arXiv:math/0505125v1 [math.CA] 7 May 2005 RAMANUJAN’S FORMULA FOR THE LOGARITHMIC DERIVATIVE OF THE GAMMA FUNCTION David Bradley Abstract. We prove a remarkable formula of Ramanujan for the logarithmic deriv- ative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in the notebooks [5]. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan’s formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler’s constant, γ. Ramanujan was evidently fond of series expansions and representations of spe- cial functions. The unorganized material in the second and third notebooks contain many interesting formulae involving the gamma function, Bessel functions, and hy- pergeometric functions, to name a few examples. The following remarkable formula for the logarithmic derivative of the gamma function has some very interesting con- sequences, and should be contrasted with the well-known series representation ψ(x + 1) := Γ ′ Γ (x + 1) = −γ + ∞  n=1  1 n − 1 x + n  , [1, p.259, 6.3.16]. Theorem. For all x> 0, ψ(x + 1) = π 3 log x + 1 2x − 1 4πx 2 + π cot(πx) e 2πx − 1 + π log |2 sin(πx)| 2 sinh 2 (πx) + ∞  k=1 2k (e 2πk − 1)(k 2 − x 2 ) − π 2 ∞  k=1 log |k 4 − x 4 | sinh 2 (πk) − 2π ∞  k=1 e −2πkx  k 2 ∞  n=1 sin(2πnx) k 2 + n 2 − k 3 ∞  n=1 cos(2πnx) n(k 2 + n 2 )  . (1) The theorem is recorded as formula 2, p. 280 in Ramanujan’s notebooks [5]. As per usual, Ramanujan gives no proof or explanation. In the sequel, we give a proof, and offer a plausible explanation of how Ramanujan may have arrived at formula (1). But first, we examine the formula more closely, with a view to gaining a better 1991 Mathematics Subject Classification. 33B15, 11Y60. Presented at the 4th conference of the Canadian Number Theory Association, Halifax, 1994 Typeset by A M S-T E X 1