How Would Riemann Evaluate $\Zeta (2N) $?

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∞ s−1 τ − 1−s τ dτ = (1 2 )Γ(s)ζ(s). (6) Z0 e +1 Now, for complex z = x + iy and integer s, let f(z) = zs−1/(ez − 1) and, given R> 0, let ΓR be the rectangular contour of vertices 0, R, R + iπ and iπ in the complex plane. By Cauchy’s theorem, we have that

f(z)dz =0. (7) ZΓR It is easy to see that the integral over the right vertical side of the contour tends to zero when R → ∞. So, when R → ∞, equation (7) gives

∞ s−1 ∞ s−1 π s−1 x − (x + iπ) (iy) x dx x+iπ dx = i iy dy. (8) Z0 e − 1 Z0 e − 1 Z0 e − 1 For the particular case s = 2, using (2) and (6), this equation reads ∞ 1 1 π y i π y sin y ζ(2) + ζ(2) + iπ x dx = dy + dy. 2 Z0 e +1 Z0 2 2 Z0 1 − cos y Taking the real part of both sides one ﬁnds the desired result ζ(2) = π2/6. Similarly, in the case of even integer s = 2n, using (2) and (6), after some manipulations the real part of (8) gives

n−1 2n − π Γ(2n)ζ(2n)+ α(n, k)ζ(2n − 2k) = (−1)n 1 (9) 4n Xk=0 where − 2n − 1 α(n, k)= 1 − 21 2n+2k (−π2)k Γ(2n − 2k). (10) 2k This allows us to obtain the values ζ(2n) recursively, the solution being then expressible in closed form with the use of Bernoulli numbers. It is reasonable to speculate that Riemann could have noticed this procedure while working out the analytic extension of ζ(s) by means of (3) using (2) as

2 a starting point. It is worth pointing out that, as a side result, taking the imaginary part of (8) for even s or the real part for odd s, one obtains integral and series representations for the values ζ(2k+1). This also suggests an intuitive interpretation of the diﬀerent nature of ζ(2n) and ζ(2n+1), being the quantities associated respectively with the imaginary and the real parts of the function f(z)= z/(ez − 1) on the imaginary axis.

References

[1] R. Chapman, Evaluating ζ(2) (1999), available at http://empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf

[2] E. De Amo, M. D´ıaz Carrillo and J. Fernández-Sánchez, Another proof of Euler’s formula for ζ(2k), Proc. Amer. Math. Soc., 139 (2011) 1441-1444. [3] H. M. Edwards,Riemann’s Zeta Function. Academic Press, New York, 1974. [4] G. F. B. Riemann, Uber¨ die anzahl der primzahlen unter einer gegebenen grösse, Monatsber. Berlin. Akad., (1859) 671-680.

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