[seqfan] Re: Incorrect bound for the prime counting function?

[Don Reble]

> https://arxiv.org/abs/1409.1780 states that: [version 7, bottom of page 5]
>
> If x >= 5.43, then PrimePi(x) < x / (log(x) - 1 - (1.17/log(x))).
>
> Wolfram Alpha says: primepi(4096000) = 289511
> Axler's bound gives: 289497.0
>
> Can anyone confirm my observation...?

    Yes. Solving for the "1.17" in that equation, one gets
	log(x) * (log(x) - 1 - x / PrimePi(x))
    which reaches a local maximum of 1.23103 at x=3445943, PrimePi=246651.
    (The primes are slightly denser than expected from 3440807-3445943.)
    That's not exceeded for x < 2,000,000,000.
    Maybe the formula would work, with 1.23104 instead of 1.17.

-- 
Don Reble  djr at nk.ca