function such that

• ${\displaystyle D(n)=1}$ if n prime, and
• ${\displaystyle D(mn)=mD(n)+D(m)n}$ (the product rule)

Summation functions

Given an arithmetic function a(n), its summation function A(x) is defined by $${\displaystyle A(x):=\sum _{n\le x}a(n).}$$ A can be regarded as a function of a real variable. Given a positive integer m, A is constant along open intervals m < x < m + 1, and has a jump discontinuity at each integer for which a(m) ≠ 0.

Since such functions are often represented by series and integrals, to achieve pointwise convergence it is usual to define the value at the discontinuities as the average of the values to the left and right: $${\displaystyle A_0(m):=\frac{1}{2}\left(\sum _{n<m}a(n)+\sum _{n\le m}a(n)\right)=A(m)-\frac{1}{2}a(m).}$$

Individual values of arithmetic functions may fluctuate wildly – as in most of the above examples. Summation functions "smooth out" these fluctuations. In some cases it may be possible to find asymptotic behaviour for the summation function for large x.

A classical example of this phenomenon is given by the divisor summatory function, the summation function of d(n), the number of divisors of n: $${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}d(n)=2}$$ $${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}\frac{\log d(n)\log \log n}{\log n}=\log 2}$$ $${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{d(1)+d(2)+\cdots +d(n)}{\log (1)+\log (2)+\cdots +\log (n)}=1.}$$

An average order of an arithmetic function is some simpler or better-understood function which has the same summation function asymptotically, and hence takes the same values "on average". We say that g is an average order of f if $${\displaystyle \sum _{n\le x}f(n)\sim \sum _{n\le x}g(n)}$$

as x tends to infinity. The example above shows that d(n) has the average order log(n).

Dirichlet convolution

Given an arithmetic function a(n), let F_a(s), for complex s, be the function defined by the corresponding Dirichlet series (where it converges): $${\displaystyle F_a(s):=\sum _{n=1}^{\mathrm{\infty }}\frac{a(n)}{n^s}.}$$ F_a(s) is called a generating function of a(n). The simplest such series, corresponding to the constant function a(n) = 1 for all n, is ζ(s) the Riemann zeta function.

The generating function of the Möbius function is the inverse of the zeta function: $${\displaystyle \zeta (s)\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n^s}=1,\mathrm{\Re }s>1.}$$

Consider two arithmetic functions a and b and their respective generating functions F_a(s) and F_b(s). The product F_a(s)F_b(s) can be computed as follows: $${\displaystyle F_a(s)F_b(s)=\left(\sum _{m=1}^{\mathrm{\infty }}\frac{a(m)}{m^s}\right)\left(\sum _{n=1}^{\mathrm{\infty }}\frac{b(n)}{n^s}\right).}$$

It is a straightforward exercise to show that if c(n) is defined by $${\displaystyle c(n):=\sum _{ij=n}a(i)b(j)=\sum _{i\mid n}a(i)b\left(\frac{n}{i}\right),}$$ then $${\displaystyle F_c(s)=F_a(s)F_b(s).}$$

This function c is called the Dirichlet convolution of a and b, and is denoted by ${\displaystyle a\ast b}$.

A particularly important case is convolution with the constant function a(n) = 1 for all n, corresponding to multiplying the generating function by the zeta function: $${\displaystyle g(n)=\sum _{d\mid n}f(d).}$$

Multiplying by the inverse of the zeta function gives the Möbius inversion formula: $${\displaystyle f(n)=\sum _{d\mid n}\mu \left(\frac{n}{d}\right)g(d).}$$

If f is multiplicative, then so is g. If f is completely multiplicative, then g is multiplicative, but may or may not be completely multiplicative.

Relations among the functions

There are a great many formulas connecting arithmetical functions with each other and with the functions of analysis, especially powers, roots, and the exponential and log functions. The page divisor sum identities contains many more generalized and related examples of identities involving arithmetic functions.

