Convergence of \psi(x)/x and \pi(x)/Li(x) to 1: How Fast?

  • Thread starter lokofer
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In summary: For x-->oo, the limit:\frac{\Omega(\sqrt (x))}{\sqrt(x)}=h(x) does tend to 0, as does the integral:\int_{0}^{\infty}dxh(x) though there is a "limiting case" where they diverge.
  • #1
lokofer
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hello..following the PNT we know that

[tex] \frac{\psi(x)}{x}\rightarrow 1 [/tex] and

[tex] \frac{\pi(x)}{Li(x)}\rightarrow 1 [/tex]

my question is "how fast" do the expressions:

[tex] |\frac{\psi(x)}{x}-1|=|f(x)| [/tex] and

[tex] |\frac{\pi(x)}{Li(x)}-1|=|g(x)| [/tex] tend to 0 ?

in the sense that for example will the expressions...

[tex] f(x)x^{1/2} [/tex] and [tex] g(x)x^{1/2} [/tex] tend to 0 or will they tend to infinite?...:rolleyes: :rolleyes: (to give a clearer explanation)
 
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  • #2
You're just asking the Riemann hypothesis now.
 
  • #3
Just about any discussion of the prime number theorem will have something to say about the error term, it shouldn't be hard for you to look this up. Something like:

[tex]\psi(x)=x+O\left(x\exp\left(-C\frac{(\log x)^{3/5}}{(\log\log x)^{1/5}} \right)\right)[/tex]

for a constant C>0 is known. Much better can be had assuming RH of course, or even larger zero free regions (the above bound comes from a zero free region).

In the other direction you have:

[tex]\psi(x)=x+\Omega(\sqrt{x}) [/tex]

though slightly better is known by a factor of some logloglog(x) I think, you can look it up to check the number of log's.
 
  • #4
In that case as "Shmoe" posted it would be to see if the limit:

[tex] \frac{\Omega(\sqrt (x))}{\sqrt(x)}=h(x) [/tex]

tends to 0 for x-->oo, or if the integral [tex] \int_{0}^{\infty}dxh(x) [/tex] is finite.

As far as i know i have seen "graphs" of [tex] \psi(x)-x [/tex] and seems (don't know if there is a well math theorem) that has the function [tex] x^{1/2} [/tex] as and "upper" and "lower" limit...( depending on what sign you take when take the square root of x ) i would be interested in knowing if the integral:

[tex] \int_{c}^{\infty}dx|x^{1/2}(\frac{d\psi}{dx}-1)|^{2} [/tex] exist so it's on an L(c,oo) space... c=oo or c=0..thanks.
 
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  • #5
lokofer said:
[tex] |\frac{\psi(x)}{x}-1|=|f(x)| [/tex] and

[tex] |\frac{\pi(x)}{Li(x)}-1|=|g(x)| [/tex] tend to 0 ?

in the sense that for example will the expressions...

[tex] f(x)x^{1/2} [/tex] and [tex] g(x)x^{1/2} [/tex] tend to 0 or will they tend to infinite?...:rolleyes: :rolleyes: (to give a clearer explanation)

The RH is equivilent to

[tex]|\operatorname{Li}(x)-\pi(x)|\le c\sqrt x \ln x[/tex] for some constant c. You're asking if

[tex]|\operatorname{Li}(x)-\pi(x)|\le c\operatorname{Li}(x)x^{-1/2}\sim c\sqrt x \ln x[/tex] for some constant c.
 
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  • #6
In fact if we define the "trace" of a certain operator (Hamiltonian ) by:

[tex] Z=Tr[e^{iuH}]=\sum_{n=-\infty}^{\infty}e^{iuE_{n}} [/tex] (1)

differentiating V. Mangodlt formula..

[tex] -\frac{d\psi}{dx}+1-\frac{1}{x^{3}-x}=\sum_{\rho}x^{\rho -1} [/tex]

If we put...[tex] \rho_{n} = 1/2+iE_{n} [/tex] ,multiplying both sides by [tex] \sqrt (x) [/tex] and letting

x=exp(u) the V.Mangoldt formula becomes just a "trace"..
 
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FAQ: Convergence of \psi(x)/x and \pi(x)/Li(x) to 1: How Fast?

What is the definition of convergence?

Convergence is the process by which a sequence or series of numbers approaches a specific value as the number of terms increases.

How is the convergence of \psi(x)/x and \pi(x)/Li(x) to 1 related to prime numbers?

The convergence of \psi(x)/x and \pi(x)/Li(x) to 1 is related to the distribution of prime numbers as these functions represent the number of primes below a given value. As the value of x increases, these functions approach 1, indicating that the number of primes approaches the value of x.

What is the significance of the convergence rate of \psi(x)/x and \pi(x)/Li(x) to 1?

The convergence rate of \psi(x)/x and \pi(x)/Li(x) to 1 is significant as it provides insight into the density and distribution of prime numbers. A faster convergence rate indicates a more evenly distributed set of primes, while a slower convergence rate may indicate clusters of primes.

How is the convergence of \psi(x)/x and \pi(x)/Li(x) to 1 calculated?

The convergence of \psi(x)/x and \pi(x)/Li(x) to 1 can be calculated using mathematical techniques such as the Euler-Maclaurin summation formula or the Cauchy convergence criterion.

What factors can affect the convergence of \psi(x)/x and \pi(x)/Li(x) to 1?

The convergence of \psi(x)/x and \pi(x)/Li(x) to 1 can be affected by the size of x, the chosen mathematical technique, and the density and distribution of prime numbers. Additionally, computational limitations and rounding errors may also impact the convergence rate.

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