# Behaviour of $\zeta(1-it)/\zeta(1+it)$?

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, Also the paper referenced in the answer to this question.) But what can we say about this quotient, at least on average, say?

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...

• +1 Thank you for the very nice reference! Sep 25, 2015 at 23:41
• In what sense use you the word regular? "... the argument of zeta of the line $\sigma=1$ becomes very regular". Of course it is a real analytic function. But, for $0<t<T$ there are $cT$ points where it is $=0$. But it is not bounded. The probability to be greater than $2\pi$ being very small, to be $\ge 4\pi$ almost incredibly small, and so on. I will not call this behavior "regular".
– juan
Sep 26, 2015 at 10:30
• @juan You mean $\sigma=\frac12$ I believe? Sep 26, 2015 at 11:57
• @ მამუკა ჯიბლაძე @paul garrett No, I mean $\sigma=1$, I speak of the function $\arg \zeta(1+i t)$. This is real analytic vanishes at $c T$ points (think on the x-ray, but this can be proved ). Also we know how to compute the probabilities I speak about.
– juan
Sep 26, 2015 at 12:12
• Oh sorry I just did not notice the word "argument" :D Sep 26, 2015 at 12:22