mersenneforum.org Gaps between non-consecutive primes
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 2022-08-01, 18:44 #56 mart_r     Dec 2008 you know...around... 11000010112 Posts Error terms, prime number scarcities, and trains Just an intermediate result that made me go "hmmmm...". Suppose we assume $$CSG=1+O(1)$$ for the gaps between non-consecutive primes, then, if I did the math right, this would imply that we also assume $$\pi(x)=Li(x)+O(\sqrt{x})$$, i.e. the error term is smaller by a factor log x compared to the RH prediction. Correct [y/n]? Code: Outline from my train of thought: p_1 = 2 (or set p_0 = 0, say) p_k = x k = pi(x)-1 ~ pi(x) gap = x-2 ~ x m = Gram(x)-Gram(2)-k+1 ~ Gram(x)-pi(x) CSG = m*|m|/gap - but for simplicity suppose that m is positive (means we assume a scarcity instead of an abundance of primes; the error term works both ways anyway): CSG = m^2/gap ~ (Gram(x)-pi(x))^2/x CSG ~ 1 --> (Gram(x)-pi(x))^2 ~ x --> Gram(x)-pi(x) ~ sqrt(x) OTOH, if Gram(x)-pi(x) = O(sqrt(x)*log(x)), then CSG = m^2/gap ~ O((x*log²x)/x) ~ O(log²x)
 2022-08-07, 19:05 #57 Bobby Jacobs     May 2018 26610 Posts That is correct. By the way, you should submit the sequences in this thread to OEIS.

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