but I sure missed it. Pi 1 4 2 E (-)! 1 + 2 n 4 Sum[------------------------------, {n, 0, oo}] == ----------- 2 n Pi -2 (1 + 2 n) Pi 3 E (1 - E ) Pi I got this as the q -> E^(-2 Pi) special case of Sum[DivisorSigma[1,2*n+1] q^n,{n,0,∞}]== Sum[(1 + 2 n) q^n/(1 - q^(1 + 2 n)), {n, 0, ∞}] == ( I (Derivative[1][DedekindEta][-((I Log[q])/(4 \[Pi]))]/ DedekindEta[-((I Log[q])/(4 \[Pi]))] - ( 3 Derivative[1][DedekindEta][-((I Log[q])/(2 \[Pi]))])/ DedekindEta[-((I Log[q])/(2 \[Pi]))] + ( 2 Derivative[1][DedekindEta][-((I Log[q])/\[Pi])])/ DedekindEta[-((I Log[q])/\[Pi])]))/(2 \[Pi] Sqrt[q]) As suggested by Sunday's "notational modularity" item, I have a bunch more special values of eta and eta' to crank into this. (I bet you can hardly wait.) Also, eta[q], eta[q^2], and eta[q^4] are polynomially related by Jacobi's aequatio identica satis abstrusa <https://www.google.com/search?client=safari&rls=en&q=aequatio+identica+satis+abstrusa&ie=UTF-8&oe=UTF-8> , which should have an eta' analog. --rwg Vaguely related: ArcLength[{t, Sqrt[Cos[t]]}, {t, -π/2, π/2}] == Integrate[Sqrt[1 + (1/4)*Sin[t]*Tan[t]], {t, -Pi/2, Pi/2}] but the ISC says HUH? Really? (Connection: Area under that arc = π^(3/2)/(2 Sqrt[2] (1/4)!^2). )