Interpretation of Some Divergent Series via the Zeta and Eta Functions

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The Riemann zeta function is defined by the Dirichlet series:



A related function is the alternating zeta function, also known as the Dirichlet eta function:


Both series are clearly divergent for .

Analytic continuation of the zeta function over the whole complex plane implies the functional relations:



For or a negative integer, the following absurd results are implied:




and so on. However, in certain specialized mathematical contexts, finite values can be assigned to such divergent series. Ramanujan was a famous proponent of such series. The sum of the natural numbers given by the last formula is, in fact, known as the Ramanujan summation. The subject received a burst of popular interest in 2014 as a result of a YouTube video [1] and an article in The New York Times [2]. The formula turns out to be very significant in dealing with infinities in string theory [3] and also in the theory of the Casimir effect for the vacuum energy of the electromagnetic field.

This Demonstration shows the results of a number of divergent series represented by zeta and eta functions with negative arguments, with , , , and . We also consider the case , which does not give a simple fraction.


Contributed by: S. M. Blinder (August 25)
Open content licensed under CC BY-NC-SA



[1] Astounding: 1 + 2 + 3 + 4 + 5 + ... = -1/12 [Video] (May 17, 2023)

[2] D. Overbye, "In the End, It All Adds Up to -1/12," The New York Times, Feb 3, 2014.

[3] J. G. Polchinski, String Theory: An Introduction to the Bosonic String, New York: Cambridge University Press, 2005 p. 22.


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