Brian,
>
>>>> I want to be certain that p(y)/y tends towards phi(m)
>>>> \times y/(prod{p : p inP})).
>
>>> Your terminology is incorrect: one does not normally speak
>>> of a function f(y) tending towards another function g(y).
>>> Do you mean that you want to be certain that the ratio of
>>> the two function of y approaches 1 as y increases?
>
>> Yes.
>
>>> Next, you haven't defined your notation. What is m?
>
>> Same as the m you were using in your term phi (m).
>
>>> Is your P here the P(y) that I defined above?
>
>> The set of primes less than or equal to the square root of y.
>
> 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). 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.
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. This is
found in both the case for the phi function and also for the approximate
number of primes found as by the prime number theorem. So surely the two
values, the one found by phi(m) y/(prod{p : p inP})) and the other found
by
n/log(n) (if I am correct in thinking that is the formula usually quoted
for
obtaining pi(n) by the PNT) should converge on one another.
With thanks.
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