Arithmetic nature of the non-trivial zeros of the Riemann zeta function and of their power sums $f(x)=\sum_\rho x^\rho/\rho$ at algebraic arguments

The distribution of the non-trivial zeros of the Riemann zeta function is a central theme of number theory, yet nothing is known about their algebraic nature. Gun, Murty and Rath (Mathematika 64 (2018) 875–897) initiated the study of the arithmetic of zero-power sums $f(x)=\sum_\rho x^\rho/\rho$ and proved, unconditionally, that $f$ takes algebraic values at most once on the algebraic points of $(1,\infty)$, and that under Schanuel's conjecture it takes none. The paper leaves open — by explicit question or as a direct residual of its conditional theorems — whether any non-trivial zero is algebraic, whether the field generated by the zeros has positive transcendence degree, whether the imaginary parts of the zeros are linearly independent over the rationals (the folklore LI conjecture), whether $f(1)=1+\gamma/2-\tfrac12\log 4\pi$ is irrational, and whether the exceptional algebraic values allowed by the 'at most one' theorems can be eliminated without Schanuel's conjecture. All of these components remain open. Determine, unconditionally, the algebraic nature of the non-trivial zeros and of their sums $\sum_\rho R(\rho)x^\rho$ at algebraic $x$.

Background

The Riemann zeta function $\zeta(s)$ extends meromorphically to the whole complex plane with a single simple pole at $s=1$, and its non-trivial zeros $\rho$ (those in the critical strip $0<\Re(s)<1$) control the fine distribution of prime numbers through explicit formulae. By the functional equation, if $\rho$ is a zero then so are $1-\rho$ and $\bar\rho$. While the distribution of the $\rho$ has been studied intensively since Riemann, essentially nothing is known about their arithmetic: it is not known whether a single non-trivial zero is algebraic or transcendental, and the folklore linear-independence (LI) conjecture asserts that the imaginary parts of the zeros in the upper half-plane are linearly independent over $\mathbb{Q}$; the source records this conjecture, citing as standard supporting evidence Theorem A of Ingham (1942), a linear-independence statement for the ordinates that Ingham deduced from his own hypothesis on the statistical distribution of pairs of zero ordinates (a pair-correlation conjecture of the kind later formulated by Montgomery), and LI is customarily invoked only as a hypothesis.

A bridge between the zeros and arithmetic constants comes from sums over zeros. Bombieri and Lagarias, in their analysis of Li's criterion for the Riemann hypothesis (the criterion saying that the Riemann hypothesis is equivalent to non-negativity of certain coefficients $\lambda_n$ built from the Laurent expansion of $\zeta'/\zeta$ at $s=1$), derived the identity $\sum_\rho 1/\rho = 1+\gamma/2-\tfrac12\log 4\pi \approx 0.0230957$, where $\gamma$ is Euler's constant and the sum runs over the non-trivial zeros (symmetrically regularized); Gun–Murty–Rath note the accompanying curiosity that the Riemann hypothesis is equivalent to $\sum_\rho |\rho|^{-2} = 2+\gamma-\log 4\pi$. Gun, Murty and Rath then studied the function

$f(x) \;=\; \sum_\rho \frac{x^\rho}{\rho} \;=\; \lim_{T\to\infty}\sum_{|\Im\rho|<T}\frac{x^\rho}{\rho}, \qquad x>0,$

and more generally $\sum_\rho R(\rho)x^\rho$ for rational functions $R=A/B$ with $A\in\overline{\mathbb{Q}}[t]$ and $B\in\mathbb{Q}[t]$ having only simple rational roots and $\deg B>\deg A$ (for $x>1$ the general theorem additionally excludes the roots $1,-2,-4,-6,\dots$, and for $0<x<1$ the roots $0,1,3,5,7,\dots$). Their proofs rest on the explicit formulae of prime number theory together with transcendence theory. A Baker period is a $\mathbb{Q}$-linear combination of logarithms of non-zero algebraic numbers; Baker's theorem on linear forms in logarithms states that every non-zero Baker period is transcendental, and an extended Baker period additionally allows a rational multiple of $1$ in the combination. From von Mangoldt's explicit formula one gets, for algebraic $x>1$, that $f(x)-x+\log 2\pi$ is a Baker period; this yields that the value set $\{f(x):x\in(1,\infty)\cap\overline{\mathbb{Q}}\}$ contains at most one algebraic number, that $f$ (which changes sign infinitely often on $(1,\infty)$) has infinitely many zeros there of which at most one is algebraic, and — under Schanuel's conjecture, the far-reaching conjecture of transcendence theory asserting that $\mathbb{Q}$-linearly independent $\alpha_1,\dots,\alpha_n$ force $\operatorname{trdeg}_{\mathbb{Q}}\mathbb{Q}(\alpha_1,\dots,\alpha_n,e^{\alpha_1},\dots,e^{\alpha_n})\ge n$ — that this exceptional algebraic value does not exist. The boundary point behaves differently: $f(1)=1+\gamma/2-\tfrac12\log 4\pi$ combines two constants ($\gamma$ and $\log\pi$) whose irrationality is itself famously open. For $0<x<1$, Ingham's form of the explicit formula shows $f(x)-x-\gamma$ is a Baker period, so algebraicity of some $f(x)$ at algebraic $x\in(0,1)$ would force $\gamma$ to be transcendental, and this exceptional possibility apparently cannot be excluded even under Schanuel's conjecture; a curious non-vanishing theorem (using Nesterenko's algebraic-independence results on periods of CM elliptic curves, i.e. elliptic curves with extra endomorphisms, and the Chowla–Selberg formula, which expresses certain products of gamma-values at rational arguments in terms of $L$-values) states that a rational zero of $f(\pi\sqrt{d}\,x)$ in $(0,1/\pi\sqrt{d})$ implies $L'(1,\chi_{-d})\neq 0$ for the quadratic Dirichlet $L$-function of $\mathbb{Q}(\sqrt{-d})$. Analogous statements are proved for the Selberg class, the axiomatized class of Dirichlet series containing $\zeta(s)$ and standard $L$-functions. The paper closes by asking whether transcendence theory can be connected to the (grand) Riemann hypothesis at all — suggesting the zero-sum identities above as a possible link — and observing that the available tools (Baker's and Nesterenko's theorems) already appear insufficient for the remaining arithmetic questions.

