Argument of the Riemann zeta function on Re(s)=1

Question

I refer to the lovely answer to this question. Is there an exact formula for the argument of the Riemann zeta function? Specifically, I would like to know the arguments along the line Re$(s)=1$. Some numerical computations/visualizations would be nice too.

Answer 1

(first : Glad you liked the answer!)

I'll use the standard notation $\;s:=\sigma+it\;$ (with $\sigma$ and $\,t>0$ real values).

Let's begin with a picture of the complex orbits $\;t\in (2,42)\mapsto\zeta(\sigma+it)\;$ for $\,\sigma=0, \dfrac 12, 1,\dfrac 32$ respectively :

Qualitative discussion :

• for $\;\sigma<\dfrac 12\;$ the loops turn around the origin $0$ (for negative values of $\sigma$ the loops look more and more like an outgoing spiral)
• as $\;\sigma\;$ approaches $\dfrac 12$ all the loops miraculously cross (or seem to cross...) the origin (to simplify the 'wording' I'll suppose the R.H. true!)
• for larger values of $\;\sigma\;$ the loops will concentrate more and more near the value $1$.

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If we imagine a line going from the origin to the complex point $\zeta(\sigma+it)$ we may conjecture that for $\,\sigma\le \dfrac 12\,$ the (continuous) phase will decrease (the rotation being clockwise for $t\gg 1$) indefinitely as $t\,$ grows while for larger values of $\sigma$ the phase will only oscillate between $-\pi$ and $+\pi$. The following green, red and blue curves show the principal argument for $\,\sigma=0, \dfrac 12, 1\,$ :

The principal argument is illustrated instead of the (continuous) phase (both were illustrated in your link for $\,\sigma=\dfrac 12$). This implies that there will be a jump of $\pm\pi$ for every zero of $\,\zeta\,$ and a jump from $-\pi$ to $+\pi$ after each loop for $\sigma<\dfrac 12$.

In the case $\sigma=\dfrac 12\,$ the phase is known as the very regular Riemann–Siegel theta function $-\theta(t)$ from $\,\displaystyle\zeta\left(\frac 12+it\right)=Z(t)\,e^{-\large{i\theta(t)}}\;$ (c.f. your link or the demonstration to follow). Anyway, from the picture, we may doubt that such a simple solution exists in the case $\,\sigma=1\,$ or even $\,\sigma\neq\frac 12$...
The phase for $\,\sigma=1\,$ appears to 'follow' the principal argument of $\,\zeta\,$ for $\sigma=\dfrac 12$ and thus to have the complexity of the amplitude $Z(t)$ rather than the simplicity of the phase $-\theta(t)$.

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Quantitative discussion :

Let's start with the symmetric version of the functional equation $\;\xi(s)=\xi(1-s)\;$ as presented by Riemann with $\,\xi(s)$ defined by : $$\tag{1}\xi(s)=\frac {s(s-1)}2 \,\Gamma\left(\frac s2\right)\,\pi^{-s/2}\,\zeta(s)$$

Since $\,\dfrac {s(s-1)}2=\dfrac {(1-s)(1-s-1)}2\,$ we may remove this common factor from $\;\xi(s)=\xi(1-s)\;$ and expand the logarithm of the result as :

$$\log\Gamma\left(\frac s2\right)-\log(\pi)\,\frac s2+\log \zeta\left(s\right)=\log\Gamma\left(\frac {1-s}2\right)-\log(\pi)\,\frac {1-s}2+\log \zeta\left(1-s\right)$$ or (putting the $\,\log\zeta\;$ terms at the left) : $$\tag{2}\log \zeta\left(s\right)-\log \zeta\left(1-s\right)=\log\Gamma\left(\frac {1-s}2\right)-\log\Gamma\left(\frac s2\right)+\log(\pi)\left(s-\frac 12\right)$$

With the functional equation some usual relations like $\tag{3}\log\zeta(\overline{s})=\overline{\log\zeta(s)},\;\log\Gamma(\overline{s})=\overline{\log\Gamma(s)}\;$ and other useful properties of $\Gamma$ will be helpful.

