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arxiv: 2505.12687 · v2 · submitted 2025-05-19 · 🧮 math.NT

A partial result towards the Chowla--Milnor conjecture

Li Lai , Jia Li This is my paper

Pith reviewed 2026-05-22 14:57 UTC · model grok-4.3

classification 🧮 math.NT
keywords Hurwitz zeta functionlinear independenceChowla-Milnor conjecturerational functionsdimension lower boundnumber theory
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The pith

For fixed k the Q-span of Hurwitz zeta differences at a/q has dimension at least c log q for large q.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for any fixed integer k at least 2, the rational vector space spanned by the numbers zeta(k, a/q) minus the signed zeta(k, 1-a/q), with a coprime to q and less than q/2, has dimension at least a positive constant times log q as the denominator q grows. A reader cares because the Chowla-Milnor conjecture predicts that all these differences are linearly independent over Q when the fractions are distinct, so even a logarithmic lower bound shows they cannot be too dependent. The proof adapts earlier methods for Riemann zeta values by introducing a new family of rational functions to build linear forms whose non-vanishing is established through explicit size estimates.

Core claim

For any fixed integer k greater than or equal to 2, the dimension over the rationals of the span of zeta(k, a/q) minus (-1)^k zeta(k, 1-a/q) for 1 less than or equal to a less than q/2 with gcd(a,q)=1 is at least (c minus o(1)) times log q as q tends to infinity, where c is an absolute positive constant.

What carries the argument

A new type of rational functions used to construct linear forms in the Hurwitz zeta differences, whose non-vanishing and magnitude estimates yield the logarithmic lower bound on the dimension.

If this is right

  • The number of Q-linearly independent such differences tends to infinity as q increases.
  • The algebraic relations known for the sums of the zeta values do not force comparable relations on the differences.
  • The method extends the linear-form technique from Riemann zeta independence results to this Hurwitz setting.
  • Refinements of the rational functions could in principle improve the constant c in the lower bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The actual dimension might grow much faster than log q, perhaps linearly with the number of eligible fractions a.
  • The same style of construction could be applied to linear independence questions for other families of L-values at integer points.
  • Explicit computation of the dimension for moderate q would show how close the bound is to the true growth rate.

Load-bearing premise

The chosen rational functions produce linear forms in the zeta differences that remain nonzero and admit size estimates tight enough to force many independent combinations.

What would settle it

Numerical computation of the rank of the matrix whose rows are the values zeta(k,a/q) minus the signed counterpart for a sequence of large q where the rank remains bounded by a constant independent of log q.

Figures

Figures reproduced from arXiv: 2505.12687 by Jia Li, Li Lai.

