In the half plane $\sigma$=Re(s) > 1 , one can find bounds for the Riemann zeta function $\zeta$(s) using either its convergent series or product formula.$\,$ From the Dirichlet series we get the upper bound $\qquad$ $\;$ $\;$ $\vert$ $\zeta$($\sigma$ + it) $\vert$ $\le$ $\zeta$($\sigma$)$\;$ and from the Euler product we get the lower bound $\vert$ $\zeta$ ( $\sigma$ + it ) $\vert$ $\ge$ $\,$$\zeta$(2$\sigma$)/$\zeta$($\sigma$). In addition, the positivity of the coefficients implies that for real s > 1,$\,$ $\zeta$(s) is monotonically decreasing from $\zeta$(1) = $\infty$$\,$ to$\,$ $\zeta$($\infty$) = 1 .
Turning to Dirichlet L-functions L(s,$\chi$)$\;$ [$\chi$ is a real character throughout], $\;$ we still have the series and product expansions for $\sigma$>1, but lose the positivity on the coefficients.$\;$ Following the same reasoning as above leads to hybrid bounds for L in terms of $\zeta$, viz. $\;$$\zeta$(2$\sigma$)/$\zeta$($\sigma$)$\le$$\vert$L($\sigma$+it,$\chi$)$\vert$ $\le$$\zeta$($\sigma$), but apparently not to L in terms of itself. Is this the best one can do?
Questions: (1) The particular case $\chi$$_4$ [the nontrivial character mod 4] has a simple structure which makes it useful as a test case.$\;$ [ $\chi$$_4$ takes the values (1,0,-1,0) at (1,2,3,4) ].$\;$ For example, an approximate calculation shows that $\vert$L(2+3i,$\chi$$_4$)$\vert$=1.1 > .91 =L(2,$\chi$$_4$). $\, $ Therefore, the naïve guess that $\vert$ L($\sigma$+it,$\chi$)$\vert$ $\le$L($\sigma$,$\chi$) is false in general.$\;$ Nevertheless, is it somehow possible to find bounds for$\;$$\; $ $\;$$\vert$L($\sigma$+it,$\chi$)$\vert$ purely in terms of L($\sigma$,$\chi$) and perhaps other L values on the real line?
(2) An alternating series argument shows that L'(s,$\chi$$_4$)>0 for s>1 and therefore that L(s,$\chi$ $_4$) increases on this interval. Now let $\chi$$_7$ be the character defined modulo 7 which takes the values (1,1,-1,1,-1,-1,0) at
(1,2,3,4,5,6,7). Note that L(1,$\chi$$_7$)>1. Is it true that
L'(s,$\chi$$_7$)<0 for s>1, i.e. that L is decreasing over s>1?
(3) What general results concerning concavity or monotonicity are available for L(s,$\chi$) when s>1?
Thanks
Let $$F(s) = \prod_p \frac{1}{1-e^{i \theta_p} p^{-s}},\qquad \Re(s) > 1, \theta_p \in \mathbb{R}$$ Then for every $a, \Re(a) > 1, \epsilon > 0$ you can find some $s, \Re(s) = \Re(a)$ such that $|F(s)-\zeta(a)| < \epsilon$, and conversely such that $|\zeta(s)-F(a)| < \epsilon$.
This is because $$p^{-s}=p^{-\sigma} e^{-i\, (t \log p\, \bmod 2\pi)}$$ and by definition of the primes, the $\log p$ are linearly independent over $\mathbb{Z}$.
Thus we can approximate $\prod_{p \le k} \frac{1}{1-e^{i \theta_p} p^{-a}}$ with an arbitrary precision, and by absolute convergence the tail $\prod_{p > k} \frac{1}{1-e^{i \theta_p} p^{-a}}$ can be made arbitrary close to $1$ for $k$ large enough.
This observation is the core of the proof of the universality of $\zeta(s)$ on $\Re(s) \in (1/2,1)$.