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Question: D.H.J. POLYMATH wrote in the paper Deterministic Methods To Find Primes the following statement:

"$\ldots$ the key observation is that the parity of the prime counting function $\pi(x)$ is closely connected to the divisor sum function, $\sum_{n \leq x}\tau(n)$."

I am asking for help in briefly describing the close connection. I was not able to follow the paper well enough to pick it up. Also am I correct in assuming parity in this context is referring to evenness and oddness of the number $\pi(x)$ ?

Anthony
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  • It is explained p.6. – reuns Apr 07 '21 at 15:21
  • @reuns yes I want more. I cannot exactly follow what was written on p.6 whence me asking for help understanding the statement. For some reason I was expecting "punch-line" to describe the connection. – Anthony Apr 07 '21 at 15:31
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    It is explained p.6 $$2\pi(x)\equiv 1+\sum_{n\le x}\mu(n)\tau(n) \bmod 4$$ Related instead of closely related is probably more accurate. – reuns Apr 07 '21 at 15:38
  • @reuns could you elaborate? Suppose $\pi(x)\equiv 1 \pmod 2$. What does that tell us about $\sum_{n\leq x}\tau(n).$ – Anthony Apr 07 '21 at 15:44
  • No, everything is explained in the paper. – reuns Apr 07 '21 at 15:45
  • @reuns I read the paper again and I just do not see the connection the author is suggesting. – Anthony Apr 07 '21 at 21:18
  • @reuns You first comment is rude. How many times does this asker need to repeat "I just don't see the connection"? Before you get it? Please try to demonstrate more patience, maybe? – amWhy Apr 08 '21 at 18:19
  • @amWhy I was patient enough to look at the text, decipher small excerpts and tell the OP that everything is explained p.6, starting with the obvious relation $2\pi(x)\equiv -1+\sum_{n\le x}\mu(n)\tau(n) \bmod 4$ (sorry for the sign) – reuns Apr 08 '21 at 18:30
  • :-) ${}{}{}{}{}$ – amWhy Apr 08 '21 at 18:39

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