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REVIEW 3 major objections 5 minor 20 references

Discrete Measures and the Extended Riemann Hypothesis

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Pith's one-line read For an algebraic number field K, the extended Riemann hypothesis for its Dedekind zeta function is equivalent to a single convergence-rate statement for discrete totient-weighted measures.

desk verdict A genuine but rough extension of Verjovsky's discrete-measure criterion to all number fields; the main equivalence is almost certainly right, but the converse direction has a repairable omitted separation argument and Lemma 4.3 has a false identity. read the letter →

arxiv 1908.03658 v2 pith:XWIQRPNL submitted 2019-08-10 math.NT

Samuel Estala-Arias This is my paper · ORCID
classification math.NT MSC 11M2611N3711R42
keywords DedekindzetafunctionExtendedRiemannHypothesisdiscretemeasuresMellintransformEulertotientalgebraicnumberfieldrateofconvergencezero-freeregions
topics The Riemann Hypothesis
open problems The Riemann Hypothesis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the extended Riemann hypothesis for the Dedekind zeta function $\zeta_K(s)$ of an algebraic number field $K$ is exactly a rate-of-convergence statement. For a field of degree $n=[K:\mathbb{Q}]$, define discrete measures $m_q(f)=\sum_{\mathfrak{a}} q\,\varphi_K(\mathfrak{a})\,f(q^{1/2}N(\mathfrak{a}))$ over integral ideals $\mathfrak{a}$; the claim is that $\zeta_K$ satisfies the Riemann hypothesis if and only if $m_q(f)$ approaches $m(f)=\frac{\kappa}{\zeta_K(2)}\int f(q)\,q\,dq$ with error $o(q^{3/4-\varepsilon})$ as $q\to 0$ for every compactly supported test function with at least $n+1$ derivatives and every $\varepsilon<1/4$. The paper also proves an unconditional $o(q^{1/2})$ error term at lower regularity, a quantitative version tying error exponents to zero-free half-planes, and optimality examples showing rougher test functions cannot reach the $1/2$ exponent. A sympathetic reader should care because the result turns an analytic conjecture about zeros into a concrete distributional statement about arithmetic objects.

What carries the argument

The central object is the family of discrete measures $m_q$ on $\mathbb{R}_+$, defined arithmetically by $m_q(f)=\sum_{\mathfrak{a}} q\,\varphi_K(\mathfrak{a})\,f(q^{1/2}N(\mathfrak{a}))$, together with its Mellin transform $M_f(s)$. The load-bearing identity is $M_f(s)=2\,\frac{\zeta_K(2s-1)}{\zeta_K(2s)}\int_0^\infty f(q)q^{2s-1}dq$: it identifies the analytic continuation of the transform with a ratio of Dedekind zeta functions, so a zero of $\zeta_K(2s)$ at $s_0$ creates a pole of $M_f$ unless the test function's Mellin integral vanishes there. The proof then uses two gears: integration by parts, which makes $M_f$ decay on vertical lines when $f$ has enough derivatives (the threshold $\ell\ge n+1$), and the Riemann–Lebesgue lemma, which converts that decay into the claimed error term after Mellin inversion.

What would settle it

To settle the converse, check the missing separation property at a hypothetical zero: fix a field $K$ and a point $s_0$ with $\mathrm{Re}(s_0)>1/4$ and $\zeta_K(2s_0)=0$, and compute $\int_0^\infty f(q)q^{2s_0-1}dq$ for a spanning family of $C^{n+1}_c(\mathbb{R}_+)$. If every such integral vanished, the 'every $f$' error bound would hold despite ERH failing; if any is nonzero, the pole argument contradicts the assumed error bound. The paper's own family $f(t)=(1-t)^{n+2}$ on $(0,1)$ and $0$ elsewhere gives the integral $(n+2)!/(2s_0(2s_0+1)\cdots(2s_0+n+2))$, which is nonzero at every such $s_0$.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 1.1(C): the Riemann hypothesis for the Dedekind zeta function holds if and only if for every $f\in C^{\ell}_c(\mathbb{R}_+)$ with $n+1\le \ell\le\infty$, one has $m_q(f)=m(f)+o(q^{3/4-\varepsilon})$ as $q\to 0$, for all $0<\varepsilon<1/4$. The same theorem gives an interpolating version: for $\alpha\in(1/2,3/4)$, the error $o(q^{\alpha-\varepsilon})$ holds for all such $f$ if and only if $\zeta_K(s)$ has no zeros in the half-plane $\mathrm{Re}(s)>2(1-\alpha)$. The mechanism is the Mellin transform identity $M_f(s)=2\,\frac{\zeta_K(2s-1)}{\zeta_K(2s)}\int_0^\infty f(q)q^{2s-1}dq$, which converts zeros of $\zeta_K$ into poles of $M_f$; Mellin inversion and contour shifting then convert holomorphy past $\mathrm{Re}(s)=1/4$ into the asserted error term. Before the equivalence, the paper establishes an unconditional error term $o(q^{1/2})$ for test functions with at least $\lfloor n/2\rfloor+2$ derivatives, and shows that characteristic functions of intervals have error with $\limsup_{q\to0} q^{-\alpha}|m_q(f)-m(f)|=\infty$ for every $\alpha>1/2$.

