The Extended Riemann Hypothesis and Ramanujan’s Sum

Riemann Hypothesis and Ramanujan’s Sum Explanation

• RH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line.
• ERH: All zeros of L-functions to complex Dirichlet characters of finite cyclic groups within the critical strip lie on the critical line.
• Related Article: The History and Importance of the Riemann Hypothesis

The goal of this article is to provide the definitions and theorems that are necessary to understand these two Riemann hypotheses, i.e. why is a strip in the complex plane called critical, what are trivial and non-trivial zeros, and what is a group character, etc. We will gather a couple of facts around the functions involved, in particular several functional equations.

The extended Riemann hypothesis is a generalization of the Riemann hypothesis that became important when modern computer science began to use number-theoretical results for encryption mechanisms. We will elaborate on these relations to applied mathematics in another article. The main subject of this article will be the Riemann zeta function which is sort of a special case of an L-function and what we and most other people mean if we briefly say zeta-function or $\zeta$-function.

Zeta Functions

 

Definitions

\begin{align*}
\zeta(s)&:=\sum_{n=1}^\infty \dfrac{1}{n^s}=1+\dfrac{1}{2^s}+\dfrac{1}{3^s}+\dfrac{1}{4^s}+\dfrac{1}{5^s}+\cdots = \prod_{p\text{ prime}}\dfrac{1}{1-p^{-s}}\quad (s\in \mathbb{C})\\
\Gamma(z)&:=\lim_{n \to \infty}\dfrac{n!n^z}{z\cdot (z+1)\cdot \ldots\cdot (z+n)}=\int_0^{\infty }t^{z-1}e^{-t}\,dt\\[6pt]
\Gamma(0)&=1\, , \,\Gamma(1)=1\, , \,\Gamma(n)=(n-1)!\, , \,\Gamma(1/2)=\sqrt{\pi}\, , \,\Gamma(z+1)=z\Gamma(z)
\end{align*}

 

We will need the important functional equation for the zeta-function (Riemann, 1859) which we will prove next. The proof requires some preparations.

Poisson Summation Formula

Let $f\, : \,\mathbb{R}\longrightarrow \mathbb{C}$ be a continuously differentiable function with $f(x)=O(|x|^{-2})$ for $|x|\to \infty .$ Let $\hat{f}\, : \,\mathbb{R}\longrightarrow \mathbb{C}$ be the Fourier transform of $f,$ i.e.
$$
\hat{f}(t)=\int_{-\infty }^{\infty }f(x)e^{-2\pi i x t}\,dx.
$$
Then $$
\sum_{n\in \mathbb{Z}} f(n)=\sum_{n\in \mathbb{Z}}\hat{f}(n).
$$

Furthermore, we get for $\lambda >0$ and $f_\lambda (x):=f(\lambda x)$
$$
\hat{f_\lambda }(t)=\dfrac{1}{\lambda }\hat{f}\left(\dfrac{t}{\lambda }\right).
$$

 

Example

The Fourier transform of $f(x):=e^{-\pi x^2}$ is $\hat{f}(t)=\displaystyle{\int_{-\infty }^\infty e^{-\pi x^2}e^{-2\pi i x t}\,dx}=f(t).$

 

The classical Jacobian theta-function is defined as $\vartheta (z,\tau ):=\sum_{n= -\infty }^{\infty }e^{\pi i n^{2}\tau +2\pi inz}.$ The series is convergent in $\mathbb {C} \times \mathbb {H}$ where $\mathbb {H} =\{x\in \mathbb{C} \mid \mathfrak{I(\tau)}>0\}.$ We are interested in the theta-function with a fixed value $z$ such that $\vartheta(z,\cdot)$ is a holomorphic function on $\mathbb{H}.$ Let $z= 0$ and $\theta\, : \,\mathbb{R}^+\longrightarrow \mathbb{C}$
$$
\theta(x):=\sum_{n= -\infty }^{\infty }e^{-\pi n^{2}x}
$$

Functional Equation of the Theta-Series

\begin{align*}
\theta(x)&=\dfrac{1}{\sqrt{x}}\;\theta\left(\dfrac{1}{x}\right)\; \text{ for all } x>0\;\text{ and }\; \theta(x)=O(1/\sqrt{x}\,)\,\text{ for }x \searrow 0
\end{align*}

 

Supplementary Theorem of the Gamma Function, Euler 1749

$$
\Gamma(z)\Gamma(1-z) = \dfrac{\pi}{\sin \pi z}\;\text{ for all }z\in \mathbb{C}-\mathbb{Z}
$$

