I was wondering whether this result is in the bibliography and how one might go about proving it: $$ \lim_{x\to \infty} \frac{\log x }{x} \sum_{p \leq x \atop p\text{ prime }}\mu(p-1)=0.$$ The sum is taken over all primes $p\leq x $.
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3$\begingroup$ I have a sinking feeling that this might be twin-prime-conjecture hard. $\endgroup$– Greg MartinCommented Aug 23, 2019 at 22:57
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$\begingroup$ If I'm not mistaken, this may imply that if there are two non negative integers $a$ et $b$ such that the number of negative values of the summand is $\sim ax$ and the number of positive values thereof is $\sim bx$, then $a=b$. This may be related to RH. $\endgroup$– Sylvain JULIENCommented Aug 24, 2019 at 8:28
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$\begingroup$ The same conclusion holds replacing mutatis mutandis $x$ by $\pi(x)$ in the considered asymptotics. $\endgroup$– Sylvain JULIENCommented Aug 24, 2019 at 9:27
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2$\begingroup$ I recommend to search for "multiplicative functions on shifted primes" I found a paper by Hildebrand, Additive and Multiplicative Functions on Shifted Primes, Proc London Math. Soc. 1989, doi.org/10.1112/plms/s3-59.2.209 where it is mentioned that this type of problem with the Liouville $\lambda$ function is open, but might follow from a Halasz Thm on shifted primes. Theorems 1 and 3 go a step in the direction you want. $\endgroup$– Christian ElsholtzCommented Aug 26, 2019 at 9:38
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