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321, no. 15) and [34, pp. 5)] 1 1 ζ (s) = − − log(2π )s + · · · . 17) where in the last step we used the equality αβ = π 2 . 14) to complete the proof. 30 Bruce C. 1 (p. 254). If a > 0, ∞ x(e2π x 0 = a π2 dx =2 − 1)(e2πa/x − 1) ∞ n=1 ∞ √ d(n)K 0 (4π an ) n=1 1 1 + 4 4π 2 a d(n) log(a/n) 1 − γ− 2 a2 − n2 log a − log(2π ) . 1) Proof. 1). 1) was actually first proved in print in 1966 by Soni [32]. 4). We use her idea to prove the second major claim of Ramanujan on page 254. In contrast to the claims on the top and bottom thirds of page 254, the one claim in the middle of page 254 seems to be missing one element, and so we shall proceed as we think Ramanujan might have done.

Selberg, Collected Papers, Vol. I, Springer-Verlag, Berlin, 1989. 30. A. Selberg and S. Chowla, On Epstein’s zeta-function (I), Proc. Nat. Acad. Sci. (USA) 35 (1949), 371–374. 31. A. Selberg and S. Chowla, On Epstein’s zeta-function, J. Reine Angew. Math. 227 (1967), 86–110. 32. K. Soni, Some relations associated with an extension of Koshliakov’s formula, Proc. Amer. Math. Soc. 17 (1966), 543–551. 33. A. Terras, Harmonic Analysis on Symmetric Spaces and Applications I, Springer-Verlag, New York, 1985.

8. C. G. J. Jacobi. De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis. Journal f¨ur die Reine und Angewandte Mathematik, 2:247–257, 1833. Reprinted in C. G. J. Jacobi: Gesammelte Werke. Vol. 3, pp. 191–268. Berlin: Georg Reimer, 1884. 9. Vladimir E. Korepin, Nikolai M. Bogoliubov, and Anatoli G. Izergin. Quantum inverse scattering method and correlation functions. Cambridge University Press, Cambridge, UK, 1993. 10. Greg Kuperberg.