Here are a few examples:

Dirichlet convolutions

${\displaystyle \sum _{\delta \mid n}\mu (\delta )=\sum _{\delta \mid n}\lambda \left(\frac{n}{\delta }\right)|\mu (\delta )|=\{\begin{array}{ll}1& \text{if }n=1\\ 0& \text{if }n\ne 1\end{array}}$ where λ is the Liouville function.

${\displaystyle \sum _{\delta \mid n}\phi (\delta )=n.}$

${\displaystyle \phi (n)=\sum _{\delta \mid n}\mu \left(\frac{n}{\delta }\right)\delta =n\sum _{\delta \mid n}\frac{\mu (\delta )}{\delta }.}$ Möbius inversion

${\displaystyle \sum _{\delta \mid n}{J}_k(\delta )=n^k.}$

${\displaystyle J_k(n)=\sum _{\delta \mid n}\mu \left(\frac{n}{\delta }\right){\delta }^k=n^k\sum _{\delta \mid n}\frac{\mu (\delta )}{{\delta }^k}.}$ Möbius inversion

${\displaystyle \sum _{\delta \mid n}{\delta }^s{J}_r(\delta )J_s\left(\frac{n}{\delta }\right)=J_{r+s}(n)}$

${\displaystyle \sum _{\delta \mid n}\phi (\delta )d\left(\frac{n}{\delta }\right)=\sigma (n).}$

${\displaystyle \sum _{\delta \mid n}|\mu (\delta )|=2^{\omega (n)}.}$

${\displaystyle |\mu (n)|=\sum _{\delta \mid n}\mu \left(\frac{n}{\delta }\right)2^{\omega (\delta )}.}$ Möbius inversion

${\displaystyle \sum _{\delta \mid n}{2}^{\omega (\delta )}=d(n^2).}$

${\displaystyle 2^{\omega (n)}=\sum _{\delta \mid n}\mu \left(\frac{n}{\delta }\right)d({\delta }^2).}$ Möbius inversion

${\displaystyle \sum _{\delta \mid n}d({\delta }^2)=d^2(n).}$

${\displaystyle d(n^2)=\sum _{\delta \mid n}\mu \left(\frac{n}{\delta }\right)d^2(\delta ).}$ Möbius inversion

${\displaystyle \sum _{\delta \mid n}d\left(\frac{n}{\delta }\right)2^{\omega (\delta )}=d^2(n).}$

${\displaystyle \sum _{\delta \mid n}\lambda (\delta )=\{\begin{array}{ll}& 1\text{ if }n\text{ is a square }\\ & 0\text{ if }n\text{ is not square.}\end{array}}$ where λ is the Liouville function.

${\displaystyle \sum _{\delta \mid n}\mathrm{\Lambda }(\delta )=\log n.}$

${\displaystyle \mathrm{\Lambda }(n)=\sum _{\delta \mid n}\mu \left(\frac{n}{\delta }\right)\log (\delta ).}$ Möbius inversion

Sums of squares

For all ${\displaystyle k\ge 4,r_k(n)>0.}$ (Lagrange's four-square theorem).

${\displaystyle r_2(n)=4\sum _{d\mid n}\left(\frac{-4}{d}\right),}$

where the Kronecker symbol has the values

${\displaystyle \left(\frac{-4}{n}\right)=\{\begin{array}{ll}+1& \text{if }n\equiv 1(\mathrm{mod}4)\\ -1& \text{if }n\equiv 3(\mathrm{mod}4)\\ 0& \text{if }n\text{ is even}.\end{array}}$

There is a formula for r₃ in the section on class numbers below. $${\displaystyle r_4(n)=8\sum _{\stackrel{d\mid n}{4\nmid d}}d=8(2+(-1{)}^n)\sum _{\stackrel{d\mid n}{2\nmid d}}d=\{\begin{array}{ll}8\sigma (n)& \text{if }n\text{ is odd }\\ 24\sigma \left(\frac{n}{2^{\nu }}\right)& \text{if }n\text{ is even }\end{array},}$$ where ν = ν₂(n). $${\displaystyle r_6(n)=16\sum _{d\mid n}\chi \left(\frac{n}{d}\right)d^2-4\sum _{d\mid n}\chi (d)d^2,}$$ where ${\displaystyle \chi (n)=\left(\frac{-4}{n}\right).}$

