REVIEW 2 major objections 3 minor 15 references
Normality of algebraic numbers and the Riemann zeta function
T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read The paper claims that an algebraic irrational number is normal to a base exactly when a family of Riemann zeta averages vanishes.
desk verdict A genuine and largely clean equivalence between normality of algebraic numbers and vanishing of zeta averages; the small ψ1 slip is presentational, not load-bearing. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is a two-sided transform. The functional equation of $\zeta$ converts $\zeta(-k+2\pi i d n)$ into a sum over integers $m$ of $e(d n \log(me/(dn)) + \theta n)$, and the stationary phase method then evaluates the exponential sum over $n$. The Fourier expansion of the periodic Bernoulli polynomial $\psi_{k+1}$ turns the result into the sum $-\frac{1}{\log^k b}\sum_{0<h<\log_b N}\psi_{k+1}(b^h\alpha)$. For $k=0$, Ridout's theorem supplies the lower bound $\|b^h\alpha\|\gg_\gamma b^{-\gamma h}$ that makes the boundary term $o(\log N)$.
What would settle it
For a fixed algebraic irrational $\alpha$ and base $b$ (for instance $\alpha=\sqrt[3]{2}$, $b=2$), compute the boundary sum $G_N=\sum_{0<h<\log_b N}\min(1,b^h/(\|b^h\alpha\|N))$ at $N=10^5,10^6,\ldots$; the proof requires $G_N=o(\log N)$. If these numbers grow like a positive multiple of $\log N$, the $k=0$ case of Theorem 1.1 fails. Alternatively, evaluate the zeta average in Corollary 1.2 at $k=0$; if it tends to zero while $b^h\alpha$ is visibly not uniformly distributed, the claimed equivalence fails.
Extended reading notes
Core claim
The central claim is that a positive algebraic irrational $\alpha$ is normal to base $b$ if and only if, for every integer $k\ge 0$, $$\lim_{N\to\infty}\frac{1}{\log N}\sum_{1\le |n|\le N} \zeta\left(-k+\frac{2\pi i n}{\log b}\right)\frac{$e^{{2\pi i n \log \alpha/\log b}}$}{$n^{{k+1}}$}=0.$$ This is Corollary 1.2. It is proved through Theorem 1.1, an asymptotic identity expressing the left-hand side as $-\frac{1}{\log^k b}\sum_{0<h<\log_b N}\psi_{k+1}(b^h\alpha)+o(\log N)$, where $\psi_{k+1}$ is the $(k+1)$-st periodic Bernoulli polynomial. Since $\alpha$ is normal to base $b$ exactly when the sequence $b^h\alpha$ is uniformly distributed modulo one, and uniform distribution is equivalent to the averages of all periodic Bernoulli polynomials tending to zero, the identity turns a digit-counting property into a statement about the Riemann zeta function on vertical arithmetic progressions.
Load-bearing premise
The proof for the $k=0$ case rests on the deep fact that for an algebraic irrational $\alpha$, the fractional part of $\alpha b^h$ stays away from $0$ by at least a small exponential amount for every $h$; if that uniformity fails, the boundary term is no longer $o(\log N)$ and the equivalence for $k=0$ collapses.
Editorial extensions
If this is right
- If Corollary 1.2 is correct, proving the classical conjecture for a given algebraic $\alpha$ is equivalent to checking that all displayed zeta averages vanish; no digit-counting is required.
- Theorem 1.1 gives a quantitative form: digit-counting averages and zeta averages agree up to $o(\log N)$, so a non-normal algebraic number would force at least one zeta average to fail to vanish.
- For $\alpha=b^{p/q}$ with $\gcd(p,q)=1$ and $b$ not a $q$-th power of an integer, normality to base $b$ is equivalent to vanishing of the same zeta averages with $e^{2\pi i n p/q}$ in place of the $\alpha$-dependent exponential.
- For integer $\alpha$, the zeta averages converge to explicit constants: nonzero only when $k$ is odd, in which case the value is proportional to $\zeta(k+1)/\log^{k+1} b$; the even-$k$ case corresponds to trivial zeros of $\zeta$.
- The $k=0$ case of the theorem reproduces the earlier characterization of simple normality of $2^{p/q}$, so the new result contains that earlier one as a special case.
Reading between the lines
- Because the asymptotic identity in Theorem 2.1 is stated for arbitrary real $\theta$, the zeta-average to Bernoulli-average connection holds for transcendental numbers as well; only the boundary term, which for algebraic $\alpha$ is controlled by Ridout's theorem, would need a different estimate.
- An effective version of the criterion would follow from replacing the qualitative lower bound on $\|\alpha b^h\|$ with an explicit one, turning the equivalence into a finitary test for normality up to a given number of digits.
- If the equivalence is correct, it relocates the normality problem entirely: one would need to prove that the displayed averages vanish for every algebraic irrational, a statement about $\zeta$ that does not mention digits.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves an asymptotic identity (Theorem 1.1) relating weighted partial sums of the Riemann zeta function at the arithmetic progression s = -k + 2πin/log b to averages of periodic Bernoulli polynomials at b^h α. For positive algebraic irrational α the error is o(log N), and the authors deduce Corollary 1.2: α is normal to base b if and only if all the displayed zeta averages vanish. The proof is built on two general results (Theorems 2.1 and 2.2) established through the functional equation, Euler-Maclaurin summation, van der Corput estimates, stationary phase, and Ridout's theorem for the k=0 boundary term.
