Riemann Hypothesis

The Riemann Hypothesis is a famous conjecture in analytic number theory that states that all nontrivial zeros of the Riemann zeta function have real part $1/2$. From the functional equation for the zeta function, it is easy to see that $\zeta(s)=0$ when $s=-2,-4,-6,\ldots$. These are called the trivial zeros. This hypothesis is one of the seven millenium questions.

The Riemann Hypothesis is an important problem in the study of prime numbers. Let $\pi(x)$ denote the number of primes less than or equal to x, and let $\mathrm{Li}(x)=\int_2^x \frac{1}{\ln t}\; dt$. Then an equivalent statement of the Riemann hypothesis is that $\pi(x)=\mathrm{Li}(x)+O(x^{1/2}\ln(x))$.

One fairly obvious method to prove the Riemann Hypothesis is to consider the reciprocal of the zeta function, $\frac{1}{\zeta(s)}{=}\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$, where $\mu(n)$ refers to the Möbius function. Then one might try to show that $\frac{1}{\zeta(s)}$ admits an analytic continuation to $\Re(s)>\frac{1}{2}$. Let $M(n)=\sum_{i=1}^n \mu(i)$ be the Mertens function. It is easy to show that if $M(n)\le\sqrt{n}$ for sufficiently large $n$, then the Riemann Hypothesis would hold. The Riemann Hypothesis would also follow if $M(n)\le C\sqrt{n}$ for any constant $C$.

Some equivalent statements of the Riemann Hypothesis are

  • The zeta function has no zeros with real part between $\frac{1}{2}$ and 1
  • $\zeta_a(s)$ has all nontrivial zeros on the line $Re(s)=\frac{1}{2}$
  • All nontrivial zeros of all L-series have real part one half where an L-series is of the form $\sum_{n=1}^\infty \frac{a_n}{n^s}$. This is the generalized Riemann Hypothesis because in the Riemann Hypothesis, $a_n$ is 1 for all n
  • $|M(x)|\le cx^{1/2+\epsilon}$ for a constant c and where $M(x)=\sum_{n\le x}\mu(n)$
  • $\sigma(n) < e^\gamma n\ log_{log_n}$ for all $n > 5040$. This being equivalent to the Riemann Hypothesis is Robin's Theorem, proved in 1984 by Guy Robin.


Purported Disproof of the Mertens Conjecture