The Chebyshev function

Idea: when Gauss was a boy (by the dates found on his notes he was approximately 16) he noticed that the primes appear with density $\frac{1}{\log x}$ around $x$ . Then, instead of counting primes and looking at the function $π (x)$ , lets weight by the natural density and look at $\sum_{p \leq x} \log p$ . Since we are weighting by what we think is the density, we expect it to be asymptotic to be $x$ . This is the first Chebyshev function.

The second Chebyshev function, denoted by $ψ (x)$ , is defined in terms of the von Mangoldt function $Λ (n)$ :

ψ (x) = \sum_{n \leq x} Λ (n),

where

Λ (n) = {\begin{cases} \log p & if n = p^{k} for some prime p and k \geq 1, \\ 0 & otherwise . \end{cases}

This means that $ψ (x)$ includes all prime powers $p^{k} \leq x$ , not just primes, but weights each one by $\log p$ .

Why $ψ (x)$ Counts Prime Numbers

Although $ψ (x)$ includes powers of primes ( $p^{2}, p^{3}, \dots$ ), those contribute very little compared to the primes themselves.

Most contributions to the sum come from $k = 1$ , i.e., the actual primes.
Higher powers $p^{k}$ become rare as $k$ increases.
Every prime $p \leq x$ contributes $\log p$ to the sum via $n = p$ , and possibly a smaller number of contributions via $p^{k} \leq x$ .

So, $ψ (x)$ behaves like a smoothed or weighted version of the prime counting function $π (x)$ . While $π (x)$ simply counts 1 for each prime $\leq x$ , $ψ (x)$ counts $\log p$ for each prime and its powers $\leq x$ .

It turns out this weighting is exactly what fits analytically with the Riemann zeta function, and is therefore more natural in complex analysis.

In fact, the prime number theorem is often first proved using $ψ (x)$ , because its analytic representation is easier to handle.

Prime number theorem via $ψ (x)$

The Prime Number Theorem in this formulation states:

ψ (x) \sim x as x \to \infty .

That is, the total weight of primes and their powers up to $x$ behaves like $x$ . Since powers contribute little, this tells us that the primes themselves are distributed with density $\sim 1 / \log x$ .

Connection to the Riemann Zeta Function

The Chebyshev function $ψ (x)$ is directly connected to the logarithmic derivative of the Riemann zeta function:

- \frac{ζ^{'} (s)}{ζ (s)} = \sum_{n = 1}^{\infty} \frac{Λ (n)}{n^{s}}, for Re (s) > 1.

This identity allows us to write $ψ (x)$ as a complex integral via Perron's formula:

ψ (x) = \frac{1}{2 π i} \int_{c - i \infty}^{c + i \infty} (- \frac{ζ^{'} (s)}{ζ (s)}) \cdot \frac{x^{s}}{s} d s, c > 1.

This representation is central in explicit formulas that express $ψ (x)$ in terms of the zeros of the Riemann zeta function, making it a key analytic bridge between prime numbers and complex analysis.

The Chebyshev function

Why ψ(x) Counts Prime Numbers

Prime number theorem via ψ(x)

Connection to the Riemann Zeta Function

Why $ψ (x)$ Counts Prime Numbers

Prime number theorem via $ψ (x)$