Lambert Series

$S(q)=\backslash sum\_\{n=1\}^\backslash infty\; a\_n\; \backslash frac\; \{q^n\}\{1-q^n\}$

$S(q)=\backslash sum\_\{n=1\}^\backslash infty\; a\_n\; \backslash sum\_\{k=1\}^\backslash infty\; q^\{nk\}\; =\; \backslash sum\_\{m=1\}^\backslash infty\; b\_m\; q^m$

$b\_m\; =\; (a*1)(m)\; =\; \backslash sum\_\{n|m\}\; a\_n$

$\backslash sum\_\{n=1\}^\{\backslash infty\}\; q^n\; \backslash sigma\_0(n)\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{q^n\}\{1-q^n\}$

$\backslash sum\_\{n=1\}^\{\backslash infty\}\; q^n\; \backslash sigma\_\backslash alpha(n)\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{n^\backslash alpha\; q^n\}\{1-q^n\}$

$\backslash sigma\_\backslash alpha(n)\; =\; (\backslash textrm\{Id\}\_\backslash alpha*1)(n)\; =\; \backslash sum\_\{d|n\}\; d^\backslash alpha$

Related topics

• Erds-Borwein_constant
• Lambert series for Euler's totient

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