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# 7.2: Chebyshev's Functions

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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

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

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

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

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

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

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

1. Show that $\psi(x)=\theta(x)+\theta(x^{1/2})+ \theta(x^{1/3})+...\theta(x^{1/m})$ where $$m\leq log_2x$$.

2. 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.

3. 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.