Editing Incidence algebra (section)

=== Divisor poset and Dirichlet series ===
Consider the poset ''D'' of positive integers ordered by [[divisibility]], denoted by <math>a\,|\,b.</math> The reduced incidence algebra consists of functions <math>f(a,b)</math> that are invariant under multiplication: <math>f(ka,kb) = f(a,b)</math> for all <math>k\ge 1.</math> (This multiplicative equivalence of intervals is a much stronger relation than poset isomorphism; e.g., for primes ''p'', the two-element intervals [1,''p''] are all inequivalent.) For an invariant function, ''f''(''a'',''b'') depends only on ''b''/''a'', so a natural basis consists of invariant delta functions <math>\delta_n</math> defined by <math>\delta_n(a,b) = 1</math> if ''b''/''a'' = ''n'' and 0 otherwise; then any invariant function can be written <math>\textstyle f = \sum_{n\geq 0} f(1,n)\, \delta_n.</math>

The product of two invariant delta functions is:
:<math>(\delta_n \delta_m)(a,b) = \sum_{a|c|b} \delta_n(a,c)\,\delta_m(c,b) = \delta_{nm}(a,b),</math>
since the only non-zero term comes from ''c'' = ''na'' and ''b'' = ''mc'' = ''nma''. Thus, we get an isomorphism from the reduced incidence algebra to the ring of formal [[Dirichlet series]] by sending <math>\delta_n</math> to <math>n^{-s}\!,</math> so that ''f'' corresponds to <math display="inline">\sum_{n\geq 1} \frac{f(1,n)}{n^s}.</math>

The incidence algebra zeta function ζ<sub>''D''</sub>(''a'',''b'') = 1 corresponds to the classical [[Riemann zeta function]] <math>\zeta(s)=\textstyle \sum_{n\geq 1}\frac{1}{n^s},</math> having reciprocal <math display="inline">\frac{1}{\zeta(s)} = \sum_{n\geq 1}\frac{\mu(n)}{n^s},</math> where <math>\mu(n)=\mu_D(1,n)</math> is the classical [[Möbius function]] of number theory. Many other [[arithmetic function]]s arise naturally within the reduced incidence algebra, and equivalently in terms of Dirichlet series. For example, the [[divisor function]] <math>\sigma_0(n)</math> is the square of the zeta function, <math>\sigma_0(n) = \zeta^2\!(1,n),</math> a special case of the above result that <math>\zeta^2\!(x,y)</math> gives the number of elements in the interval [''x'',''y'']; equivalenty, <math display="inline">\zeta(s)^2 = \sum_{n\geq 1} \frac{\sigma_0(n)}{n^s}.</math>

The product structure of the divisor poset facilitates the computation of its Möbius function. [[Fundamental theorem of arithmetic|Unique factorization into primes]] implies ''D'' is isomorphic to an infinite Cartesian product <math>\N\times\N \times \dots</math>, with the order given by coordinatewise comparison:  <math>n = p_1^{e_1}p_2^{e_2}\dots</math>, where <math>p_k</math> is the ''k''<sup>th</sup> prime,  corresponds to its sequence of exponents <math>(e_1,e_2,\dots).</math> Now the Möbius function of ''D'' is the product of the Möbius functions for the factor posets, computed above, giving the classical formula: 
:<math>\mu(n) = \mu_D(1,n) = \prod_{k\geq 1}\mu_{\N}(0,e_k)
\,=\,\left\{\begin{array}{cl}(-1)^d & \text{for } n \text{ squarefree with } d \text{ prime factors}\\
0 & \text{otherwise.}
\end{array}\right.</math>

The product structure also explains the classical [[Euler product]] for the zeta function. The zeta function of ''D'' corresponds to a Cartesian product of zeta functions of the factors, computed above as <math display="inline">\frac{1}{1-t},</math> so that <math display="inline">\zeta_D \cong \prod_{k\geq 1}\!\frac{1}{1-t},</math> where the right side is a Cartesian product. Applying the isomorphism which sends ''t'' in the ''k''<sup>th</sup> factor to <math>p_k^{-s}</math>, we obtain the usual Euler product.