# Euler product approximation of the Riemann zeta function

It is well-known that the Riemann zeta function can be approximated in the critical strip by a finite Dirichlet polynomial. More precisely, $$\zeta(\sigma+it) = \sum_{n \leq X} n^{-\sigma-it} + \frac{X^{1-\sigma-it}}{\sigma+it-1} + O ( X^{-\sigma})$$ uniformly for all $X$ satisfying $2 \pi X / C \geq |t|$, where $C$ is a fixed constant greater than 1. (This is Theorem 4.11 in Titchmarsh's book). In particular this holds for $\sigma = 1/2$.

Question: is there a similar (simple) approximation of zeta by a finite Euler product?

• PS: I want to avoid the large correction term of order $\approx X^{1-\sigma}$. This large correction term corresponds to the large number of summands in the Dirichlet polynomial. The idea is that an Euler product approximation would probably be "sparse" when written as a Dirichlet polynomial, thus not requiring this large correction term. – Kurisuto Asutora Mar 16 '17 at 8:05

We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function $$\zeta$$ is well approximated by short $$X$$-term truncations $$\zeta_X$$ of its Euler product. Conversely, if the approximation by products is good in this region, the zeta-function has at most finitely many zeros in it. 