Problem Statement

As stated and left open by Gun–Murty–Rath (2018), determine — unconditionally, i.e. without assuming Schanuel's conjecture, the Riemann hypothesis, or any other unproved hypothesis — the algebraic nature of the non-trivial zeros $\mathbb{X}$ of the Riemann zeta function and of their power sums at algebraic arguments. The single scientific objective (understanding the arithmetic of the zeros and their summatory transforms, which all components serve) decomposes into the following explicitly stated component questions:

1. (Algebraic nature of the zeros) As asked in the source, determine both of the following: (a) is the transcendence degree over $\mathbb{Q}$ of the field $\mathbb{Q}(\mathbb{X})$ generated by all non-trivial zeros at least one — equivalently, is at least one non-trivial zero of $\zeta(s)$ transcendental? and (b) is any individual non-trivial zero algebraic? Nothing is known about the algebraic nature of the zeros: both questions are open. The two are not equivalent: a negative answer to either settles the other affirmatively (for instance, 'no non-trivial zero is algebraic' forces every zero to be transcendental), whereas an affirmative answer to one leaves the other open.

2. (Linear independence) Determine whether the imaginary parts of the non-trivial zeros in the upper half-plane are linearly independent over $\mathbb{Q}$ (the folklore LI conjecture). The source also asks the seemingly easier question whether the $\mathbb{Q}$-vector space spanned by the imaginary parts of all non-trivial zeros has dimension at least two, but that weaker bound is implied by LI and does not by itself settle the linear-independence question.

3. (The boundary value) Is $f(1)=\sum_\rho 1/\rho = 1+\tfrac12\gamma-\tfrac12\log 4\pi$ irrational?

4. (Exceptional values, $x>1$) Is $f(x)$ transcendental for every algebraic $x>1$ — i.e. does the single algebraic value allowed by the unconditional “at most one” theorem not exist, as is known only under Schanuel's conjecture? Similarly for the general family: for $R=A/B$ with $A\in\overline{\mathbb{Q}}[t]$ and $B\in\mathbb{Q}[t]$ having only simple rational roots outside $\{1,-2,-4,-6,\dots\}$ and satisfying $\deg B>\deg A$, the source's Theorem 4.2 (Section 4) already proves that the corrected combination $g(x)=\sum_\rho\frac{A(\rho)}{B(\rho)}x^\rho+\sum_i\frac{A(\alpha_i)}{B'(\alpha_i)}\frac{\zeta'}{\zeta}(\alpha_i)x^{\alpha_i}$ is an extended Baker period for algebraic $x>1$ and, when $\sum_i\frac{A(\alpha_i)}{B'(\alpha_i)(1-\alpha_i)}\neq 0$, has at most one algebraic zero on $(1,\infty)$; determine the open remainder, namely whether that exceptional algebraic zero can be eliminated and whether the raw sum $\sum_\rho\frac{A(\rho)}{B(\rho)}x^\rho$ is transcendental for every algebraic $x>1$ (in the case $\sum_i\frac{A(\alpha_i)}{B'(\alpha_i)(1-\alpha_i)}=0$ the source proves only that at least one of the raw sum and $\sum_i\frac{A(\alpha_i)}{B'(\alpha_i)}\frac{\zeta'}{\zeta}(\alpha_i)x^{\alpha_i}$ is transcendental).

5. (Exceptional values, $0<x<1$) Does $f(x)$ take any algebraic value at an algebraic $x\in(0,1)$ — an event that would force $\gamma$ to be transcendental and that the source could not rule out even under Schanuel's conjecture?

References

1. Sanoli Gun, M. Ram Murty, Purusottam Rath, “Transcendental sums related to the zeros of zeta functions”, Mathematika 64 (2018), no. 3, 875–897. DOI: 10.1112/s0025579318000293; arXiv:1807.11201, https://arxiv.org/abs/1807.11201
2. A. E. Ingham, “On Two Conjectures in the Theory of Numbers”, American Journal of Mathematics 64 (1942), 313–319. DOI: 10.2307/2371685, https://doi.org/10.2307/2371685
3. Enrico Bombieri, Jeffrey C. Lagarias, “Complements to Li's Criterion for the Riemann Hypothesis”, Journal of Number Theory 77 (1999), no. 2, 274–287. DOI: 10.1006/jnth.1999.2392, https://doi.org/10.1006/jnth.1999.2392
4. Shekhar Suman, Raman Kumar Das, “A note on series equivalent of the Riemann hypothesis”, Indian Journal of Pure and Applied Mathematics 54 (2023), 117–119. DOI: 10.1007/s13226-022-00237-6, https://doi.org/10.1007/s13226-022-00237-6
5. Anup B. Dixit, M. Ram Murty, “On Ihara's conjectures for Euler–Kronecker constants”, Acta Arithmetica 210 (2023), 95–123. DOI: 10.4064/aa220729-19-3, https://doi.org/10.4064/aa220729-19-3
6. Vincent Nguyen, “Some series equivalent to the extended Riemann hypothesis for Dedekind zeta functions”, arXiv:2509.21518 (2025), https://arxiv.org/abs/2509.21518