Define the function σ_k^*(n) as $${\displaystyle {\sigma }_k^{\ast }(n)=(-1{)}^n\sum _{d\mid n}(-1{)}^d{d}^k=\{\begin{array}{ll}\sum _{d\mid n}{d}^k={\sigma }_k(n)& \text{if }n\text{ is odd }\\ \sum _{\stackrel{d\mid n}{2\mid d}}{d}^k-\sum _{\stackrel{d\mid n}{2\nmid d}}{d}^k& \text{if }n\text{ is even}.\end{array}}$$

That is, if n is odd, σ_k^*(n) is the sum of the kth powers of the divisors of n, that is, σ_k(n), and if n is even it is the sum of the kth powers of the even divisors of n minus the sum of the kth powers of the odd divisors of n.

${\displaystyle r_8(n)=16{\sigma }_3^{\ast }(n).}$

Adopt the convention that Ramanujan's τ(x) = 0 if x is not an integer.

${\displaystyle r_{24}(n)=\frac{16}{691}{\sigma }_{11}^{\ast }(n)+\frac{128}{691}\left\{(-1{)}^{n-1}259\tau (n)-512\tau \left(\frac{n}{2}\right)\right\}}$

Divisor sum convolutions

Here "convolution" does not mean "Dirichlet convolution" but instead refers to the formula for the coefficients of the product of two power series:

${\displaystyle \left(\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n\right)\left(\sum _{n=0}^{\mathrm{\infty }}{b}_n{x}^n\right)=\sum _{i=0}^{\mathrm{\infty }}\sum _{j=0}^{\mathrm{\infty }}{a}_i{b}_j{x}^{i+j}=\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{i=0}^n{a}_i{b}_{n-i}\right)x^n=\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n.}$

The sequence ${\displaystyle c_n=\sum _{i=0}^n{a}_i{b}_{n-i}}$ is called the convolution or the Cauchy product of the sequences a_n and b_n.
These formulas may be proved analytically (see Eisenstein series) or by elementary methods.

${\displaystyle {\sigma }_3(n)=\frac{1}{5}\left\{6n{\sigma }_1(n)-{\sigma }_1(n)+12\sum _{0<k<n}{\sigma }_1(k){\sigma }_1(n-k)\right\}.}$

${\displaystyle {\sigma }_5(n)=\frac{1}{21}\left\{10(3n-1){\sigma }_3(n)+{\sigma }_1(n)+240\sum _{0<k<n}{\sigma }_1(k){\sigma }_3(n-k)\right\}.}$

${\displaystyle \begin{array}{cc}{\sigma }_7(n)& =\frac{1}{20}\left\{21(2n-1){\sigma }_5(n)-{\sigma }_1(n)+504\sum _{0<k<n}{\sigma }_1(k){\sigma }_5(n-k)\right\}\\ & ={\sigma }_3(n)+120\sum _{0<k<n}{\sigma }_3(k){\sigma }_3(n-k).\end{array}}$

${\displaystyle \begin{array}{cc}{\sigma }_9(n)& =\frac{1}{11}\left\{10(3n-2){\sigma }_7(n)+{\sigma }_1(n)+480\sum _{0<k<n}{\sigma }_1(k){\sigma }_7(n-k)\right\}\\ & =\frac{1}{11}\left\{21{\sigma }_5(n)-10{\sigma }_3(n)+5040\sum _{0<k<n}{\sigma }_3(k){\sigma }_5(n-k)\right\}.\end{array}}$

${\displaystyle \tau (n)=\frac{65}{756}{\sigma }_{11}(n)+\frac{691}{756}{\sigma }_5(n)-\frac{691}{3}\sum _{0<k<n}{\sigma }_5(k){\sigma }_5(n-k),}$ where τ(n) is Ramanujan's function.

