Editing Arithmetic function (section)

=== Divisor sum convolutions ===
Here "convolution" does not mean "Dirichlet convolution" but instead refers to the formula for the coefficients of the [[Power series#Multiplication and division|product of two power series]]:
: <math>  \left(\sum_{n=0}^\infty a_n x^n\right)\left(\sum_{n=0}^\infty b_n x^n\right)
= \sum_{i=0}^\infty \sum_{j=0}^\infty  a_i b_j x^{i+j}
= \sum_{n=0}^\infty \left(\sum_{i=0}^n a_i b_{n-i}\right) x^n
= \sum_{n=0}^\infty c_n x^n
.</math>

The sequence <math>c_n = \sum_{i=0}^n a_i b_{n-i}</math> is called the [[convolution]] or the [[Cauchy product]] of the sequences ''a''<sub>''n''</sub> and ''b''<sub>''n''</sub>.
{{br}}These formulas may be proved analytically (see [[Eisenstein series]]) or by elementary methods.<ref>Williams, ch. 13; Huard, et al. (external links).</ref>
: <math>
\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\}.
</math> &nbsp; &nbsp;<ref name="Ramanujan, p. 146">Ramanujan, ''On Certain Arithmetical Functions'', Table IV; ''Papers'', p. 146</ref>
: <math>
\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\}.
</math> &nbsp; &nbsp;<ref name="Koblitz, ex. III.2.8">Koblitz, ex. III.2.8</ref>
: <math>
\begin{align}
\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{align}
</math> &nbsp; &nbsp;<ref name="Koblitz, ex. III.2.8" /><ref>Koblitz, ex. III.2.3</ref>
: <math>
\begin{align}
\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{align}
</math> &nbsp; &nbsp;<ref name="Ramanujan, p. 146" /><ref>Koblitz, ex. III.2.2</ref>
: <math>
\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),
</math> &nbsp; &nbsp; where ''τ''(''n'') is Ramanujan's function. &nbsp; &nbsp;<ref>Koblitz, ex. III.2.4</ref><ref>Apostol, ''Modular Functions ...'', Ex. 6.10</ref>

Since ''σ''<sub>''k''</sub>(''n'') (for natural number ''k'') and ''τ''(''n'') are integers, the above formulas can be used to prove congruences<ref>Apostol, ''Modular Functions...'', Ch. 6 Ex. 10</ref> for the functions. See [[Ramanujan tau function]] for some examples.

Extend the domain of the partition function by setting {{math|1=''p''(0) = 1.}}
: <math>
p(n)=\frac{1}{n}\sum_{1\le k\le n}\sigma(k)p(n-k).
</math> &nbsp; &nbsp;<ref>G.H. Hardy, S. Ramannujan, ''Asymptotic Formulæ in Combinatory Analysis'', § 1.3; in Ramannujan, ''Papers'' p. 279</ref> &nbsp; This recurrence can be used to compute ''p''(''n'').