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Consider the divergent series $$\sum_{n=1}^{\infty} \frac{2^n}{n} $$

This can be seen as arising from the function $f(z) = -\ln(1-z)=\sum_{n=1}^{\infty} \frac{z^n}{n} $ and 'evaluating' that power series at $z=2$. I am interested in a divergent renormalization for this expression.

For many such divergent series we can use analytic continuation to find the divergent value. For example to evaluate $\sum_{n=0}^{\infty} 2^n = -1 $ it suffices to analytically continue $\sum_{n=0}^{\infty} z^n = \frac{1}{1-z}$ to the whole complex plane and once you do that it's easy to reason that $\sum_{n=0}^{\infty} 2^n = -1$ (although there are even easier algebraic arguments).

The problem here, is that the function $-\ln(1-z)$ has monodromy. As you analytically-continue the logarithm you end up with this multi-valued helix. And so when we ask what is $-\ln(-1)$ the function can be one of $...-3i\pi,-i\pi, i\pi, 3i\pi... $

If we apply some 'common sense' we might ask 'what are the most reasonable options' and that reduces it to $i\pi$ and $-i\pi$. Depending on if you think it's more natural to move clockwise or counterclockwise around the origin.

At this point picking ONE of those values seems extremely difficult. There's no real reason to pick clockwise over counterlockwise or vice versa and the notions of 'principal branch' are a human construct, not really opinionated by the math itself.

If there was a setting where this series arises that FORCES a choice it would be nice (for example in physics, or some other part of number theory etc...) but I don't know of any offhand. How do I actually assign a divergent summation to this? It's probably going to be one of those two choices $\pm i\pi$ but I would like a good argument for which one and why. (Or even cooler if you could show that in 2 different natural settings this series HAS to take on different values).

Sidharth Ghoshal
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  • I'm not sure if we have an appropriate tag for this, but it looks like it should be addressed in the norm on $\Bbb C_2$, the $p=2$ complex $p$-adic system. – J.G. Mar 31 '24 at 15:21
  • @Invariance i added complex geometry as a tag because of the monodromy reference here, i also do see why you would remove the tag (it was a stretch to get those folks' attention) – Sidharth Ghoshal Mar 31 '24 at 15:34
  • @J.G. Thanks for engaging! :) I put a $p$-adic number theory tag to get closer to the space of ideas you mentioned – Sidharth Ghoshal Mar 31 '24 at 15:35
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    This link discusses the identity $$\sum_{n=1}^{\infty} \frac{2^n}{n}=0$$ in $\mathbb{Q}_2$. – Sangchul Lee Mar 31 '24 at 15:43
  • Wow! thank you for finding this. At risk of sounding like an 'astrologer' this makes sense since it is really the "average" of all the possible values it could be which is vaguely reminiscent of the averaging ideas in this blog post: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ – Sidharth Ghoshal Mar 31 '24 at 15:45
  • @SangchulLee do there exist p-adic derivations of $\sum_{n=1}^{\infty} \frac{3^n}{n} = -\ln(2)$? If such a thing exists that would give me further evidence for the idea i wrote above – Sidharth Ghoshal Mar 31 '24 at 16:16
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    @SidhartGhoshal: There cannot be a $p$-adic derivation of that identity because the real number $-ln(2)$ is not a $p$-adic number in any simple way. There might be some theory that exploits $p$-adics here, but one has to be extremely careful not to mix up reals and $p$-adics in a "naive" way. Cf. https://math.stackexchange.com/a/4007515/96384 – Torsten Schoeneberg Mar 31 '24 at 19:19

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