"Jack" <jj@[EMAIL PROTECTED]
> wrote in message
news:Vqnhk.100$WT.80@[EMAIL PROTECTED]
>
> "Brian M. Scott" <b.scott@[EMAIL PROTECTED]
> wrote in message
> news:si8sg57enf2l.1wpz77twjgg32.dlg@[EMAIL PROTECTED]
>> On Tue, 22 Jul 2008 01:37:36 +0100, Jack <jj@[EMAIL PROTECTED]
>
>> wrote in <news:hZ9hk.17700$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.
I concede I should have said, instead of "the pro****tion of n in
[1,y] whose factors are exclusively q and 1", something more like "the
pro****tion of n in [1,y] for which the number of p in P such that p|n is
one...".
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