Load-bearing premise

The load-bearing premise is that the admissible smooth test functions are rich enough to detect every possible zero of $\zeta_K(2s)$ in the half-plane $\mathrm{Re}(s)>1/4$: if such a zero existed, at least one function in $C^{n+1}_c(\mathbb{R}_+)$ would have a nonzero weighted integral at that point, and the paper does not supply this separation step.

Editorial extensions

If this is right

  • If the extended Riemann hypothesis holds for $K$, then every compactly supported test function with at least $n+1$ derivatives satisfies $m_q(f)=m(f)+o(q^{3/4-\varepsilon})$ as $q\to0$, for all $\varepsilon<1/4$.
  • Conversely, if that error bound holds for every such $f$, then $\zeta_K$ has no zeros off the critical line, so the extended Riemann hypothesis follows.
  • The quantitative version couples the error exponent to zero-free half-planes: an error $o(q^{\alpha-\varepsilon})$ for all such $f$ is equivalent to $\zeta_K(s)$ having no zeros with $\mathrm{Re}(s)>2(1-\alpha)$, for $\alpha\in(1/2,3/4)$.
  • Without any unproved hypothesis, test functions with at least $\lfloor n/2\rfloor+2$ derivatives already give $m_q(f)=m(f)+o(q^{1/2})$, and this exponent cannot be improved by replacing $f$ with the characteristic function of an interval, whose error satisfies $\limsup_{q\to0} q^{-\alpha}|m_q(f)-m(f)|=\infty$ for every $\alpha>1/2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the missing separation step in the converse can be supplied by the paper's own Beta-function family; taking $f(t)=(1-t)^{n+2}$ on $(0,1)$ and $0$ elsewhere gives Mellin transform $(n+2)!/(2s(2s+1)\cdots(2s+n+2))$, which does not vanish in $\mathrm{Re}(s)>1/4$, so the 'error bound for every $f$ implies ERH' direction is repairable if the contour estimates are valid.
  • Beyond the paper: because the argument uses only the Euler product, the functional equation, and Phragmén–Lindelöf bounds for $\zeta_K$, the same measure reformulation should extend to other L-functions with these features, such as Hecke L-functions, yielding analogous 'L-function hypothesis as convergence rate' equivalences.
  • Beyond the paper: the quantitative form suggests an empirical test for a fixed number field: compute $m_q(f)-m(f)$ for the Beta-family test functions at small $q$ and compare the observed decay rate with the predicted zero-free half-plane $\mathrm{Re}(s)>2(1-\alpha)$; the relation is checkable at finite precision.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper defines, for an algebraic number field K, discrete measures m_q on R+ by m_q(f)=∑_{a⊂o_K} q φ_K(a) f(q^{1/2} N(a)), and studies their weak convergence as q→0 to the absolutely continuous measure m(f)=κ/ζ_K(2)∫ f(q) q dq. The main result, Theorem 1.1(C), asserts that the Extended Riemann Hypothesis for ζ_K(s) is equivalent to the statement that for every f∈C_c^ℓ(R+) with n+1≤ℓ≤∞, one has m_q(f)=m(f)+o(q^{3/4-ε}) for all 0<ε<1/4, together with a one-parameter refinement involving zero-free half-planes. The proof is based on the Mellin transform identity M_f(s)=2ζ_K(2s-1)/ζ_K(2s) ∫ f(q) q^{2s-1} dq and on estimates for this transform combined with Mellin inversion. Further results give conditional rates of convergence under the Lindelöf hypothesis or under a generalized circle-problem hypothesis, and optimality statements for characteristic functions and certain continuous functions.

