Chebyshev Functions

Chebyshev's Theta-function

Definition

If $x>0$ we define Chebyshev's $vartheta$ -function by the equation

\vartheta(x)=\sum_{p \le x} \log p

where $p$ runs over all primes $\le$ x

Similary we have another function that is related to the theta function.

Chebyshev's Psi-function

Definition

For $x>0$ we define Chebyshev's $\psi$ -function by the formula

\psi(x)=\sum_{n \le x} \Lambda(n)

where $\Lambda$ is the Mangoldt function.

Now with our definitions we can combine these two functions as follows,

since $\Lambda(n) = 0$ unless $n$ is a prime power we can write the definition of $\psi(x)$ as,

\psi(x) = \sum_{n \le x} \Lambda(n) = \sum_{m=1}^{\infty} \sum_{p \le x^{1/m}} \log p

Now you might notice that the sum on $m$ is in fact a finite sum. Surely enough, the sum on $p$ is empty if $x^{1/m} < 2$ , that is, if $(1/m) \log x < \log 2$ , or if,

m > \frac{\log x}{\log 2} = \log_{2}(x).

Therefore we get,

\psi(x) = \sum_{m \le \log_2(x)}\sum_{p \le x^{1/m}} \log p

Now we can use the theta function to finish it off and get,

\psi(x) = \sum_{m \le \log_2(x)}\sum_{p \le x^{1/m}} \log p = \sum_{m \le \log_2 x} \vartheta(x^{1/m})

Mangoldt Function Limits