# 8.3: The Riemann Zeta Function


The Riemann zeta function $$\zeta(z)$$ is an analytic function that is a very important function in analytic number theory. It is (initially) defined in some domain in the complex plane by the special type of Dirichlet series given by $\zeta(z)=\sum_{n=1}^{\infty}\frac{1}{n^z},$ where $$Re(z)>1$$. It can be readily verified that the given series converges locally uniformly, and thus that $$\zeta(z)$$ is indeed analytic in the domain in the complex plane $$\bf C$$ defined by $$Re(z)>1$$, and that this function does not have a zero in this domain.

We first prove the following result which is called the Euler Product Formula.

$$\zeta(z)$$, as defined by the series above, can be written in the form $\zeta(z)=\prod_{n=1}^{\infty}\frac{1}{\left(1-\frac{1}{p_n^z}\right)},$ where $$\{p_n\}$$ is the sequence of all prime numbers.

knowing that if $$|x|<1$$ then $\frac{1}{1-x}=\sum_{k=0}^{\infty}x^k,$ one finds that each term $$\frac{1}{1-\frac{1}{p_n^z}}$$ in $$\zeta(z)$$ is given by $\frac{1}{1-\frac{1}{p_n^z}}=\sum_{k=0}^{\infty}\frac{1}{p_n^{kz}},$ since every $$|1/p_n^z|<1$$ if $$Re(z)>1$$. This gives that for any integer $$N$$ \begin{aligned} \prod_{n=1}^N\frac{1}{\left(1-\frac{1}{p_n^z}\right)}&=&\prod_{n=1}^N\left(1+ \frac{1}{p_n^z}+\frac{1}{p_n^{2z}}+\cdots\right)\nonumber\\&=&\sum\frac{1}{p_ {n_1}^{k_1z}\cdots p_{n_i}^{k_jz}}\\&=&\sum\frac{1}{n^z}\nonumber\end{aligned} where $$i$$ ranges over $$1,\cdots,N$$, and $$j$$ ranges from $$0$$ to $$\infty$$, and thus the integers $$n$$ in the third line above range over all integers whose prime number factorization consist of a product of powers of the primes $$p_1=2,\cdots, p_N$$. Also note that each such integer $$n$$ appears only once in the sum above.

Now since the series in the definition of $$\zeta(z)$$ converges absolutely and the order of the terms in the sum does not matter for the limit, and since, eventually, every integer $$n$$ appears on the right hand side of 8.15 as $$N\longrightarrow\infty$$, then $$\lim_{N\to\infty}\left[\sum\frac{1}{n^z}\nonumber\right]_N=\zeta(z)$$. Moreover, $$\lim_{N\to\infty}\prod_{n=1}^N\frac{1}{\left(1-\frac{1}{p_n^z}\right)}$$ exists, and the result follows.

The Riemann zeta function $$\zeta(z)$$ as defined through the special Dirichlet series above, can be continued analytically to an analytic function through out the complex plane C except to the point $$z=1$$, where the continued function has a pole of order 1. Thus the continuation of $$\zeta(z)$$ produces a meromorphic function in C with a simple pole at 1. The following theorem gives this result.

$$\zeta(z)$$, as defined above, can be continued meromorphically in C, and can be written in the form $$\zeta(z)=\frac{1}{z-1}+f(z)$$, where $$f(z)$$ is entire.

Given this continuation of $$\zeta(z)$$, and also given the functional equation that is satisfied by this continued function, and which is $\zeta(z)=2^z\pi^{z-1}\sin\left(\frac{\pi z}{2}\right)\Gamma(1-z)\zeta(1-z),$ (see a proof in ), where $$\Gamma$$ is the complex gamma function, one can deduce that the continued $$\zeta(z)$$ has zeros at the points $$z=-2,-4,-6,\cdots$$ on the negative real axis. This follows as such: The complex gamma function $$\Gamma(z)$$ has poles at the points $$z=-1,-2,-3,\cdots$$ on the negative real line, and thus $$\Gamma(1-z)$$ must have poles at $$z=2,3,\cdots$$ on the positive real axis. And since $$\zeta(z)$$ is analytic at these points, then it must be that either $$\sin\left(\frac{\pi z}{2}\right)$$ or $$\zeta(1-z)$$ must have zeros at the points $$z=2,3,\cdots$$ to cancel out the poles of $$\Gamma(1-z)$$, and thus make $$\zeta(z)$$ analytic at these points. And since $$\sin\left(\frac{\pi z}{2}\right)$$ has zeros at $$z=2,4,\cdots$$, but not at $$z=3,5,\cdots$$, then it must be that $$\zeta(1-z)$$ has zeros at $$z=3,5,\cdots$$. This gives that $$\zeta(z)$$ has zeros at $$z=-2,-4,-6\cdots$$.

It also follows from the above functional equation, and from the above mentioned fact that $$\zeta(z)$$ has no zeros in the domain where $$Re(z)>1$$, that these zeros at $$z=-2,-4,-6\cdots$$ of $$\zeta(z)$$ are the only zeros that have real parts either less that 0, or greater than 1. It was conjectured by Riemann, The Riemann Hypothesis, that every other zero of $$\zeta(z)$$ in the remaining strip $$0\leq Re(z)\leq 1$$, all exist on the vertical line $$Re(z)=1/2$$. This hypothesis was checked for zeros in this strip with very large modulus, but remains without a general proof. It is thought that the consequence of the Riemann hypothesis on number theory, provided it turns out to be true, is immense.