I an trying to learn more about the $\zeta$ function, and I stumbled upon this very interesting paper https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Riffer-Reinert.pdf and having a few questions on some of the proofs.
$(1)$ In the proof of theorem $2.4$, I am confused about the part that states "Using theorem $3.3$ we see that $\zeta(z) =$" Since there is no Theorem $3.3$ in the paper.
$(2)$ In the proof of Lemma $3,2$, I don't understand the second equality after "It now follows that". In order for an expression to equal the real part of itself, wouldn't we have to know that the imaginary part is $0$? I don't see why that would be the case here.
$(3)$ In Theorem $4.3$, why is $\zeta$ s homomorphic at $z_0$?
$(4)$ In the proof of Theorem $5.2$, I don't understand how applying Theorem $6.1$ gives us the inequality's it states.
Well, (1) was already addressed by reuns in his first comment: we just have the functional equation withou developing the gamma function. There is certainly no lemma 3.3 in that paper. Their bad...
(2) Here observe we're taking the logarithm of the absolute value (or modulus) of that expression. This means that is the real logarithm function...
(3) is something that I suppose was proved earlier: the zeta function is holomorphic in the shole complex plane except for a simple pole at $\;z=1\;$. Thus, for any element $\;\sigma+it_0\;,\;\;t_0\neq0\;$, the zeta function is analytic there.
Number (4) I don't get it. That paper is a weird one: using future results to deduce stuff. That doesn't seem wise nor even good mathematical writing.