 mersenneforum.org Gaps between non-consecutive primes
 Register FAQ Search Today's Posts Mark Forums Read  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.   Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post mart_r Prime Gap Searches 14 2020-06-30 12:42 enzocreti enzocreti 0 2019-03-28 13:45 a1call Information & Answers 8 2017-02-06 17:30 axn Lounge 21 2016-06-05 13:00 gd_barnes Riesel Prime Search 1 2007-07-30 23:26

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