1. Let's consider $\,s=\sigma+it\,$ and study $(2)$ in the specific case $\,\sigma=\dfrac 12$ : $$\log \zeta\left(\frac 12+it\right)-\log \zeta\left(\frac 12-it\right)=\log\Gamma\left(\frac {\frac 12-it}2\right)-\log\Gamma\left(\frac {\frac 12+it}2\right)+\log(\pi)it$$ From $(3)$ we are simply subtracting a complex and its complex conjugate (twice) so that this becomes : $$2\,\Im\log \zeta\left(\frac 12+it\right)=-2\,\Im\log\Gamma\left(\frac {\frac 12+it}2\right)+\log(\pi)it$$ i.e. the classical Riemann–Siegel theta function for the case $\,\sigma=\dfrac 12$ : $$\tag{4}\boxed{-\theta(t):=\arg \zeta\left(\frac 12+it\right)=-\arg\Gamma\left(\frac 14+\frac{it}2\right) + \frac {\log(\pi)}2\;t}\quad t\in \mathbb{R}$$

2. Let's examine $(2)$ with $\,\sigma=0\,$ and thus provide a link with the case of interest $\,\sigma=1$ : $$\tag{5}\log \zeta\left(it\right)-\log \zeta\left(1-it\right)=\log\Gamma\left(\frac {1-it}2\right)-\log\Gamma\left(\frac {it}2\right)+\log(\pi)\left(it-\frac 12\right)$$ We will again deduce the argument from the imaginary part and (using the Legendre duplication formula $(6.1.18)$ for $\Gamma\,$) : \begin{align} \arg\zeta(it)+\arg\zeta(1+it)&=-\arg\left(\Gamma\left(\frac {it}2\right)\Gamma\left(\frac {1+it}2\right)\right)+\log(\pi)\,t\\ &=-\arg\left(\sqrt{\pi}\,2^{1-it}\,\Gamma(it)\right)+\log(\pi)\,t\\ \end{align} Since $\;-\arg\left(2^{-it}\right)=\log(2)\,t\;$ this becomes simply : $$\tag{6}\boxed{\arg\zeta(it)+\arg\zeta(1+it)=-\arg\Gamma\left(it\right) + \log(2\pi)\;t\,}\quad t\in \mathbb{R}$$ $$\text{(up to a $\,2\,\pi\,k\,$ constant depending of normalization)}$$ or the rather neat : $$\tag{7}\boxed{\;\arg\left((2\,\pi)^{-it}\Gamma(it)\,\zeta(it)\,\zeta(1+it)\right)=0\;}\quad t\in \mathbb{R}$$ which appears deep until noticing that the usual functional equation $$\tag{8}\zeta(1+it)=2(2\pi)^{it}\sin\left(\frac {\pi}2(1+it)\right)\Gamma(-it)\zeta(-it)$$ and some Gamma relations transform $(7)$ in the trivial $\quad\arg(\zeta(+it)\,\zeta(-it))=0$.

$\Gamma(it)\,$ is very regular (see the pictures along the line $x=0$) but of course $\;\arg\zeta(it)\,$ is not simpler than $\;\arg\zeta(1+it)$...

Can we learn something more from the real part of $(5)$ ? Well that $$\tag{9}\left|\frac{\zeta(it)}{\zeta(1+it)}\right|=\sqrt{\frac {t\;\tanh\left(\frac{\pi}2\,t\right)}{2\,\pi}}$$

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Some references to recent work concerning $\,\zeta(1+it)\,$ (for very different points of view) :

• Andersson (2012) "On the zeta function on the line $\Re(s) = 1$" (with references to earlier work)
• Granville and Soundararajan (2005) "Extreme values of $|\zeta(1+it)|$"
• Lamzouri (2008) "The two dimensional distribution of values of $\zeta(1+it)$"
• Trudgian (2012) "A new upper bound for $|\zeta(1+it)|$"
• de Reyna, Brent, van de Lune (2014) "A note on the real part of the Riemann zeta-function" (values of $t$ such that $\Re \zeta(1+it)<0$)
• following this interesting paper by van de Lune (1983) "Some observations concerning the zero-curves of the real and imaginary parts of zeta" (the argument of zeta is considered there)

An old paper of Wintner (1936) could also be of interest "The almost periodic behavior of the function $1/\zeta(1+it)$" and many others of course...