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<h1 align=center> Analytic number theory
</h1>
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The motivation for studying the zeros of the zeta-function
is the precise relationship between the zeros of $\zeta(s)$ and
the errot term in the \begin{rawhtml}<a href="../26a">prime number theorem</a>\end{rawhtml}.  
Define the Von Mangoldt function by
$$
\Lambda(n)=\begin{cases}  \log p &\hbox{ if $n=p^k$ }\cr
                       0 & otherwise, \cr
              \end{cases}
$$
where $p$ is a prime number.  Then
\begin{eqnarray*}
\psi(X):=&& \sum_{n\le X} \Lambda(n) \cr
=&& Li(X) + O(X^{\sigma_0+\varepsilon})
\end{eqnarray*}
if and only if $\zeta(s)$ does not vanish for $\sigma>\sigma_0$.
There $Li(X)$ is the logarithmic integral
$$
Li(X)=\int_2^X \frac{dt}{\log t} .
$$
In particular, the best possible error term in the prime number
theorem is $O(X^{\frac12 + \varepsilon})$, which is equivalent
to the Riemann Hypothesis.

Similarly, the \begin{rawhtml}<a href="../17a">Generalized Riemann Hypothesis</a>\end{rawhtml} is 
equivalent to the best possible error term for the counting
function of primes in arithmetic progressions.

Riemann Hypotheses for other $L$-functions have not yet been shown
to be strongly connected to the distribution of the prime numbers.

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<p> <br>
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<p> <br>
Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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