REVIEW
Value-distribution of Euler--Zagier type multiple zeta-functions with identical arguments on the critical line
Number Theory · Submitted 23 Oct 2026 · 34 references
Pith's verdict: A genuinely new conditional result with a sharp r=3/r≥4 contrast; the main theorem is coherent, but the support step behind Theorem 1.3 needs a fuller proof before I'd call it fully verified.
T0 · AI review
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Referee report
The paper
The paper claims
Assuming the Riemann Hypothesis, the values of the r-fold Euler–Zagier function with identical arguments at non-trivial zero ordinates form a dense subset of the complex plane for every r≥4.
The reading
What carries the argument
The load-bearing identity is the harmonic-product decomposition ζ_r(s)=(1/r!)∑_{Π}c(Π)ζ(s;Π), whose partition structure separates into singleton-containing terms that vanish at zero ordinates and singleton-free terms that group into a polynomial P_s(X) in the variable X=ζ(2s). At s=1/2+iγ, ζ_r(1/2+iγ)=P_{1/2+iγ}(ζ(1+2iγ)), so the whole problem becomes the value distribution of ζ(1+2iγ) under simultaneous control of finitely many angles (log p_m/2π)γ. That control is supplied by Theorem 1.3, a discrete limit theorem showing that along a near-lattice subsequence of zero ordinates the pair (ζ(1+2iγ), ($p_m^{{-iγ}}$)_{m≤M}) converges in distribution to a measure supported on all of C×($S^{1}$)^M. The polynomial converts that support property into denseness of the original function.
Extended reading notes
Core claim
The central claim is Theorem 1.1(i): under RH, the set {ζ_r(1/2+iγ): γ a non-trivial zero ordinate of ζ} is dense in C for every r≥4. The proof begins with the harmonic-product decomposition specialized to equal arguments, ζ_r(s) = (1/r!)∑_Π c(Π)ζ(s;Π) with c(Π) = (−1)^{r−ℓ}∏(|P_j|−1)!. Partitions containing a singleton block produce terms with a factor ζ(s), so at s=1/2+iγ these terms vanish because ζ(1/2+iγ)=0 under RH. The remaining partitions give a polynomial P_{1/2+iγ}(X) evaluated at X=ζ(1+2iγ). For r≥4 this polynomial has positive degree, so every complex number z has a preimage a under P_{1/2}. The auxiliary Theorem 1.3 then produces zero ordinates, with positive lower density along a suitable subsequence, for which ζ(1+2iγ) is near a and the relevant prime-angle shifts are simultaneously small; this forces ζ_r(1/2+iγ) to be near z. For r=3 the polynomial is constant and the values reduce to ζ(3/2+3iγ)/3, which is bounded by ζ(3/2)/3.
Load-bearing premise
The Riemann Hypothesis is the load-bearing premise: the proof needs every non-trivial zero to satisfy ζ(1/2+iγ)=0 so that all singleton-partition terms vanish, and a secondary support theorem is quoted from the author's forthcoming paper rather than proved here.
Editorial extensions
If this is right
- For every r≥4, {ζ_r(1/2+it):t∈R} is dense in C under the Riemann Hypothesis, since the zero-ordinal subset is already dense.
- For r=3, the set at zero ordinates is bounded and therefore not dense, with explicit bound by ζ(3/2)/3, so the critical-line story has a sharp algebraic threshold at multiplicity four.
- The simultaneous approximation in Theorem 1.3 has positive lower density along the chosen subsequence, so the dense behaviour is typical in a counting sense rather than a sparse fluke.
- For 0<σ<1/2 under RH, and for σ≤0 unconditionally, V_r(σ) is not dense, giving a complete non-density side of the analogy with the Riemann zeta-function.
Reading between the lines
- Beyond the paper: the true dividing line is the degree of P_{1/2}(X), which vanishes exactly for r=3, so the same polynomial reduction would determine density for linear combinations of equal-argument Euler–Zagier functions from the degree of their reduction polynomial.
- Beyond the paper: because ζ_3(1/2+iγ)=ζ(3/2+3iγ)/3, the r=3 exceptional set is not merely bounded but lies on the continuous curve t↦ζ(3/2+3it)/3; the present argument does not use any fine structure of that curve, and a refined statement about its shape would require new input.
- Beyond the paper: the discrete-universality theorem is stated for ζ(1+2iγ) and finitely many prime angles; replacing the target a in Theorem 1.3 by values of other absolutely convergent Euler products would allow the same polynomial mechanism to treat products and rational functions of Euler–Zagier functions at zero ordinates.
Original abstract
We consider the value-distribution of the Euler--Zagier type multiple zeta-functions with identical arguments on the critical line. Mishou showed that the set of the values of the Euler--Zagier double sum with identical arguments on the critical line is dense in the complex plane under the Riemann Hypothesis. In this paper, we generalize Mishou's result to the $r$-fold Euler--Zagier multiple zeta-functions with identical arguments for $r geq 4$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We consider the value-distribution of the Euler--Zagier type multiple zeta-functions with identical arguments on the critical line. Mishou showed that the set of the values of the Euler--Zagier double sum with identical arguments on the critical line is dense in the complex plane under the Riemann Hypothesis. In this paper, we generalize Mishou's result to the $r$-fold Euler--Zagier multiple zeta-functions with identical arguments for $r geq 4$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the value-distribution of the Euler--Zagier type multiple zeta-functions with identical arguments on the critical line. | background 0.62 | 0.41 | 0.04 | unclear 0.75 |
| s1 | Mishou showed that the set of the values of the Euler--Zagier double sum with identical arguments on the critical line is dense in the complex plane under the Riemann Hypothesis. | background 1.00 | 0.03 | 0.04 | asserted 1.00 |
| s2 | In this paper, we generalize Mishou's result to the $r$-fold Euler--Zagier multiple zeta-functions with identical arguments for $r geq 4$. | contribution 0.95 | 0.89 | 0.82 | asserted 0.97 |
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Discussion
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