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 $\pi (x)$, lets weight by the natural density and look at $\sum _{p\le x}\log p$. 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 $\psi (x)$, is defined in terms of the von Mangoldt function $\mathrm{\Lambda }(n)$:

where

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

Why $\psi (x)$ Counts Prime Numbers

Although $\psi (x)$ 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 $p\le x$ contributes $\log p$ to the sum via $n=p$, and possibly a smaller number of contributions via $p^k\le x$.

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

It turns out that this function is more natural in complex analysis than the prime counting function:

The prime number theorem in this formulation states:

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 $\sim 1/\log x$.

And the exact expression for $\psi (x)$ involves the zeroes of the Riemann zeta function. See prime counting function#Dream of a Formula for the details.