I am trying to understand the behaviour of
$$\int^\infty\_{-\infty}\frac{\xi(1-it)}{\xi(1+it)}h(t)\frac{dt}{t}$$
where $h$ is a Schwartz function on $\mathbb R$, and $\xi(s)$ the completed Riemann zeta function. Clearly it is the quotient of zeta functions that is the most difficult to study.


One knows certain things about $\zeta(1+it)$, for example the pole at $t=0$ and the nonvanishing for all $t$. (See [this question](https://mathoverflow.net/questions/57630/order-of-zeta1it), Also the paper referenced in the answer to [this question](https://mathoverflow.net/questions/80973/lower-bound-for-re-zeta1it).) But what can we say about this quotient, at least on average, say?



Behaviour of $\zeta(1-it)/\zeta(1+it)$?
For zeta itself, there is a clear result, that eventually the argument of zeta on the line $\Re(s)=1$ becomes very regular. Some more sophisticated things around this are in 


D. Hejhal, "On a result of G. Polya concerning the
Riemann $\xi$-function", J. D'analyse Math. 55 (1990), 60-95.


Also, (5.14) of that paper recalls the consequence
$$
\log \zeta(1+iu) - \log \zeta(1+it) \;=\; O\Big({\log t \over \log\log
t}\Big)\cdot (u-t)
$$
for $u\ge t$ from [Titchmarsh~1986]. In an earlier edition, this was on page 98. In
the edition revised by Heath-Brown, this is (5,17.4) on page 112.


For more generaly automorphic $L$-functions, somewhere in Iwaniec-Kowalski they show that similar results follow for self-adjoint (data for) $L$-functions assuming sufficient GRH's ... but/and it is not clear to me whether/how the delicate-but-elementary arguments for zeta and maybe Dirichlet $L$-functions over $\mathbb Q$ would generalize...