Significance. The explicit Mellin factorization M_f(s)=2φ_K(s) times the Mellin transform of f is the paper's main positive content; it cleanly exposes why smoothness of f translates into decay of M_f and hence into improved error exponents. If Theorem 1.1(C) were fully established, it would provide a natural generalization of Verjovsky's criterion for ζ(s) to Dedekind zeta functions and could serve as a useful reformulation of the ERH. The paper does not appear to assume the conclusion, and the factorization itself is a valuable contribution. However, as detailed below, two gaps in the proofs — the missing test-function separation in the implication from the error bound to RH, and an invalid identity in the proof of Lemma 4.3 — mean that the central claim is not yet rigorously established as written.

major comments (3)
  1. [Section 4.4 (Proof of Theorem D; Theorem 1.1(C), implication 'error bound for every f ⇒ RH')] The proof that m_q(f)=m(f)+o(q^{3/4-ε}) for every allowed f implies the Riemann hypothesis is incomplete. Equation (5) shows M_f(s)=2ζ_K(2s-1)/ζ_K(2s)∫f(q)q^{2s-1}dq. If ρ is a zero of ζ_K with Re(ρ)>1/2, then s0=ρ/2 lies in Re(s)>1/4 and ζ_K(2s) vanishes at s0, so M_f(s) would have a pole at s0 unless the test-function factor ∫f(q)q^{ρ-1}dq also vanishes. The manuscript never proves that there exists an allowed f with this integral nonzero. Consequently, the conclusion that M_f(s) is holomorphic for Re(s)>1/4 for every f contradicts the existence of such a zero only under an additional separation argument. This is load-bearing: it is exactly the 'only if' direction of the central equivalence. The gap is fillable, for example by choosing f_t(q)=η(q)q^t with η∈C_c^∞ positive and t outside the zero set of the entire function t↦∫η(q)q^{ρ-1+t}dq, but the argument must be supplied in the paper.
  2. [Lemma 4.3 (used in Theorem 1.1(D))] The proof of Lemma 4.3 contains a false identity. The text states that if x+1 is restricted to values that are norms of prime integral ideals, then φ_K(x+1)=|{n⊂o | N(n)≤x}|=κx+O(x^{1-1/n}). This is incorrect: for a prime ideal p with N(p)=x+1, the definition gives φ_K(p)=N(p)-1=x, not the counting function of all integral ideals of norm at most x. Moreover the notation φ_K(⌊x+1⌋) is ambiguous because φ_K is defined on integral ideals, not on real numbers. As written, the contradiction argument in Lemma 4.3 does not go through. The lemma may be salvageable by a different argument, but this proof needs to be replaced.
  3. [Theorem 1.1(C), 'furthermore' part (general α)] The two implications in the 'furthermore' part of Theorem 1.1(C), concerning an arbitrary α∈(1/2,3/4), are stated but not proved. Section 4.4 only treats the case α=3/4. The converse direction ('no zeroes in Re(s)>2(1-α) implies equation (2)') requires estimates for M_f(s) on the line Re(s)=1-α+ε analogous to those in Lemma 3.4, and the forward direction requires the same test-function separation argument as above, adapted to the zero-free region. Since these statements are part of the theorem's claim, they need either a proof or an explicit restriction of the theorem to the case α=3/4.
minor comments (5)
  1. [Section 4.1 (Proof of Theorem A)] The proof of Theorem 1.1(A) is written only for f equal to a characteristic function of an interval; the text says the remaining cases 'can be proved similarly'. For a claim about all f∈C_c^0(R+), a standard approximation by step functions requires a quantitative statement about the error under the measure m_q, so at least a brief explanation of this approximation step should be added.
  2. [Section labels (Sections 4.2–4.4)] The section titles do not match the theorem numbering: Section 4.2 proves Theorem 1.1(D), Section 4.3 proves Theorem 1.1(B), and Section 4.4 proves Theorem 1.1(C). This makes the structure difficult to follow and should be corrected.
  3. [Lemma 3.3 proof] The statement 'if 1/2 ≤ Re(s), ζ_K(2s)^{-1}=O(1)' is not literally correct on the line Re(s)=1/2, since 1/ζ_K(1+it) can grow slowly as t→∞; the standard zero-free region gives O(t^ε) for any ε>0. The final decay estimate in the lemma is unaffected, but the displayed assertion should be qualified.
  4. [Theorem 1.1(E)–(F) and Section 4.5] The definitions of the functions F in Theorem 1.1(E)–(F) contain the typo 'for ≤1', which should read 'for t≤1'. The same typo appears in Section 4.5.
  5. [Section 4.5, estimates for MF_r(s)] The claimed bound MF(σ+it)=O(1/(1+|t|)^{1+1/4}) in Theorem (E) is stated without displaying the computation; from φ_K(σ+it)=O(t^{n/2+ε}) and the Beta-function factor of degree ⌊n/2⌋+2 one obtains the even/odd-parity exponents -2+ε and -3/2+ε, which imply the stated bound but should be shown explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Mellin-transform derivation is self-contained; the converse direction has a fillable test-function separation gap, not a circular step.