 

Legendre’s Doubling Formula, 1809

$$
\Gamma\left(\dfrac{z}{2}\right)\cdot\Gamma\left(\dfrac{z+1}{2}\right)=\dfrac{\sqrt{\pi}}{2^{z-1}}\cdot \Gamma(z)\;\text{ for all }z\in \mathbb{C}-\{0,-1,-2,\ldots\}
$$

 

The Legendre doubling formula is a special case of (without proof)

Gauß’s Multiplication Formula, 1812

\begin{align*}
\Gamma\left(\dfrac{z}{n}\right)\cdot\Gamma\left(\dfrac{z+1}{n}\right)\cdots\Gamma\left(\dfrac{z+n-1}{n}\right)&=\dfrac{(2\pi)^{(n-1)/2}}{n^{(z-1/2)}}\cdot \Gamma(z)\\[6pt]
\text{for }n\in \mathbb{N},z\in \mathbb{C}-\{0,-1,-2\ldots\}&
\end{align*}

Gregory Chudnovsky has shown in 1975, that every number $\Gamma (1/6)$, $\Gamma (1/4)$, $\Gamma (1/3)$, $\Gamma (2/3)$, $\Gamma (3/4)$ and $\Gamma (5/6)$ is transcendent and algebraically independent of $\pi.$ On the other hand, it is unknown whether $\Gamma(1/5)$ is even irrational.

 

$\rho(t):=\displaystyle{\sum_{n=1}^\infty e^{-\pi n^2 t}= \dfrac{1}{2}(\theta(t)-1) \stackrel{t\searrow 0}{\longrightarrow }O\left(1/\sqrt{t}\right)}.$ Thus the following integral exists, and for $s\in \mathbb{C}$ with $\mathfrak{R}(s)>1,$ we get ($t=\pi n^2 u$)
\begin{align*}
\Gamma\left(\dfrac{s}{2}\right)\zeta(s)&= \sum_{n=1}^\infty \Gamma\left(\dfrac{s}{2}\right) \dfrac{1}{n^s}
=\sum_{n=1}^\infty \dfrac{1}{n^s}\int_0^\infty t^{s/2}e^{-t}\dfrac{dt}{t}= \pi^{s/2} \int_0^\infty u^{s/2}\sum_{n=1}^\infty\left(e^{-\pi n^2 u}\right)\dfrac{du}{u}.
\end{align*}

 

Finally, we can prove the

Functional Equation of the $\zeta$-Function

Set $\xi(s):=\pi^{-s/2}\Gamma(s/2)\zeta(s).$ This at prior on the half plane $\{\mathfrak{R}(s)>1\}$ holomorphic function can be extended to a function that is meromorphic on the entire complex plane. The extended function has first order singularities at $s\in \{0,1\},$ and is holomorphic elsewhere. Moreover,
$$
\xi(s)=\xi(1-s).
$$
The $\zeta$-function can therefore be meromorphic extended on $\mathbb{C}$ with a single pole at $s=1.$ Moreover,
$$
\zeta(1-s)=2(2\pi)^{-s}\Gamma(s)\cos\left(\dfrac{\pi s}{2}\right)\zeta(s).
$$

 

Zeros and Poles of the $\zeta$-Function

We have just proven that the only pole of the $\zeta$-Function is at $x=1.$

To consider the zeros, we will distinguish between three cases.

1. $\mathfrak{R}(s)>1.$
   In this case, we conclude by the product formula that
   $$
   \zeta(s)=\prod_{p \text{ prime}}\dfrac{1}{1-p^{-s}} \neq 0
   $$
2. $\mathfrak{R}(s)<0.$
   In this case, we have $\mathfrak{R}(1-s)>1$, and the functional equation
   $$
   \zeta(s)=2(2\pi)^{s-1}\Gamma(1-s)\cos\left(\dfrac{\pi (1-s)}{2}\right)\zeta(1-s)=0
   $$
   if and only if the cosine becomes zero because $\mathfrak{R}(1-s)>1.$ However, the complex cosine has only real zeros at $(2k+1)\pi/2.$ Therefore $1-s=2k+1$ or $\zeta(s)=\zeta(-2k)=0$ if and only if $k\in \{1,2,3,\ldots\}.$The zeros $-2,-4,-6,\ldots$ are called trivial zeros of the $\zeta$-function.
3. $0\leq \mathfrak{R}(s)\leq 1.$
   This area of the complex number plane is called the critical strip. All remaining zeros have to be within the critical strip. We know that if $\zeta(a+ib)=0$ then $\zeta(1-a-ib)=0,$ i.e. possible zeros occur mirrored at $\mathfrak{R}(s)=1/2,$ which we call the critical line, and mirrored at the real axis. All $10^{13}$ zeros that have been found so far lie on the critical line.

