Editing Arithmetic function (section)

=== ''σ''<sub>''k''</sub>(''n''), ''τ''(''n''), ''d''(''n'') – divisor sums ===
'''[[divisor function|σ<sub>''k''</sub>(''n'')]]''' is the sum of the ''k''th powers of the positive divisors of ''n'', including 1 and ''n'', where ''k'' is a complex number.

'''''σ''<sub>1</sub>(''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, '''''σ''<sub>0</sub>(''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).

<math display="block">\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).</math>

Setting ''k'' = 0 in the second product gives
<math display="block">\tau(n) = d(n) = (1 + a_{1})(1+a_{2})\cdots(1+a_{\omega(n)}).</math>