What is the expected length of the sum of vectors in a multi-dimensional sphere? Suppose we pick $m$ vectors i.i.d from the surface of a $d$-dimensional unit sphere (they all have length 1). 
What would be the expected length of their sum?
Equivalently, we can ask about the expected length of their average.
I managed to solve it for 2 vectors in 2 dimensions, but I need a generic solution...
If there's any "known solution" for this problem, even a link will be good enough.
 A: OK, to atone for my rash comment, I recalled that I needed something related in:
@article {MR2356429,
    AUTHOR = {Rivin, Igor},
     TITLE = {Surface area and other measures of ellipsoids},
   JOURNAL = {Adv. in Appl. Math.},
  FJOURNAL = {Advances in Applied Mathematics},
    VOLUME = {39},
      YEAR = {2007},
    NUMBER = {4},
     PAGES = {409--427},
      ISSN = {0196-8858},
   MRCLASS = {52A38 (28A75 33C65 60F99)},
  MRNUMBER = {2356429 (2008k:52012)},
       DOI = {10.1016/j.aam.2006.08.009},
       URL = {http://dx.doi.org/10.1016/j.aam.2006.08.009},
}
(see page 414). And found it in the book by Mathai and Provost:
@book {MR1192786,
    AUTHOR = {Mathai, A. M. and Provost, Serge B.},
     TITLE = {Quadratic forms in random variables},
    SERIES = {Statistics: Textbooks and Monographs},
    VOLUME = {126},
      NOTE = {Theory and applications},
 PUBLISHER = {Marcel Dekker Inc.},
   ADDRESS = {New York},
      YEAR = {1992},
     PAGES = {xxii+367},
      ISBN = {0-8247-8691-2},
   MRCLASS = {62H10 (62E15)},
  MRNUMBER = {1192786 (94g:62110)},
MRREVIEWER = {Donald R. Jensen},
}
It should be in the vicinity of page 62. The argument uses Laplace transforms, and is not trivial.
A: At http://www.carma.newcastle.edu.au/~jb616/walks2.pdf Jonathan Borwein and co-authors discuss $$\int_{[0,1]^n}\left|\sum_{k=1}^ne^{2\pi ix_k}\right|^s\thinspace dx$$ which you may find has some relevance to your question. That paper also has references to others where such integrals are discussed, and there's another paper on Jonathan's website on a closely related topic. 
