7.2: Chebyshev’s Functions

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$Wissam_Raji.jpg$
Contributed by Wissam Raji
Associate Professor and the Chairman (Math) at American University of Beirut

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We introduce some number theoretic functions which play important role in the distribution of primes. We also prove analytic results related to those functions. We start by defining the Van-Mangolt function

$\Omega(n)=logp$ if $n=p^m$ and vanishes otherwise.

We define also the following functions, the last two functions are called Chebyshev’s functions.

$\pi(x)=\sum_{p\leq x}1.$
$\theta(x)=\sum_{p\leq x}log p$
$\psi(x)=\sum_{n\leq x}\Omega(n)$

Notice that \[\psi(x)=\sum_{n\leq x}\Omega(n)=\sum_{m=1, \ p^m\leq x}^{\infty}\sum_p\Omega(p^m)=\sum_{m=1}^{\infty}\sum_{p\leq x^{1/m}}log p.\]

$\pi(10)=4$.
$\theta(10)=log 2+ log 3+ log 5+log 7$.
$\psi(10)=log 2+ log 2+log 2+ log 3+ log 3+ log 5+ log 7$

It is easy to see that \[\psi(x)=\theta(x)+\theta(x^{1/2})+ \theta(x^{1/3})+...\theta(x^{1/m})\] where $m\leq log_2x$. This remark is left as an exercise.

Notice that the above sum will be a finite sum since for some $m$, we have that $x^{1/m}<2$ and thus $\theta(x^{1/m})=0$.
We use Abel’s summation formula now to express the two functions $\pi(x)$ and $\theta(x)$ in terms of integrals.

For $x\geq 2$, we have \[\theta(x)=\pi(x)\log x-\int_ {2}^{x}\frac{\pi(t)}{t}dt\] and \[\pi(x)=\frac{\theta(x)}{\log x}+\int_{2}^x\frac{\theta(t)}{t\log^2t}dt.\]

We define the characteristic function $\chi(n)$ to be $1$ if $n$ is prime and $0$ otherwise. As a result, we can see from the definition of $\pi(x)$ and $\theta(x)$ that they can be represented in terms of the characteristic function $\chi(n)$. This representation will enable use to apply Abel’s summation formula where $f(n)=\chi(n)$ for $\theta(x)$ and where $f(n)=\chi(n) \log n$ for $\pi(x)$. So we have, \[\pi(x)=\sum_{1\leq n/leq x}\chi(n) \ \ \ \ \mbox{and} \ \ \ \theta(x)=\sum_{1\leq n\leq x}\chi(n)\log n\] Now let $g(x)=\log x$ in Theorem 84 with $y=1$ and we get the desired result for the integral representation of $\theta(x)$. Similarly we let $g(x)=1/\log x$ with $y=3/2$ and we obtain the desired result for $\pi(x)$ since $\theta(t)=0$ for $t<2$.

We now prove a theorem that relates the two Chebyshev’s functions $\theta(x)$ and $\psi(x)$. The following theorem states that if the limit of one of the two functions $\theta(x)/x$ or $\psi(x)/x$ exists then the limit of the other exists as well and the two limits are equal.

For $x>0$, we have \[0 \leq \frac{\psi(x)}{x}-\frac{\theta(x)}{x}\leq \frac{(\log x)^2}{2\sqrt{x}\log 2}.\]

From Remark 4, it is easy to see that \[0\leq \psi(x)-\theta(x)=\theta(x^{1/2})+ \theta(x^{1/3})+...\theta(x^{1/m})\] where $m\leq log_2x$. Moreover, we have that $\theta(x)\leq x\log x$. The result will follow after proving the inequality in Exercise 2.

Exercises

Show that \[\psi(x)=\theta(x)+\theta(x^{1/2})+ \theta(x^{1/3})+...\theta(x^{1/m})\] where $m\leq log_2x$.
Show that $0\leq \psi(x)-\theta(x)\leq (\log_2(x))\sqrt{x}\log\sqrt{x}$ and thus the result of Theorem 86 follows.
Show that the following two relations are equivalent \[\pi(x)=\frac{x}{\log x}+O\left(\frac{x}{\log^2x}\right)\] \[\theta(x)=x+O\left(\frac{x}{\log x}\right)\]

Contributors

Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution (CC BY) license.