full rationale

The derivation is self-contained and non-circular. The central identity, Equation (5), M_f(s) = 2 zeta_K(2s-1)/zeta_K(2s) * integral f(q)q^{2s-1}dq, is obtained by substituting the definition of m_q(f), Eq. (1), into the Mellin transform, Eq. (4), and summing the resulting Dirichlet series. This is a direct computation, not an assumption of the conclusion. The forward implication in Theorem 1.1(C) uses the Riemann hypothesis only through the standard zero-free region and growth estimates in Lemma 3.4, which make the Mellin-inversion contour shift legitimate; it then derives the o(q^{3/4-epsilon}) error term. The reverse implication assumes the error bound for every test function to force holomorphy of M_f(s) for Re(s) > 1/4+epsilon, then uses Eq. (5) to conclude that zeta_K(2s) is zero-free in that half-plane. That last inference is not circular, but it is incomplete as written: to rule out a zero s0 of zeta_K(2s), one must exhibit an allowed f with integral f(q)q^{2s0-1}dq nonzero, and the paper does not supply this separation argument. This is an easily filled analytic gap, not a reduction of the conclusion to its own input. There are no fitted parameters called predictions, no data subsets, and no load-bearing self-citation: the author's prior work [3] appears only as background, while the main equivalence rests on Eq. (5) and standard Dedekind-zeta facts. Therefore no step is equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper uses only standard objects of algebraic number theory and analytic number theory. It introduces no new constants, entities, or ad hoc assumptions. All axioms are classical theorems or conditional hypotheses explicitly stated in the theorems.

assumptions (5)
  • standard math Standard analytic properties of the Dedekind zeta function: meromorphic continuation, functional equation, Euler product (Section 2.1).
    Invoked throughout, e.g., in defining φ_K(s) and the Mellin transform M_f(s).
  • standard math The ideal counting estimate N(x) = κx + O(x^{1-1/n}) (used in Lemma 4.1).
    This is a known theorem for algebraic number fields, cited to [9] and used to derive the generalized Mertens theorem.
  • standard math The class number formula for the residue κ of ζ_K(s) at s=1 (Remark 2.1).
    Used to identify the main constant in the measure m.
  • standard math Phragmén-Lindelöf bounds for ζ_K(s) (Section 2.1.2).
    Used in Lemma 3.3 and Lemma 3.4 to bound the Mellin transform on vertical lines.
  • domain assumption Under the Extended Riemann Hypothesis, the bound ζ_K(2s)^{-1} = O(t^ε) for Re(s) > 1/4 (used in Lemma 3.4).
    This is a conditional assumption within the proof of the 'if' direction of Theorem 1.1(C); it is a standard consequence of ERH.

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Cite this review

Pith. "Pith review of Discrete Measures and the Extended Riemann Hypothesis." pith.science (2026). https://pith.science/paper/XWIQRPNL

@misc{pith2026190803658,
  author       = {Pith},
  title        = {Pith review of: Discrete Measures and the Extended Riemann Hypothesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XWIQRPNL}},
  note         = {Machine review of arXiv:1908.03658}
}
abstract

In this work we show that the Riemann hypothesis for the Dedekind zeta--function $\zeta_{\mathrm{K}}(s)$ of an algebraic number field $\mathrm{K}$ is equivalent to a problem of the rate of convergence of certain discrete measures defined arithmetically on the multiplicative group of positive real numbers to the measure $\zeta_{\mathrm{K}}(2)^{-1}\kappa q dq $, where $\kappa$ denotes the residue of $\zeta_{\mathrm{K}}(s)$ at $s=1$ and $dq$ the Lebesgue measure.

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