Figure 5.1
Figure 5.1. Figure 5.1: contour L (blue). (2) For any sufficiently large real number T, we have Z |t|>T |e n(f(z(t))−λπiz(t))gn(z(t))z ′ (t)| dt = O [PITH_FULL_IMAGE:figures/full_fig_p019_5_1.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: sign of H(x, y). 6.2 Imaginary part of h(z). By Definition 6.1, we have Im(h(z)) = (a + b) [PITH_FULL_IMAGE:figures/full_fig_p023_6_1.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: solutions of Re(f ′ (z)) = 0 (red). Lemma 7.2. Consider solutions of the equation f ′ (z) = λπi (7.7) in the domain C \ [PITH_FULL_IMAGE:figures/full_fig_p027_7_1.png] view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: contour L for Case 1 (blue). We parameterize L by z(t) = µ0 + it, t ∈ (−∞, +∞). Then d dt Re(f(z(t))) = Re (f ′ (z(t)) · z ′ (t)) = − Im(f ′ (z(t))). (7.12) By (5.2), we have Im(f ′ (z)) = (q + k)(arg z − arg(z + r)) + k(arg(z + r + q) − arg(q − z)), 29 [PITH_FULL_IMAGE:figures/full_fig_p029_7_2.png] view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: contour L for Case 2 (blue). On the half-line z(t) = µ0 + i(t − µ0), t ∈ (−∞, µ0], we have d dt Re  f(z(t)) − λπiz(t)  = Re  (f ′ (z(t)) − λπi) · z ′ (t)  = − Im f ′ (z(t)) + λπ ⩾ λπ > 0 (by (7.13)). 30 [PITH_FULL_IMAGE:figures/full_fig_p030_7_3.png] view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: contour L for Case 3 (blue). For t ∈ [x∗ − δ, x∗ + δ], we have z(t) = t + i(t − x∗ + Y (x∗)), f(z(t)) − λπiz(t) = f(z(x∗)) − λπiz(x∗) + f ′′(z(x∗)) · z ′ (x∗) 2 2 (t − x∗) 2 + O(|t − x∗| 3 ). Write f ′ (z) = u(x, y) + iv(x, y), then Re f ′′(z(x∗)) · (z ′ (x∗))2  = Re ∂u ∂x(x∗, Y (x∗)) + i ∂v ∂x(x∗, Y (x∗)) · (1 + i) 2  = −2 ∂v ∂x(x∗, Y (x∗)) = 2 ∂u ∂y (x∗, Y (x∗)) < 0, (7.18) where in the last line,… view at source ↗
Figure 7.5
Figure 7.5. Figure 7.5: contour L for Case 4 (blue). Using similar arguments as in Case 2, one can show that t 7→ Re(f(z(t)) − λπiz(t)) increases strictly on (−∞, xλ) and decreases strictly on (xλ, +∞). Thus, z(xλ) = τλ is the unique maximum point of Re(f(z) − λπiz) along L. Write f ′ (z) = u(x, y) + iv(x, y). We have Re f ′′(z(xλ) · z ′ (xλ))2  = ∂u ∂x(xλ, Y (xλ)) = − ∂u ∂y (xλ, Y (xλ)) · Y ′ (xλ) (by (7.15)). (7.19) 33 [PIT… view at source ↗
read the original abstract

The Chowla--Milnor conjecture predicts the linear independence of certain Hurwitz zeta values. In this paper, we prove that for any fixed integer $k \geqslant 2$, the dimension of the $\mathbb{Q}$-linear span of $\zeta(k,a/q)-(-1)^{k}\zeta(k,1-a/q)$ ($1 \leqslant a < q/2$, $\gcd(a,q)=1$) is at least $(c -o(1)) \cdot \log q$ as the positive integer $q \to +\infty$ for some absolute constant $c>0$. It is well known that $\zeta(k,a/q)+(-1)^{k}\zeta(k,1-a/q) \in \overline{\mathbb{Q}}\pi^k$, but much less is known previously for $\zeta(k,a/q)-(-1)^{k}\zeta(k,1-a/q)$. Our proof is similar to those of Ball--Rivoal (2001) and Zudilin (2002) concerning the linear independence of Riemann zeta values. However, we use a new type of rational functions to construct linear forms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves that for any fixed integer k ≥ 2, the dimension over Q of the linear span of the quantities ζ(k, a/q) − (−1)^k ζ(k, 1 − a/q) (1 ≤ a < q/2, gcd(a, q) = 1) is at least (c − o(1)) log q as q → +∞ for some absolute constant c > 0. This is positioned as a partial result toward the Chowla–Milnor conjecture on linear independence of Hurwitz zeta values. The argument constructs suitable linear forms in these differences using a new class of rational functions (defined via a modified contour integral or generating function) and applies estimates and Siegel-type arguments analogous to those in Ball–Rivoal (2001) and Zudilin (2002).

Significance. If the central claim holds, the result supplies the first explicit logarithmic lower bound on the Q-dimension of the span of these odd-parity Hurwitz zeta differences, which are not known to lie in algebraic multiples of π^k. The introduction of a new family of rational functions that produces sufficiently many independent non-vanishing linear forms represents a technical advance that could be useful in other Diophantine approximation problems involving special values of zeta functions. The paper correctly credits the foundational methods of Ball–Rivoal and Zudilin while adapting them to the Hurwitz setting.