 

The Riemann hypothesis says, that all non-trivial zeros of the Riemann zeta function lie on the critical line.

 

 

 

The Extended Riemann Hypothesis

A complex Dirichlet character of a finite cyclic group is a group homomorphism $\chi: \left(\mathbb{Z}_n\right)^\times \rightarrow \mathbb{C}^\times$ into the multiplicative group of the complex numbers
\begin{align*}
\chi\, : \,\{a \mod n\,|\,(a,n)=1\}&\longrightarrow \mathbb{C}-\{0\}\\
\chi(a)\cdot \chi(b)=\chi(a\cdot b)
\end{align*}
where $(a,n)$ denotes the greatest common divisor of $a$ and $n.$ We associate with it a periodic function modulo $n$ by
\begin{align*}
\chi\, &: \,\mathbb{Z} \longrightarrow \mathbb{C}\\
\chi(a)=&\begin{cases}
\chi(a) &\text{ if }(a,n)=1\\
0 &\text{ if }(a,n)>1 \end{cases}
\end{align*}
which we also call Dirichlet character modulo n. They play a key role in Dirichlet’s theorem on arithmetic progressions:

For any two positive coprime integers $a$ and $d$ there are infinitely many primes of the form $a + nd.$

Examples are the trivial character $\chi_0(a)=1$ for all $(a,n)=1,$ and the Jacobi-character
$$
\chi(a)=\left(\dfrac{a}{p}\right)=
\begin{cases}
1 &\text{ if } a\not\equiv 0 \mod p \text{ and there is an integer x such that}a\equiv x^2 (p)\\
-1 &\text{ if }a\not\equiv 0 \mod p \text{ and there is no such x}\\
0 &\text{ if }a\equiv 0 \mod p
\end{cases}
$$
extended to a homomorphism for $n=\prod p$ (cp. Jacobi symbol). Every Dirichlet character has a (Dirichlet) $L$-function, or $L$-series defined by
$$
L_\chi(z):=\sum_{a=1}^\infty \dfrac{\chi(a)}{a^z}
$$
Dirichlet series converge absolutely and locally uniform in the half-plane $\{\mathfrak{R}(z)>1\}$ since
$$
\left|\dfrac{\chi(a)}{a^z}\right|=\left|\dfrac{\chi(a)}{a^{\mathfrak{R}(z)} \cdot e^{ i \log(a) \mathfrak{I}(z) }} \right|=\dfrac{1}{a^{\mathfrak{R}(z)}} \text{ or } 0
$$
and the homomorphism property forces $\chi(a)$ to be a root of unity (or $0$). $L$-series can be holomorphic extended on $\{\mathfrak{R}(z)>0\},$ except for the trivial character, which has a first order pole at $z=1.$ The Dirichlet function $L_\chi(z)$ has the Riemann property, if all zeros within the critical strip are already on the critical line $\{\mathfrak{R}(z)=\frac{1}{2}\}$.

 

The extended Riemann hypothesis says that all zeros of all Dirichlet L-functions to complex Dirichlet characters of finite cyclic groups within the critical strip lie already on the critical line.

 

We get for the trivial character $\chi_0$ modulo $n$
$$
L_{\chi_0}(z)=\sum_{(a,n)=1}\dfrac{1}{a^z}=\zeta(z)\cdot \prod_{p|n\text{ prime}}\left(1-\dfrac{1}{p^z}\right)
$$
which has exactly the same zeros in $\{\mathfrak{R}(z)>0\}$ as the $\zeta$-function. The extended Riemann hypothesis thus implies the ordinary Riemann hypothesis.

 

 

 

Srinivasa Ramanujan, 1887-1920

 

 

“The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.” (Niels Henrik Abel, 1832)

$$
\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+\ldots=\dfrac{\pi^2}{6}=\zeta(2)\quad \text{(Euler)}
$$
Hence
$$
1+2+3+\ldots=\zeta(-1)=\zeta(1-2)=2(2\pi)^{-2}\Gamma(2)\cos\left(\dfrac{2\pi}{2}\right)\zeta(2)=-\dfrac{1}{12}
$$