1

I was interested in the prime zeta function and its values, so I calculated in Excel the sum of reciprocals of prime numbers squared up to $1$ MM, and tried to relate it to some known irrational numbers ($e$,$\pi$,$\phi$,...). I was surprised when I checked that

$$\frac{\frac{\phi}{2}+1}{4}-\sum_{p<1000000}{1\over p^2}=0.000006896328\dots$$

However, I read here that

$$N=\sum_p{1\over p^2}=\sum_{k=1}^{\infty}{\mu(k)\over k}\log(\zeta(2k))=0.4522474200\dots$$

And

$$\frac{\frac{\phi}{2}+1}{4}=0.452254\dots$$

¿Is this nice approximation just random? ¿Can someone give a reasonable explanation to this?

Juan Moreno
  • 831
  • 8
  • 15
  • 1
    It might be just by chance like [here](https://math.stackexchange.com/questions/724872/why-is-e-pi-pi-so-close-to-20/724966), or [here](https://math.stackexchange.com/questions/2710124/why-is-2-ln-662-pi-so-close-to-13?noredirect=1&lq=1). Or by a series like [here](https://math.stackexchange.com/questions/709049/why-is-sqrt331-so-close-to-pi?noredirect=1&lq=1). More comments are [here](https://mathoverflow.net/questions/53443/sum-of-the-reciprocal-of-the-primes-squared). – Dietrich Burde Feb 16 '21 at 15:51
  • @DietrichBurde I agree with you that most probably it will be by chance, but who know, maybe is related to some interesting series that someone recalls. – Juan Moreno Feb 16 '21 at 17:52
  • Seems to be a coincidence. But in fact, an astonishing approximation (+1) – Peter Feb 17 '21 at 14:32

0 Answers0