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

[Alexei Kourbatov]

Actually Axler's article in INTEGERS has a one-line corrigendum:
In Corollary 3.4 on page 8, replace "If x > 5.43" by "If x > 2 634 800
823".

The corrigendum can be retrieved from
http://math.colgate.edu/~integers/vol16.html

This also will necessitate a corrigendum to my article (JIS 18, 2015,
Article 15.11.2)
but only in a couple of proof details; theorem statements in JIS 15.11.2
are OK as stated.











On Sun, Mar 10, 2019 at 4:30 PM Don Reble via SeqFan <seqfan at list.seqfan.eu>
wrote:

>
> > 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
>
>
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