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32 votes
2 answers
3k views

Does the equation $1 + 2 + 3 + \dots = -\frac{1}{12}$ have a natural $p$-adic interpretation?

Consider the equation $$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$ "proved" by Ramanujan Euler. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = 1}^{\...
Frank Thorne's user avatar
  • 7,347
4 votes
0 answers
289 views

Is there a conjecture about the bounds (constant or a function) of $\sum_{n \le x} \mu(n)/\sqrt{n}$

Here $\mu(n)$ is Möbius function and $M(x)$ is Mertens function. The computations show that the partial sums $\sum_{n \le x} \frac{\mu(n)}{\sqrt{n}}$ stays between $-0.2$ and $-1.2$ when $10^1<x<...
Shree's user avatar
  • 203
3 votes
1 answer
578 views

What is the (fractional) half-derivative of zeta at $s=0$ (and how to compute it)?

(I asked this in MSE before but there was only a general reference which did not help for my specific question) I think I understood the concept of fractional derivatives applied to ...
Gottfried Helms's user avatar
2 votes
0 answers
238 views

Possible regularisation for sum of function of primes

Consider the following sum of function of primes: $$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$ Here $p$ runs through all primes and $e$ is Euler's constant. We can see that the sum ...
Zaza's user avatar
  • 149
0 votes
1 answer
259 views

Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?

[This question is copied from math.stackexchange, it didn't get answers so far] For some exercises with (divergent) summation of the Stieltjes constants,see also MSE I'm trying a formula, which ...
Gottfried Helms's user avatar