rwg>This series gives you (http://gosper.org/filds.dvi) [...] Much nicer: Integrate from 0 and use Zeta'(-1): t inf / / 2 2 [ [ s log(t + s ) I log(Gamma(x)) dx = I -------------- ds + t log(Gamma(t)) ] ] 2 %pi s / / %e - 1 0 0 2 2 t log(t) t t - --------- + -- + - + Zeta'(-1). 2 4 2 rwg>For valuations of ilg(n/4 and n/6) (and hence the integral on the
right), see www.tweedledum.com/rwg/idents.htm, (d1021) et seq. (near the end).
Mma 6.0 can now do these (but not the infinite integral) in terms of (the superfluous) Log[Glaisher] instead of Zeta'[-1]. Use FunctionExpand if you get a PolyLog[-2,...] or a Derivative[1,0][Zeta][-1,...].
The rhs integral (and Binet's) look suspiciously Abel-Plana, and might yield interesting series (or product!).
Despite many stratagems, I can't get the sum to converge. --rwg ADROITLY DILATORY IDOLATRY