====== The Prime Number Theorem with the Error Bound ======
**Yasemin Cara (Cornell University / LATP) - Jeudi 25 février - Frumam.**\\

In this talk, we will give a proof of the Prime Number Theorem with the Error Bound. Let $\pi(x)$ be the function which gives the number of primes less than or equal to $x$ and\\

$Li(x) = \int_{2}^x \frac{dt}{ln(t)}$\\

be the logarithmic integral function. Let $\epsilon > 0$ be given and let $1/2\leq\alpha<1$, where $\alpha$ is fixed. Then\\

$\pi(x) - Li(x) = O(x^{\alpha + \epsilon})$ as $x \rightarrow \infty$\\

if and only if the Riemann zeta function $\zeta(s)$ has no zeroes in the strip $\alpha<Re(s)<1$. The case $\alpha = 1/2$ corresponds to the Riemann hypothesis.