Euler's theorem states that if the real part of a complex number $z$ is larger than 1, then $\zeta(z)=\displaystyle\prod_{n=1}^\infty \frac{1}{1-p_n^{-z}}$, where $\zeta(z)=\displaystyle\sum_{n=1}^\infty n^{-z}$ is the Riemann zeta function and $\{p_n\}$ is the set of primes in increasing order.

After reading the argument on extending $\zeta(z)$ to a meromorphic function for those complex numbers $z$ whose real part is larger than $0$, I am wondering whether it is possible to write

$\zeta(z)=\displaystyle\prod_{n=1}^\infty f_n(z)\cdot\frac{1}{1-p_n^{-z}}, \hspace{0.1in} 0< \Re z <1,$

for some holomorphic functions $f_n$. I am new to the Riemann zeta function, so if this is a silly question forgive me. Thank you.


Yes and no, if you want something nontrivial. The analytically extended zeta function admits a Weierstrass factorization (called the Hadamard factorization in this case):


where the product is over the nontrivial zeros. You can also see the trivial zeros at $s=-2k$ from the Gamma function. You can further expand the product via:

$$\Gamma(1+s/2)=\frac{e^{-\gamma z}}{z}\prod_{k=1}^\infty \left(1+\frac{1+s/2}{k}\right)^{-1}e^{(s+1/2)/k}.$$

This is about as good a product as you'll get and it'll agree with the Euler formula for $s>1$.


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