The Chebyshev functions

The first Chebyshev function: when Gauss was a boy (by the dates found on his notes he was approximately 16) he noticed that the primes appear with density $\delta (x)=\frac{1}{\log x}$ 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 $\theta (x)$.
More in detail, let's analyze the derivative $\frac{d}{dx}\theta (x)$. When we go from $x$ to $x+1$ the increment in $\theta (x)$ is sometimes $\log (x)$ and sometimes $0$. In average, the increment is given by $\log (x)\cdot \delta (x)$, but this is precisely 1. So $\theta (x)\approx x$.

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 $\log p$.

Analyzing the expected increment when moving from $x$ to $x+1$, the function $\psi (x)$ increases by $\log p$ whenever $x$ hits a prime power $p^k$.Primes ($k=1$): Occur with density ${\delta }_1(x)=\frac{1}{\log x}$ and contribute weight $\log x$.$$\text{Expected increment} = \log x \cdot \frac{1}{\log x} = 1$$Squares ($k=2$): A prime square $p^2\approx x$ means $p\approx \sqrt{x}$. The density of prime squares at $x$ is:$$\delta_2(x) = \frac{d}{dx}\pi(\sqrt{x}) \approx \frac{1}{2\sqrt{x}\log\sqrt{x}} = \frac{1}{\sqrt{x}\log x}$$Each prime square adds weight $\log p=\log \sqrt{x}=\frac{1}{2}\log x$.$$\text{Expected increment} = \left(\frac{1}{2}\log x\right) \cdot \left(\frac{1}{\sqrt{x}\log x}\right) = \frac{1}{2\sqrt{x}}$$General $k$-th powers: A power $p^k\approx x$ occurs with density ${\delta }_k(x)\approx \frac{1}{k{x}^{1-1/k}\log (x^{1/k})}=\frac{1}{x^{1-1/k}\log x}$ and adds weight $\frac{1}{k}\log x$.$$\text{Expected increment} = \left(\frac{1}{k}\log x\right) \cdot \left(\frac{1}{x^{1 - 1/k}\log x}\right) = \frac{1}{k x^{1 - 1/k}}$$Summing the expected increments from all powers $k$:$$\frac{d}{dx}\psi(x) \approx 1 + \frac{1}{2\sqrt{x}} + \frac{1}{3 x^{2/3}} + \dots$$Integrating this expected rate of change from $1$ to $x$:$$\psi(x) \approx \int_1^x \left(1 + \frac{1}{2\sqrt{t}} + O\left(t^{-2/3}\right)\right) dt \approx x + \sqrt{x} + O(x^{1/3}) \approx x$$

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 $\le x$.

Why to use it?

It turns out that this function is more natural in complex analysis than the prime counting function or the first Chebyshev function above: see Riemann hypothesis, for the details.