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The prime counting function conjecture, help to compute formula

Posted 1 month ago
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I was wondering if somebody can be kind enough to test this conjecture with a large enough, say $x>50$, using a powerful computer and Mathematica:

$\pi(x) \sim -16\sum _{h=1}^{\infty}\frac{x^{2h+1}}{2h+1}\sum _{i=1}^h \log\zeta(2i)\sum _{v=i}^{h}\frac{(-1)^{h-v}(4\pi )^{2h-2v}}{\zeta(2v-2i)(2h+2-2v)!} \text{, if }x\text{ is sufficiently large.}$

Note this formula assumes 1 is not a prime, as it should. Notice that $x=50$ will require increasing the precision and the number of terms, $M$.

$MaxExtraPrecision = 500; M = 230; x = 12; 
  N[{PrimePi[x], x1 = -16*Sum[(x^(2*h + 1)/(2*h + 1))*
              Sum[Log[Zeta[2*i]]*Sum[((-1)^(h - v)*(4*Pi)^(2*h - 2*v))/
                      (Zeta[2*v - 2*i]*(2*h + 2 - 2*v)!), {v, i, 
          h}], {i, 1, h}], 
            {h, 1, M}]}, 10]

The reasoning behind it can be found here.

Posted 1 month ago

Tips on how to improve the speed of the formula also welcome.

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