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The infinite sums

  • $1 + 2 + 4 + 8 + ...$

  • $1 + 3 + 9 + 27 + ...$

  • $1 + 5 + 25 + 125 + ...$

  • $1 + 7 + 49 + 343 + ...$

  • ...

of powers of primes do not converge in the usual sense. However, by analytically continuing the expression

  • $1 + x + x^2 + x^3 + ... = \frac{1}{1-x}$

we can assign the values $-1, -\frac{1}{2}, -\frac{1}{4}$ to these series by simply putting 2, 3, and 5 into this expression.

However, there is another way to get the same result. Rather than using analytic continuation on the reals, we can simply take the result in different rings of $p$-adic integers. Then we get:

  • The first one converges in the 2-adic integers to $-1$.
  • The second one converges in the 3-adic integers to $-\frac{1}{2}$.
  • The third one converges in the 5-adic integers to $-\frac{1}{4}$.

This is somewhat interesting, because it would seem that analytic continuation in the reals and p-adic numbers have little to do with one another.

However, in this case, the two are related because by changing the metric on $\Bbb Q$ to a $p$-adic one, the radius of convergence changes, so that the expression $1 + x + x^2 + x^3 + ... = \frac{1}{1-x}$ is valid in the appropriate ring for the values described.

In other words, if you start with the ring of p-adic integers in mind, it is easy to see that the formal power series converges in that ring.

My question is, is it possible to go the other way - starting with a formal power series, to derive a nontrivial, "natural" choice of ring in which the series converges?

Here are some particular examples:

  • Suppose we want to evaluate the infinite sum $1 + x + x^2 + x^3 + ...$ and set $x$ equal to some non-integer value $r$. There is no ring of $r$-adic integers for arbitrary rational $r$, but we can use the analytic continuation to "assign" the result $\frac{1}{1-r}$ to this expression. But, is there also a nontrivial, "natural" ring in which this sum does directly converge to that result? What if $r$ is allowed to be real rather than rational?

  • Likewise, rather than needing a different ring for each $r$ there a ring in which that power series converges for all values of $r \neq 1$? If not, then at least for all integer or rational $r$?

  • Is there a general method to associate such rings to power series, particularly ones besides the simple $1 + x + x^2 + ...$ one listed?

Mike Battaglia
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    I don't think there's much to say here beyond the $p$-adic case, e.g. because of Ostrowski's theorem (https://en.wikipedia.org/wiki/Ostrowski%27s_theorem). If $x$ is rational you can work in the $p$-adic integers where $p$ is a prime dividing the numerator but not the denominator of $x$ in lowest terms. – Qiaochu Yuan Mar 28 '19 at 08:01
  • One way to think about this question is to split it up into a formal identity in a ring of formal power (or more generally, Laurent) series, and then asking whether there exist certain "evaluation homomorphisms" from those rings into the base rings. That last question, however, will quickly boil down to good old metric "annulus of convergence" considerations. For this specific series, you need precisely a kind of norm on your ring w.r.t. which your $r$ is smaller than $1$. – Torsten Schoeneberg Mar 28 '19 at 12:05
  • @QiaochuYuan that theorem only applies to absolute values, which are "multiplicative" metrics. We wouldn't need multiplicativity for this, would we? Would we even need for it to be a metric (e.g. obey the triangle inequality)? Perhaps a different topology is all that's really needed. – Mike Battaglia Mar 28 '19 at 14:48
  • @TorstenSchoeneberg I guess the same question applies - would it need to be a "norm" on the field, e.g. an absolute value (which is a multiplicative metric)? – Mike Battaglia Mar 28 '19 at 15:04
  • Is the next sequence in your list $1+7\ldots$ or $1+9\ldots$? – Robert Frost Apr 26 '19 at 03:52
  • It was $1 + 7 + ...$. – Mike Battaglia Apr 26 '19 at 03:59
  • Maybe linked https://math.stackexchange.com/questions/141971/divergent-series-and-p-adics – Liam Jan 16 '25 at 13:42

1 Answers1

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The p-adic Riemann zeta function ζp(s) values at negative odd integers are those of the Riemann zeta function at negative odd integers.

A p-adic L-function, is a function analogous to the Riemann zeta function, but whose domain and target are p-adic. See https://en.m.wikipedia.org/wiki/P-adic_L-function.

Your intuition is informally similar to the main conjecture of the Iwasawa theory, which was proven true in 1984. See below:

https://en.m.wikipedia.org/wiki/Main_conjecture_of_Iwasawa_theory

Nate Busa
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    Not really related to the question. If you know $p$-adic L-functions then there are certainly more interesting questions you can answer to.. – reuns Apr 13 '21 at 16:24
  • @reuns ... just started on math.stackexchange.com : I do hope so! Thanks. – Nate Busa Apr 23 '21 at 03:40