- In a message dated 20/05/01 02:23:59 GMT Daylight Time, Punkish301 writes:

<< In a message dated 20/05/01 01:59:28 GMT Daylight Time,

tech_newsletters@... writes:

Dear Michael,

<<

Do we know what the value of a is?

Since Riemann's hypothesis is generally believe to be true, should

one use that value of pi(x)?

Li(x) <=> integral from 0 to x of x/log x, right?

What does "O(z)" mean as it is used in the statements of pi(x) above?

How would one use the PNT or its improved version to determine the

density of Primes of particular forms over arbitrary ranges? (fixed

k, fixed n, or variable k,n)>>

Considering you've been looking for primes for less than a month, you've

done well understanding a bit of PNT, which I havent really got to grips with

after five years...lol.

Couple of things, the integral Li(x) is 1/log t dt between 0 and x, rather

than x/log x, and I'm not really sure if mathematicians would be so

interested in the value of the constant A in the error term of PNT - and what

exactly would we learn if we did? We know the error term and that is what

counts (do people on the board agree here, or not? I might not have defined

the integral too well...sorry)

O(z) - I think O(x*exp(-A*sqrt(log(x))) means "of the order of magnitude of

this formula" ie the average of Li(x) minus pi(x), it's meant to be taken as

x tends to infinity, not for a particular value of x - I think (will anyone

like to correct me if I'm wrong).

I don't think mathematicians like to rely too heavily on Riemann, since it's

not known conclusively - nothing concrete yet, despite lots of evidence by

various people that it *might* be true, there still might exist a complex

zero that doesn't fit the conjecture which would spoil the whole thing! - and

I don't understand what you mean by "should one use that value of pi(x)" can

you explain this a bit?

I'll leave the rest to Yves...he knows more about the other formulae than I

do,

Thanks, from Guy >>

[Non-text portions of this message have been removed] - --- In primenumbers@y..., Punkish301@a... wrote:
> In a message dated 20/05/01 02:23:59 GMT Daylight Time, Punkish301

writes:

>

I can't saw I truely understand it, but I am trying since I am a

> << In a message dated 20/05/01 01:59:28 GMT Daylight Time,

> tech_newsletters@y... writes:

> Dear Michael,

> <<

> Do we know what the value of a is?

> Since Riemann's hypothesis is generally believe to be true,

> should one use that value of pi(x)?

> Li(x) <=> integral from 0 to x of x/log x, right?

> What does "O(z)" mean as it is used in the statements of pi(x)

> above? How would one use the PNT or its improved version to

> determine the density of Primes of particular forms over

> arbitrary ranges? (fixed k, fixed n, or variable k,n)

> >>

> Considering you've been looking for primes for less than a month,

> you've done well understanding a bit of PNT, which I havent really

> got to grips with after five years...lol.

prospective math major :)

> O(z) - I think O(x*exp(-A*sqrt(log(x))) means "of the order of

Ah, So the O( ... ) is for Order of magnitude like in complexity

> magnitude of this formula" ie the average of Li(x) minus pi(x),

> it's meant to be taken as x tends to infinity, not for a

> particular value of x - I think (will anyone like to correct me if

> I'm wrong).

analysis, which programmers among other use.

> I don't think mathematicians like to rely too heavily on Riemann,

Of course, but I had gotten the impression that the Riemann's

> since it's not known conclusively - nothing concrete yet, despite

> lots of evidence by various people that it *might* be true, there

> still might exist a complex zero that doesn't fit the conjecture

> which would spoil the whole thing!

hypothesis is generally believed to be true, by the same token as

Fermat's Last Theorem was believed to be true in the early 1990's.

Never the less, I concur that we should not 'count' on it to be true.

Would it be wise to use it in a probability calculation?

> I don't understand what you mean by "should one use that value of

Yes, I was unclear. I meant should one use pi(x) = Li(x) + O(x^(1/2)

> pi (x)" can you explain this a bit?

log x) to derive a density function for primes. Since the O(x) is

just the error term, my question is probably mute, unless one would

need to use the error term when making a particularly accurate

statement of the probability of finding prime(s) in a range.

Additionally I am assuming that the formula Gallot references for

determining the density of prime(s) is derived in some way from the

Prime Number Theorem (PNT). Is this a valid deduction?