Since σ_k(n) (for natural number k) and τ(n) are integers, the above formulas can be used to prove congruences for the functions. See Ramanujan tau function for some examples.

Extend the domain of the partition function by setting p(0) = 1.

${\displaystyle p(n)=\frac{1}{n}\sum _{1\le k\le n}\sigma (k)p(n-k).}$ This recurrence can be used to compute p(n).

Class number related

Peter Gustav Lejeune Dirichlet discovered formulas that relate the class number h of quadratic number fields to the Jacobi symbol.

An integer D is called a fundamental discriminant if it is the discriminant of a quadratic number field. This is equivalent to D ≠ 1 and either a) D is squarefree and D ≡ 1 (mod 4) or b) D ≡ 0 (mod 4), D/4 is squarefree, and D/4 ≡ 2 or 3 (mod 4).

Extend the Jacobi symbol to accept even numbers in the "denominator" by defining the Kronecker symbol: $${\displaystyle \left(\frac{a}{2}\right)=\{\begin{array}{ll}0& \text{ if }a\text{ is even}\\ (-1{)}^{\frac{a^2-1}{8}}& \text{ if }a\text{ is odd. }\end{array}}$$

Then if D < −4 is a fundamental discriminant $${\displaystyle \begin{array}{cc}h(D)& =\frac{1}{D}\sum _{r=1}^{|D|}r\left(\frac{D}{r}\right)\\ & =\frac{1}{2-\left(\frac{D}{2}\right)}\sum _{r=1}^{|D|/2}\left(\frac{D}{r}\right).\end{array}}$$

There is also a formula relating r₃ and h. Again, let D be a fundamental discriminant, D < −4. Then $${\displaystyle r_3(|D|)=12\left(1-\left(\frac{D}{2}\right)\right)h(D).}$$

Prime-count related

Let ${\displaystyle H_n=1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}}$ be the nth harmonic number. Then

${\displaystyle \sigma (n)\le H_n+e^{H_n}\log H_n}$ is true for every natural number n if and only if the Riemann hypothesis is true.

The Riemann hypothesis is also equivalent to the statement that, for all n > 5040, $${\displaystyle \sigma (n)<e^{\gamma }n\log \log n}$$ (where γ is the Euler–Mascheroni constant). This is Robin's theorem.

${\displaystyle \sum _p{\nu }_p(n)=\mathrm{\Omega }(n).}$

${\displaystyle \psi (x)=\sum _{n\le x}\mathrm{\Lambda }(n).}$

${\displaystyle \mathrm{\Pi }(x)=\sum _{n\le x}\frac{\mathrm{\Lambda }(n)}{\log n}.}$

${\displaystyle e^{\theta (x)}=\prod _{p\le x}p.}$

${\displaystyle e^{\psi (x)}=\mathrm{lcm}[1,2,\dots ,\lfloor x\rfloor ].}$

Menon's identity

In 1965 P Kesava Menon proved $${\displaystyle \sum _{\stackrel{1\le k\le n}{gcd(k,n)=1}}gcd(k-1,n)=\phi (n)d(n).}$$

This has been generalized by a number of mathematicians. For example,

• B. Sury $${\displaystyle \sum _{\stackrel{1\le k_1,k_2,\dots ,k_s\le n}{gcd(k_1,n)=1}}gcd(k_1-1,k_2,\dots ,k_s,n)=\phi (n){\sigma }_{s-1}(n).}$$
• N. Rao $${\displaystyle \sum _{\stackrel{1\le k_1,k_2,\dots ,k_s\le n}{gcd(k_1,k_2,\dots ,k_s,n)=1}}gcd(k_1-a_1,k_2-a_2,\dots ,k_s-a_s,n{)}^s=J_s(n)d(n),}$$ where a₁, a₂, ..., a_s are integers, gcd(a₁, a₂, ..., a_s, n) = 1.
• László Fejes Tóth $${\displaystyle \sum _{\stackrel{1\le k\le m}{gcd(k,m)=1}}gcd(k^2-1,m_1)gcd(k^2-1,m_2)=\phi (n)\sum _{\stackrel{d_1\mid m_1}{d_2\mid m_2}}\phi (gcd(d_1,d_2))2^{\omega (\mathrm{lcm}(d_1,d_2))},}$$ where m₁ and m₂ are odd, m = lcm(m₁, m₂).

