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add an equivalent series using the polylogarithm |
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A [[Lambert series]] involving the divisor function is: {{sfnp|Hardy|Wright|2008|pp=338-341|loc=§17.10}}
:<math>\sum_{n=1}^\infty q^n \sigma_a(n) = \sum_{n=1}^\infty \sum_{j=1}^\infty n^a q^{j\,n} = \sum_{n=1}^\infty \frac{n^a q^n}{1-q^n} = \sum_{n=1}^\infty \operatorname{Li}_{-a}(q^n)</math>
for arbitrary [[complex number|complex]] |''q''| ≤ 1 and ''a'' (<math>\operatorname{Li}</math> is the [[polylogarithm]]). This summation also appears as the [[Eisenstein series#Fourier series|Fourier series of the Eisenstein series]] and the [[Weierstrass elliptic functions#Invariants|invariants of the Weierstrass elliptic functions]].
For <math>k>0</math>, there is an explicit series representation with [[Ramanujan sum]]s <math> c_m(n) </math> as :<ref>{{cite book |author=E. Krätzel |title=Zahlentheorie |publisher=VEB Deutscher Verlag der Wissenschaften |place =Berlin |year=1981 |pages=130}} (German)</ref>
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