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Old 2022-08-01, 18:44   #56
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Dec 2008
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Default 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]?

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)
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