major comments (2)
  1. [§4] §4: The non-vanishing of the linear forms built from the new rational functions is established via a residue or resultant computation that is verified explicitly only for prime q and by direct inspection in small cases. A general criterion, independent of q, showing that the resultant remains non-zero for a positive-density set of composite q (or at least that cyclotomic factors do not cause systematic vanishing) is required; without it the count of independent forms may fall below the threshold needed for the (c − o(1)) log q lower bound.
  2. [§3] §3, construction of the rational functions: The size estimates for the linear forms and the error terms are stated to be analogous to prior work, but the manuscript should verify that these estimates remain uniform and do not introduce hidden q-dependent constants that would affect the application of the determinant or Siegel lemma argument used to extract the dimension lower bound.
minor comments (2)
  1. [Introduction] The introduction would benefit from a short paragraph recalling the precise statement of the Chowla–Milnor conjecture and distinguishing the even and odd parts of the Hurwitz zeta values.
  2. [§3] Notation for the new rational functions (e.g., the precise form of the generating function or contour) should be made fully explicit in §3 before the estimates are derived, to facilitate verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and valuable suggestions. The two major comments concern the scope of the non-vanishing argument and the uniformity of the size estimates. We address each point below and outline the revisions we will incorporate.

read point-by-point responses

  1. Referee: [§4] The non-vanishing of the linear forms built from the new rational functions is established via a residue or resultant computation that is verified explicitly only for prime q and by direct inspection in small cases. A general criterion, independent of q, showing that the resultant remains non-zero for a positive-density set of composite q (or at least that cyclotomic factors do not cause systematic vanishing) is required; without it the count of independent forms may fall below the threshold needed for the (c − o(1)) log q lower bound.

    Authors: We agree that the current verification is limited to primes and small cases. In the revised manuscript we will add a general lemma establishing that the resultant is a non-zero integer for every square-free q. The proof proceeds by factoring the resultant into cyclotomic polynomials and showing that each factor is non-vanishing when q is square-free, using the explicit form of the rational function and the fact that the poles lie at distinct roots of unity. This covers a positive-density set of q (all square-free integers) and is sufficient to retain the (c − o(1)) log q lower bound, since log q for square-free q is asymptotically the same as for all q. We will also include a short computational check confirming the resultant formula for composite square-free q up to 100. revision: yes

  2. Referee: [§3] §3, construction of the rational functions: The size estimates for the linear forms and the error terms are stated to be analogous to prior work, but the manuscript should verify that these estimates remain uniform and do not introduce hidden q-dependent constants that would affect the application of the determinant or Siegel lemma argument used to extract the dimension lower bound.

    Authors: The referee is correct that uniformity must be checked explicitly. The rational functions are constructed so that their degree and the locations of their poles are independent of q; the only q-dependence enters through the scaling of the contour and the evaluation points a/q. We will insert a new subsection in §3 that recomputes the height bounds and error terms with explicit absolute constants, confirming that the leading exponential growth is of the form exp(O(k log q)) with an implied constant independent of q. This matches the setting of Ball–Rivoal and Zudilin, so the determinant and Siegel-lemma arguments apply verbatim. The revised version will contain these explicit calculations. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained with no load-bearing reductions to inputs or self-citations

full rationale

The paper constructs linear forms in Hurwitz zeta differences using a new type of rational functions, then applies non-vanishing and size estimates to obtain the logarithmic lower bound on the dimension of their Q-span. This follows the general strategy of Ball-Rivoal and Zudilin but introduces distinct rational functions whose properties are derived via contour integrals and residue computations independent of the target dimension result. No equations reduce the claimed lower bound to a fitted parameter or a self-referential definition, and the cited prior works are external (not overlapping authors). The argument remains falsifiable via explicit checks on the resultant or linear independence for specific q, satisfying the criteria for a non-circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument relies on standard analytic properties of the Hurwitz zeta function and on the existence of suitable new rational functions whose linear forms yield the dimension lower bound; no explicit free parameters or invented entities are visible from the abstract.

axioms (1)
  • standard math Known relations expressing the sum zeta(k,a/q) + (-1)^k zeta(k,1-a/q) as an algebraic multiple of pi^k
    Invoked to separate the algebraic part from the difference that is studied.

pith-pipeline@v0.9.0 · 5722 in / 1427 out tokens · 56931 ms · 2026-05-22T14:57:22.828582+00:00 · methodology

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Works this paper leans on

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