Chebyshev function

The Chebyshev function $\psi$ is defined by

$\psi (x)=\sum _{n\leq x}\Lambda (n)$,

where $\Lambda$ is the von Mangoldt function. The Chebyshev function is often used in proofs related to prime numbers, because it is typically simpler to work with than the prime counting function, $\pi (x)$.

The exact formula

Based on a conjecture by Riemann, von Mangoldt proved an explicit for $\psi (x)$ whose main term is a sum over the nontrivial zeros of the Riemann zeta function:

$\psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-{\frac {\zeta '(0)}{\zeta (0)}}-{\frac {1}{2}}\log(1-x^{-2})$

Here $\rho$ runs over the nontrivial zeros of the zeta function, and

$\psi _{0}(x)={\begin{cases}\psi (x)-{\frac {1}{2}}\Lambda (x)&x=p^{m}{\mbox{, p prime, m an integer}}\\\psi (x)&{\mbox{otherwise.}}\end{cases}}$

From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of $-x^{\omega }/{\omega }$ over the nontrivial zeros of the zeta function, $\omega =-2,-4,-6,\cdots$, i.e.

$\sum _{k=1}^{\infty }{\frac {x^{2k}}{2k}}={\frac {1}{2}}\log(1-x^{-2})$.

Relation to the prime counting function

The Chebyshev function can be related to the prime counting function as follows. Define

$\pi _{1}(x)=\sum _{n\leq x}{\frac {\Lambda (n)}{\log n}}$

Then

$\pi _{1}(x)=\sum _{n\leq x}\Lambda (n)\int _{n}^{x}{\frac {dt}{t\log ^{2}t}}+{\frac {1}{\log x}}\sum _{n\leq x}\Lambda (n)=\int _{2}^{x}{\frac {\psi (t)dt}{t\log ^{2}t}}+{\frac {\psi (x)}{\log x}}$.

The transition from $\pi _{1}$ to the prime counting function, $\pi$, is made through the equation

$\pi _{1}(x)=\pi (x)+{\frac {1}{2}}\pi (x^{1/2})+{\frac {1}{3}}\pi (x^{1/3})+\cdots$

Certainly $\pi (x)\leq x$, so for the sake of approximation, this last relation can be recast in the form

$\pi (x)=\pi _{1}(x)+O({\sqrt {x}})$.

The Riemann hypothesis

The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part 1/2. In this case, $|x^{\rho }|={\sqrt {x}}$. Then

$|\sum _{\rho }{\frac {x^{\rho }}{\rho }}|\leq \left(\sum _{\gamma }{\frac {1}{\gamma }}\right){\sqrt {x}}=O({\sqrt {x}}\log ^{2}x)$,

where $\gamma$ runs over the imaginary parts of the nontrivial roots of the zeta function. Then by the above, it can readily be shown that

$\pi (x)=\operatorname {li} (x)+O({\sqrt {x}}\log x).$