Editing Dirichlet convolution (section)

==Properties and examples==

In these formulas, we use the following [[arithmetical function]]s:
* <math>\varepsilon</math> is the multiplicative identity: <math>\varepsilon(1) = 1</math>, otherwise 0 (<math>\varepsilon(n)=\lfloor \tfrac1n \rfloor</math>).
* <math>1</math> is the constant function with value 1:  <math>1(n) = 1</math> for all <math>n</math>. Keep in mind that <math>1</math> is not the identity. (Some authors [[Incidence algebra#Special_elements|denote this]] as <math>\zeta</math> because the associated Dirichlet series is the [[Riemann zeta function]].)
* <math>1_C</math> for <math>C \subset \mathbb{N}</math> is a set [[indicator function]]:  <math>1_C(n) = 1</math> iff <math>n \in C</math>, otherwise 0.
*<math>\text{Id}</math> is the identity function with value ''n'':  <math>\text{Id}(n) = n</math>.
*<math>\text{Id}_k</math> is the ''k''th power function:  <math>\text{Id}_k(n)=n^k</math>.

The following relations hold:

* <math>1 * \mu = \varepsilon</math>,&nbsp;the Dirichlet inverse of the constant function <math>1</math> is the [[Möbius function]] (see [[Möbius_function#Proof_of_the_formula_for_the_sum_of_μ_over_divisors|proof]]). Hence:
*<math>g = f * 1</math>  if and only if  <math>f = g * \mu</math>, the [[Möbius inversion formula]].
*<math>\sigma_k = \text{Id}_k * 1</math>, the [[divisor function|kth-power-of-divisors sum function]] ''σ''<sub>''k''</sub>.
*<math>\sigma = \text{Id} * 1</math>, the sum-of-divisors function {{nowrap|1=''σ'' = ''σ''<sub>1</sub>}}.
*<math>\tau = 1 * 1</math> , the number-of-divisors function {{nowrap|1=''τ''(''n'') = ''σ''<sub>0</sub>}}.
*<math>\text{Id}_k = \sigma_k * \mu</math>,&nbsp; by Möbius inversion of the formulas for ''σ''<sub>''k''</sub>, ''σ'', and ''τ''.
*<math>\text{Id} = \sigma * \mu</math>
*<math>1 = \tau * \mu</math>
*<math>\phi * 1 = \text{Id}</math> , proved under [[Euler's totient function#Divisor sum|Euler's totient function]].
*<math>\phi = \text{Id} * \mu</math> , by Möbius inversion.
*<math>\sigma = \phi * \tau</math> &nbsp;, from convolving 1 on both sides of <math>\phi * 1 = \text{Id}</math>.
*<math>\lambda * |\mu| = \varepsilon</math> &nbsp;where ''λ'' is [[Liouville's function]].
*<math>\text{Id} * \phi = P </math>, where <math> P </math> is [[Pillai's arithmetical function]], also known as the gcd-sum function.
*<math>\lambda * 1 = 1_{\text{Sq}}</math>&nbsp;where Sq = {1, 4, 9, ...} is the set of squares.
*<math>\text{Id}_k * (\text{Id}_k \mu) = \varepsilon </math>
*<math>\tau^3 * 1 = (\tau * 1)^2</math>
*<math>J_k * 1 = \text{Id}_k</math>,  [[Jordan's totient function]].
*<math>(\text{Id}_s J_r) * J_s = J_{s + r} </math>
*<math>\Lambda * 1 = \log</math>, where <math>\Lambda</math> is [[von Mangoldt function|von Mangoldt's function]].
*<math>|\mu| \ast 1 = 2^{\omega},</math> where <math>\omega(n)</math> is the [[prime omega function]] counting ''distinct'' prime factors of ''n''.
*<math>\Omega \ast \mu = 1_{\mathcal{P}}</math>, the characteristic function of the prime powers. 
*<math>\omega \ast \mu = 1_{\mathbb{P}}</math> where <math>1_{\mathbb{P}}(n) \mapsto \{0,1\}</math> is the characteristic function of the primes. 

This last identity shows that the [[prime-counting function]] is given by the summatory function 

:<math>\pi(x) = \sum_{n \leq x} (\omega \ast \mu)(n) = \sum_{d=1}^{x} \omega(d) M\left(\left\lfloor \frac{x}{d} \right\rfloor\right)</math> 

where <math>M(x)</math> is the [[Mertens function]] and <math>\omega</math> is the distinct prime factor counting function from above. This expansion follows from the identity for the sums over Dirichlet convolutions given on the [[divisor sum identities]] page (a standard trick for these sums).<ref>{{cite book|title=Apostol's Introduction to Analytic Number Theory|last1=Schmidt |first1=Maxie}} This identity is a little special something I call "croutons". It follows from several chapters worth of exercises in Apostol's classic book.</ref>
<!-- * ''μ'' ∗ 1 = ''ε'' ∗ (''μ'' ∗ Id<sub>''k''</sub>) ∗ Id<sub>''k''</sub> = ''ε'' (generalized Möbius inversion) -->