In fact, if f is any arithmetical function $${\displaystyle \sum _{\stackrel{1\le k\le n}{gcd(k,n)=1}}f(gcd(k-1,n))=\phi (n)\sum _{d\mid n}\frac{(\mu \ast f)(d)}{\phi (d)},}$$ where ${\displaystyle \ast }$ stands for Dirichlet convolution.

Miscellaneous

Let m and n be distinct, odd, and positive. Then the Jacobi symbol satisfies the law of quadratic reciprocity: $${\displaystyle \left(\frac{m}{n}\right)\left(\frac{n}{m}\right)=(-1{)}^{(m-1)(n-1)/4}.}$$

Let D(n) be the arithmetic derivative. Then the logarithmic derivative $${\displaystyle \frac{D(n)}{n}=\sum _{\stackrel{p\mid n}{p\text{ prime}}}\frac{v_p(n)}{p}.}$$ See Arithmetic derivative for details.

Let λ(n) be Liouville's function. Then

${\displaystyle |\lambda (n)|\mu (n)=\lambda (n)|\mu (n)|=\mu (n),}$ and

${\displaystyle \lambda (n)\mu (n)=|\mu (n)|={\mu }^2(n).}$

Let λ(n) be Carmichael's function. Then

${\displaystyle \lambda (n)\mid \varphi (n).}$ Further,

${\displaystyle \lambda (n)=\varphi (n)\text{ if and only if }n=\{\begin{array}{l}1,2,4;\\ 3,5,7,9,11,\dots \text{ (that is, }{p}^k\text{, where }p\text{ is an odd prime)};\\ 6,10,14,18,\dots \text{ (that is, }2p^k\text{, where }p\text{ is an odd prime)}.\end{array}}$

See Multiplicative group of integers modulo n and Primitive root modulo n.

${\displaystyle 2^{\omega (n)}\le d(n)\le 2^{\mathrm{\Omega }(n)}.}$

${\displaystyle \frac{6}{{\pi }^2}<\frac{\varphi (n)\sigma (n)}{n^2}<1.}$

${\displaystyle \begin{array}{cc}{c}_q(n)& =\frac{\mu \left(\frac{q}{gcd(q,n)}\right)}{\varphi \left(\frac{q}{gcd(q,n)}\right)}\varphi (q)\\ & =\sum _{\delta \mid gcd(q,n)}\mu \left(\frac{q}{\delta }\right)\delta .\end{array}}$ Note that ${\displaystyle \varphi (q)=\sum _{\delta \mid q}\mu \left(\frac{q}{\delta }\right)\delta .}$

${\displaystyle c_q(1)=\mu (q).}$

${\displaystyle c_q(q)=\varphi (q).}$

${\displaystyle \sum _{\delta \mid n}{d}^3(\delta )={\left(\sum _{\delta \mid n}d(\delta )\right)}^2.}$ Compare this with 1³ + 2³ + 3³ + ... + n³ = (1 + 2 + 3 + ... + n)²

${\displaystyle d(uv)=\sum _{\delta \mid gcd(u,v)}\mu (\delta )d\left(\frac{u}{\delta }\right)d\left(\frac{v}{\delta }\right).}$

${\displaystyle {\sigma }_k(u){\sigma }_k(v)=\sum _{\delta \mid gcd(u,v)}{\delta }^k{\sigma }_k\left(\frac{uv}{{\delta }^2}\right).}$

${\displaystyle \tau (u)\tau (v)=\sum _{\delta \mid gcd(u,v)}{\delta }^{11}\tau \left(\frac{uv}{{\delta }^2}\right),}$ where τ(n) is Ramanujan's function.

First 100 values of some arithmetic functions