We consider Ramanujan's contribution to formulas for Euler's constant fl. For example, in his second notebook Ramanujan states that (in modern notation) 1 X k=1 (\Gamma1) k\Gamma1 nk ` x k k! ' n = ln x + fl + o(1) as x ! 1. This is known to be correct for the case n = 1, but incorrect for n ? 2. We consider the case n = 2. We also suggest a different, correct generalization of the case n = 1. 1.

Whether ’tis nobler in the mind to suffer The slings and arrows of outrageous fortune, Or to take arms against a sea of troubles And by opposing end them? Hamlet 3/1, by W. Shakespeare In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computer-assisted proofs will be give a special attention. Finally, in a highly speculative part, we will anticipate the evolution of proofs under the assumption that the quantum computer will materialize. We will argue that there is little ‘intrinsic ’ difference between traditional and ‘unconventional ’ types of proofs. 2 Mathematical Proofs: An Evolution in Eight Stages Theory is to practice as rigour is to vigour. D. E. Knuth Reason and experiment are two ways to acquire knowledge. For a long time mathematical

doi:10.4169/193009809X468689 (∑n The mathematical constant γ = limn→∞ k=1 1 k − ln(n)) = 0.5772156..., known as Euler’s constant, is not as well known as its cousins π, e, i, but is still important enough to warrant serious consideration in the circles of applied mathematics, calculus, and number theory. Some authors will occasionally refer to γ as the Euler-Mascheroni constant, so named after the Italian geometer Lorenzo Mascheroni (1750–1800), who actually introduced the symbol γ for the constant (although there is controversy about this claim) and also computed, though with error, the first 32 digits [16, 34]. Sometimes one will find in older texts the symbols C (this was Euler’s constant of integration) and A (also from Mascheroni) to represent the constant, but these notations seem to have disappeared in the modern era [27]. Our aim in this article is to present a survey of γ that is both manageable by, and enlightening to, those who favor mathematics at the undergraduate level. To try and follow in the footsteps of the big boys π and e is quite a chore, but this brief historical description of γ and colorful portfolio of applications and surprising appearances in a multitude of settings is both impressive and mathematically educational. Defining and evaluating the constant Calculus students can approximate the integral ∫ n (1/x) dx = ln(n) by inscribed and 1 circumscribed rectangles, and hence obtain the inequalities (for any integer n> 1) 1 n < n ∑ 1 − ln(n) <1, k so if the limit exists, 0 ≤ lim n→∞ k=1 n∑ k=1

The object of mathematical rigour is to sanction and