Significance. If correct, this is a striking and substantial contribution: it converts the notoriously intractable property of digit normality for algebraic numbers into concrete analytic vanishing conditions for zeta values on vertical arithmetic progressions. The paper is largely self-contained, the error terms are optimized explicitly, and the application of Ridout's theorem to control the k=0 boundary term is legitimate. The results also give clean corollaries for rational powers and for integer α. However, one step in the proof of the general identities uses an inequality that is false as stated, so the main theorems are not yet fully established.
major comments (2)
- [Section 9, Eq. (9.3)] The uniform bound |∑_{a≤|m|≤b} e(mx)/m| ≪ 1 is false. For example, with a=1, b=N, and x=1/N, the sum equals 2∑_{m=1}^N cos(2πm/N)/m, which is asymptotic to 2 log N. Lemma 6.4 bounds the sine partial sums, not the symmetric exponential sums. This inequality is used immediately after (9.4) to discard the interval H2(1)-2d ≤ h < dλ+θ at a cost of O(log log N), and that step is needed for the proofs of both Theorems 2.1 and 2.2. A replacement estimate is required; for k=0 the partial sums can grow logarithmically, and Theorem 2.1 makes no Diophantine assumption on θ or d.
- [Corollary 1.2, proof] The proof states that every real polynomial on [0,1] can be represented as a finite linear combination of the ψ_k because 'each ψ_k is a polynomial of degree k'. This is false for k=1: ψ1 is defined to vanish at integers, so it is not identical to the polynomial B1(x) on all of [0,1]. The argument can be repaired by noting that for algebraic irrational α, the numbers b^h α are never integers, so ψ1(b^h α) = B1({b^h α}) on the sequence in question; the proof should state this explicitly.
minor comments (3)
- [Throughout] The paper uses e(x) for e^{2πix} but also writes e^{(h+θ)/d} for the natural exponential. This double use is very confusing in formulas such as e(me(θ-h)/d) in Section 9; exp(...) should be used for the natural exponential.
- [Section 3, heuristic] The heuristic sketch knowingly chooses η=0 in Lemma 3.2 although the lemma requires η>0. The eventual rigorous argument does not use this choice, but the sketch would be clearer if it noted that the rigorous proof supplies the missing η parameter.
- [Theorem 2.1, statement] The two displayed forms of the theorem in (2.1) and (2.2) would be easier to read if the notation for the natural exponential in the summation limits were defined explicitly at that point.
Circularity Check
No significant circularity: the zeta criterion is derived from standard analytic number theory, not fitted to normality.
full rationale
The paper's central claim (Corollary 1.2) reduces normality of an algebraic irrational α to the vanishing of a zeta average, but nothing in that zeta average is fitted to normality data and no parameter is tuned to force the equivalence. The derivation chain is Corollary 1.2 ⇐ Theorem 1.1 ⇐ Theorems 2.1 and 2.2, proved in Section 9 from the zeta functional equation (Lemma 3.1), Euler–Maclaurin summation, the stationary-phase method, and, for the k = 0 boundary term, Ridout's theorem (Theorem 2.3), an external deep result. The output quantity ∑ ψ_{k+1}(b^h α) is exactly what normality requires to vanish by the standard Bugeaud/Kuipers–Niederreiter uniform-distribution criterion, not an input imposed by construction. The self-citation of [KS23] is contextual: the authors say Theorem 1.1 extends their earlier formula, but the proof does not invoke [KS23] as a black box, and the earlier result is subsumed rather than load-bearing. No uniqueness theorem is imported from the authors, and no empirical quantity is renamed as a prediction. The only lapse is presentational: ψ1 is not continuous on [0,1], so the polynomial-span argument in Corollary 1.2 should cite Riemann integrability of the periodic Bernoulli functions rather than continuity; since α is algebraic irrational, b^h α is never an integer and the step survives. This does not make the claim circular.
Assumptions & free parameters
assumptions (5)
- domain assumption Ridout's theorem (Theorem 2.3) provides lower bounds |α - a/b| ≥ C b^{-1-ε} for algebraic irrational α and denominators b that are S-units.
- standard math Functional equation and approximate functional equation for the Riemann zeta function (Lemmas 3.1, 4.2, 4.4).
- standard math Euler-Maclaurin summation formula (Lemma 5.2) and van der Corput / stationary phase integral estimates (Lemma 8.2).
- domain assumption Normality of α to base b is equivalent to the uniform distribution modulo 1 of (b^h α).
- ad hoc to paper The set of periodic Bernoulli functions can represent polynomials in the sense used for the Weierstrass approximation reduction.
Cite this review
Pith. "Pith review of Normality of algebraic numbers and the Riemann zeta function." pith.science (2026). https://pith.science/paper/LJG34LWU
@misc{pith2026241202337,
author = {Pith},
title = {Pith review of: Normality of algebraic numbers and the Riemann zeta function},
year = {2026},
howpublished = {\url{https://pith.science/paper/LJG34LWU}},
note = {Machine review of arXiv:2412.02337}
}
abstract
A real number is called simply normal to base $b$ if every digit $0,1,\ldots ,b-1$ should appear in its $b$-adic expansion with the same frequency $1/b$. A real number is called normal to base $b$ if it is simply normal to every base $b, b^2, \ldots$. In this article, we discover a relation between the normality of algebraic numbers and a mean of the Riemann zeta function on vertical arithmetic progressions. Consequently, we reveal that a positive algebraic irrational number $\alpha$ is normal to base $b$ if and only if we have \[ \lim_{N\to \infty}\frac{1}{\log N} \sum_{1\leq |n|\leq N} \zeta\left(-k+\frac{2\pi i n}{\log b} \right) \frac{e^{2\pi i n \log \alpha /\log b}}{n^{k+1}} =0 \] for every integer $k\geq 0$.
Reference graph
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Reviewed August 11, 2026 · model on record in the stance chip above.
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