Background:
In the 1700s, Euler solved the Basel Problem, which was to solve $\sum_{n=1}^\infty\frac{1}{n^2}$ in closed-form. Euler showed that it was equal to $\frac{\pi^2}{6}$ by first expressing $\frac{\sin(x)}{x}$ in Taylor series form, and then writing it as the normalized product of the linear factors given by its roots.
$\frac{\sin(x)}{x} = (1 – \frac{x}{\pi})(1 – \frac{x}{-\pi})(1 – \frac{x}{2\pi})(1 – \frac{x}{-2\pi})…$
When multiplied out and the x^2 terms are collected, they come out to $-\frac{1}{\pi^2}(\sum_{n=1}^∞\frac{1}{n^2})$, which corresponds to x^2's coefficient of -1/6 in the Taylor Series expansion. Thus, the infinite sum comes out to $\frac{\pi^2}{6}$.
My question
I have a function with an infinite number of complex zeroes, is continuous and differentiable everywhere, and has no infinities, complex or otherwise, for finite input (I believe this is called a holomorphic function, but I have yet to take a complex analysis class, so I'm not 100% sure). For reference, my function is $(x-1)\zeta(x)$. At x=1, this function is equal to 1, according to Mathematica.
Please note
At x = 0, $(x-1)\zeta(x) = \frac{1}{2}$
If z is a non-trivial zero of $\zeta(x)$, so is 1-z
The trivial zeroes of $\zeta(x)$ are the negative even numbers
In the same fashion that Euler described sin(x)/x as the normalized product of its linear factors, can this function also be expressed as the normalized product of its linear factors? Namely, if $z_k$ is the kth non-trivial zero of the Riemann zeta function with "positive" imaginary component, then is this true:
$(x-1)\zeta(x) = (1/2)\prod_{n=1}^\infty(1-\frac{x}{z_n})(1-\frac{x}{1-z_n})(1+\frac{x}{2n})$
If not, why not?
Additional question
In general, the only two classes of functions I can think of without any complex zeroes are non-zero constants and exponentials. So, can all functions be expressed as a product of a constant, an exponential, and its normalized linear factors?
If I need to make anything clearer, please let me know. I'm hastily posting this from a school computer and I might not be as clear as I would like.
Best Answer
Hello Gabriel,
I think you should indeed have a look at the theory of (entire) holomorphic/meromorphic functions, since $x\mapsto (x-1)\zeta(x)$ belongs to that class. You more particularly wish to learn about Hadamard or Weierstrass factorization theorems see Wikipedia, as pointed out in the comment by Andres. There you should reasonnably find the answers to your questions.