Zeros of the Zeta Function: Exploring $\rho$ Values

[mathworker]

In an article it is given that,

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\(\displaystyle \zeta(s)=\text{exp}
(\sum_{n=1}^\infty\frac{\Lambda{(n)}}{\text{log}(n)}n^{-s})\)​

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$\zeta(s)$ has pole at $s=1$ and zeroes at several $s=\rho$.
here i think he considered the function inside the exponential rather than whole exponential to obtain poles and zeroes but I think we should consider it along with exponential or can we?.Or does he consider the entire function if so how does it have pole at s=1 and zeroes at \(\displaystyle s=\rho\)
what are those several $\rho$'s?

 

[Jhoneine]

The expression $\zeta(s)$ given in the article is a complex analytic function, so it can have both poles and zeroes. The pole at $s=1$ is due to the fact that the function inside the exponential (i.e. $\sum_{n=1}^\infty\frac{\Lambda{(n)}}{\text{log}(n)}n^{-s}$) has a singularity at $s=1$. The zeroes at several $s=\rho$ are due to the fact that the same function inside the exponential has zeroes at certain values of $s$. These values of $s$ are the several $\rho$'s referred to in the article.