"Brian M. Scott" <b.scott@[EMAIL PROTECTED]
> wrote in message
news:si8sg57enf2l.1wpz77twjgg32.dlg@[EMAIL PROTECTED]
> On Tue, 22 Jul 2008 01:37:36 +0100, Jack <jj@[EMAIL PROTECTED]
>
> wrote in <news:hZ9hk.17700$A42.13504@[EMAIL PROTECTED]
> in
> alt.algebra.help:
>
> [...]
>
>>> So for y in N we let P(y) be the set of primes not exceeding
>>> sqrt(y) and m(y) = prod(P(y)); p(y) is the number of primes
>>> not exceeding y, i.e., the number usually denoted by pi(y).
>>> You say that you want to be sure that
>
>>> lim_{y --> oo}{[p(y)/y]/[y * phi(m(y))/m(y)]} = 1,
>
>>> where phi is the Euler totient function.
>
>>> This appears not to be true. Heuristically speaking, the
>>> denominator y * phi(m(y))/m(y) ought to be roughly equal to
>>> the number of integers in [1, y] that are not divisible by
>>> any member of P(y). The integers in [1, y] that are not
>>> divisible by any member of P(y) are precisely the primes in
>>> the interval (sqrt(y), y], of which there are
>>> p(y) - p(sqrt(y)). Thus, the ratio
>
>>> [p(y)/y]/[y * phi(m(y))/m(y)]
>
>>> ought to be about
>
>>> [p(y)/y]/[p(y) - p(sqrt(y))]
>
>>> for large y. Consider the reciprocal, which on your view
>>> should also approach 1:
>
>>> [p(y) - p(sqrt(y))]/[p(y)/y] =
>>> y * [1 - p(sqrt(y))/p(y)].
>
>>> I noted before that p(sqrt(y))/p(y) is about 2/sqrt(y) for
>>> large y, so y * [1 - p(sqrt(y))/p(y)] is about
>
>>> y * [1 - 2/sqrt(y)] = y * [sqrt(y) - 2]/sqrt(y) =
>>> sqrt(y) * [sqrt(y) - 2],
>
>>> which is on the rough order of y, not 1.
>
>> As you know I find the equations difficult to follow. But
>> I do not see in them any mention of ln(n).
>
> So what? In any case, it's there, in the statement 'I noted
> before that p(sqrt(y))/p(y) is about 2/sqrt(y) for large y'.
>
>> I should say also that the number theorist I had worked
>> with was so adamant that the claim holds that I wonder
>> whether we are speaking at cross purposes.
>
> I'm simply responding to what you tell me. Unfortunately,
> that changes from post to post. As a result, at this point
> I have no idea what the number theorist actually told you.
>
>> As y increases, each new prime, p, of value <sqrt(y) is,
>> on average, further apart in absolute terms than the
>> previous (i.e. the prime that is adjacent but of lower
>> value than p), but closer in pro****tionate terms to the
>> previous; and the value of 1/p, p in P gets smaller and
>> smaller; furthermore, for any p,q in P, where q < p and
>> p and q are adjacent primes, the pro****tion of n in
>> [1,y] whose factors are exclusively q and 1 gets ever
>> closer to the pro****tion of n whose factors are exclusive
>> p and 1.
>
> Bollocks. First, the only positive integer whose only
> factors are 1 and q is q; I assume that you mean numbers
> whose only prime factor is q. Those are simply the powers
> q^n for n > 0.
Bollocks. Take y=19^2. Now consider 3*23. 23<19^2 and 3*23 has only one
factor in P.
Cheers.
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