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Re: Re: Question on the Limiting Value of Ratios of Consecuative Primes...

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  • Subject: [mg88587] Re: [mg88555] Re: Question on the Limiting Value of Ratios of Consecuative Primes...
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 9 May 2008 03:24:28 -0400 (EDT)
  • References: <fvpcpn$mge$1@smc.vnet.net> <200805080814.EAA14338@smc.vnet.net>

On 8 May 2008, at 17:14, David W.Cantrell wrote:

> "Richard Palmer" <rhpalmer at gmail.com> wrote:
>> Is there some analytic limit to the ratio of consecuative primes?
>
> Yes. The limit is 1.
>
> Since there are infinitely many twin primes, it's obvious that, if the
> limit exists, it must be 1.
>
> I don't know how to show nicely that the limit exists.

This result follows from the following one proved in H.L.  Montgomery  
"Topics in Multiplicative Number Theory" (Springer 1971):

For any epsilon >0 and x > x0(epsilon), there is a prime in the  
interval [x,x+x^(3/5 + epsilon)].

This means that for arbirarily small epsilon and a sufficiently large  
Prime[n], the ratio Prime[n+1]/Prime[n] is less than (Prime[n] 
+Prime[n]^(3/5+epsilon))/Prime[n] = 1 + Prime[n]^(epsilon-2/5), which  
can be made arbitrarily close to 1 (for epsilon <2/5).

Actually, there has been a lot of interest in the long term behaviour  
of the difference of consecutive primes. Montgomery mentions the  
following which he attributes to Cramer:

Limit[(Prime[n+1]-Prime[n])/Log[Prime[n]]^2,n->Infinity] == 1

(this is not meant to be a Mathematica formula, since Mathematica does  
not have the notion of the limit of a sequence but only a mathematical  
statement written in the Mathematica notation)

but I am not sure if that is a theorem or only a conjecture.

Andrzej Kozlowski


>
>
>> The expression  Limit[Prime[i]/Prime[i+1],{i,->Infinity}] returns
>> unevaluated.
>
> I suspect that is just as well. My understanding is that Mathematica  
> is not
> designed to deal with limits of sequences and that, had you gotten an
> answer, it should not have been trusted.
>
>> Plotting Table[ Prime[i]/Prime[i+1],{i,1,20000}] shows a lot
>> of structure with a minimum of 3/5.
>
> I suppose I see the "structure" to which you refer: various strings of
> points which could be visualized as lying on smooth curves. The points
> which form the uppermost string correspond to the ratios of the twin
> primes. The points which form the next string down correspond to the  
> ratios
> of primes having a difference of 4. And then the points which form  
> the next
> string down correspond to the ratios of primes having a difference  
> of 6.
> Etc.
>
> David W. Cantrell
>



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