Denseness of the values of $\log\zeta(1/2+it)$ on the critical line: the remaining case of the Bohr–Courant value-distribution problem

For fixed real $\sigma$, the sets $\{\zeta(\sigma+it)\mid t\in\mathbb{R}\}$ and $\{\log\zeta(\sigma+it)\mid t\in\mathbb{R}\}$ are classical objects of value-distribution theory. Bohr and Courant (1914) and Bohr (1916) proved both are dense in $\mathbb{C}$ for $1/2<\sigma\leq 1$, and work of Garunkštis and Steuding shows that for $\sigma<1/2$ the values of $\zeta$ are not dense under the Riemann Hypothesis, leaving the critical line $\sigma=1/2$ as the sole unresolved case. Endo and Inoue (Forum Mathematicum, 2021) posed this remaining case as their Problem 1: is the set $\{\log\zeta(1/2+it)\mid t\in\mathbb{R}\}$ dense in the complex plane? They resolved the analogous questions for iterated integrals of $\log\zeta$ — under the Riemann Hypothesis the first vertical integral has dense values, and for $m\geq 2$ denseness of the $m$-th iterated integral is equivalent to the Riemann Hypothesis — but left the base case $m=0$ open. No later work settles the denseness question for $\log\zeta$ (or even for $\zeta$) at $\sigma=1/2$, so the problem remains open.

Background

The Riemann zeta-function is $\zeta(s)=\sum_{n\geq 1}n^{-s}$ for complex $s=\sigma+it$ with $\sigma>1$, extended by analytic continuation to a meromorphic function on the plane. The vertical strip $0<\sigma<1$ is called the critical strip, and the line $\sigma=1/2$ the critical line; the Riemann Hypothesis (RH) asserts that all nontrivial zeros of $\zeta$ lie on this line. Since $\zeta$ has zeros on the critical line itself, $\log\zeta(1/2+it)$ is understood, as is standard in this literature, at ordinates $t$ with $\zeta(1/2+it)\neq 0$, the logarithm being fixed by continuous variation of the argument; the denseness question asks whether these values come arbitrarily close to every point of $\mathbb{C}$.

A classical program studies how the values of $\zeta$ and $\log\zeta$ on vertical lines are spread out in the plane. Bohr and Courant (1914) proved that for every fixed $1/2<\sigma\leq 1$ the set $\{\zeta(\sigma+it)\}$ is dense in $\mathbb{C}$, and Bohr (1916) strengthened this to denseness of $\{\log\zeta(\sigma+it)\}$ in $\mathbb{C}$ for the same $\sigma$-range. The method represents $\log\zeta(\sigma+it)$, well inside the strip, through its Euler product over primes, so that its behavior is governed by the frequencies $e^{-it\log p}$ for primes $p$; by Kronecker's theorem — the classical fact that a linear flow with rationally independent frequencies is dense on a torus — the prime phases can be tuned almost independently, forcing the values to be everywhere dense. Exactly at $\sigma=1/2$ this mechanism collapses: the prime sum is no longer absolutely convergent, the approximating product loses the uniformity the argument needs, and the zeros of $\zeta$ sitting on the line itself create logarithmic singularities along the path of variation. Bohr and Jessen (1930–1932) developed this into an exact limit theorem for the value distribution of $\log|\zeta(\sigma+it)|$ for $\sigma>1/2$, the starting point of the probabilistic theory of the zeta-function.

The picture off the line is otherwise complete. Garunkštis and Steuding (2014) showed that, assuming RH, for fixed $\sigma<1/2$ the values $\zeta(\sigma+it)$ are not dense in $\mathbb{C}$, and that the pair $(\zeta(1/2+it),\zeta'(1/2+it))$ is not dense in $\mathbb{C}^{2}$; hence the critical line is the only vertical line whose value-denseness is undecided. The closest approach to it is due to Kowalski and Nikeghbali (2012), who studied the distribution of $\log\zeta(1/2+it)$ through mod-Gaussian convergence — a probabilistic scheme in which normalized characteristic functions of a sequence of random variables converge to a Gaussian kernel — and gave an explicit sufficient condition (non-vanishing of a certain limiting function, their Corollary 9) under which the values $\zeta(1/2+it)$ would be dense in $\mathbb{C}$; the condition is strong and remains unverified, so it settles nothing.

Endo and Inoue (2021) introduced the $m$-times iterated integrals of $\log\zeta$ on vertical lines, $\eta_{m}(\sigma+it)=\int_{0}^{t}\eta_{m-1}(\sigma+it')\,dt'+c_{m}(\sigma)$ with $\eta_{0}=\log\zeta$, precisely to obtain new information on this last case. Their Theorem 2 shows unconditionally that the analogous horizontal iterated integrals have dense values for $1/2\leq\sigma<1$. Their Theorem 1 shows that if $\zeta$ has at most finitely many zeros with real part greater than $\sigma$ — in particular under RH at $\sigma=1/2$ — then the values of the first vertical integral $\int_{0}^{t}\log\zeta(1/2+it')\,dt'$ are dense in $\mathbb{C}$, and that for every $m\geq 2$ the denseness of $\{\eta_{m}(1/2+it)\}$ is equivalent to RH. These results give strong indirect evidence but leave the original question — the case $m=0$ — open, which is precisely the problem stated here.

Problem Statement

Endo–Inoue Problem 1 (verbatim from the source, with the branch convention made explicit): Is the set

$\{\log\zeta(1/2+it)\mid t\in\mathbb{R}\}$

dense in the complex plane? Here $\log\zeta(1/2+it)$ is understood at ordinates $t$ with $\zeta(1/2+it)\neq 0$, the logarithm being fixed by continuous variation of the argument along the critical line, and denseness means: for every $w\in\mathbb{C}$ and every $\varepsilon>0$ there exists a real $t$ with $\zeta(1/2+it)\neq 0$ such that the continuously-varied value $\log\zeta(1/2+it)$ lies within $\varepsilon$ of $w$. This is the $\sigma=1/2$ case — the only one not settled by Bohr (1916) for $1/2<\sigma\leq 1$ and by Garunkštis–Steuding (2014) for $\sigma<1/2$ under RH — of the classical problem on denseness of the values of $\log\zeta$ on vertical lines. An answer is either a proof that the value set is dense in $\mathbb{C}$, or a proof that it is not (i.e., that some nonempty open subset of $\mathbb{C}$ contains no value $\log\zeta(1/2+it)$). A positive answer would in particular imply denseness of the values $\zeta(1/2+it)$ in $\mathbb{C}\setminus\{0\}$, the companion classical open problem for $\zeta$ itself.

References

1. Kenta Endo, Shōta Inoue, 'On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness', Forum Mathematicum 33(1) (2021) 167–176. DOI: 10.1515/forum-2020-0075, https://doi.org/10.1515/forum-2020-0075 (preprint arXiv:2004.00823, https://arxiv.org/abs/2004.00823)
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