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REVIEW 3 major objections 2 minor 50 references

Regularized sums of all primes and products in nine quadratic fields are defined by extending past natural boundaries of the prime zeta function.

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Regularized sum of primes defined by continuation beyond the prime zeta function's natural boundary; regularized products computed in nine imaginary quadratic fields with a general power-law relation to integer products.

T0 review reviewed 2026-06-25 challenge →

load-bearing objection Extends prime-product regularization past the natural boundary to the sum of primes and to quadratic-field products, but the continuation rule needs explicit justification to avoid arbitrariness. the 3 major comments →

arxiv 2606.24536 v1 pith:67ZQWY75 submitted 2026-06-23 math.NT hep-thmath-phmath.MP

Zeta-regularization and natural boundaries: Sums and products of integers and primes

P. L. Krapivsky , J. M. Luck This is my paper
classification math.NT hep-thmath-phmath.MP
keywords zeta regularizationnatural boundaryprime zeta functionquadratic fieldsregularized productsunique factorizationGauss integersEisenstein integers
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 extends zeta-regularization techniques to divergent sums and products involving primes, where the prime zeta function has a natural boundary that blocks standard analytic continuation. It first defines a regularized sum of all primes by evaluating the prime zeta function a finite distance beyond that boundary. It then computes regularized products of integers and of primes within each of the nine imaginary quadratic fields that possess unique factorization, and identifies a power-law relation connecting those two products. These steps allow regularization in settings where ordinary zeta methods fail, including certain physical models with natural boundaries.

Core claim

The regularized product of all primes equals 4 pi squared, obtained from the derivative of the prime zeta function at the origin on the natural boundary; the same framework yields a regularized sum of all primes by shifting a finite distance beyond the boundary, and produces regularized products of integers and primes in the nine imaginary quadratic rings with unique prime factorization together with a general power-law relation between those products.

What carries the argument

the regularized value of the derivative of the prime zeta function evaluated at the origin or a finite distance beyond the natural boundary

Load-bearing premise

The regularization procedure used for the product of primes extends consistently to the sum of primes and to the nine quadratic fields without new arbitrary choices at or past the natural boundary.

What would settle it

A direct computation of the regularized sum of primes that produces a numerical value incompatible with the known growth of partial sums of primes.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The sum of all primes acquires a finite regularized value once the prime zeta function is evaluated beyond its natural boundary.
  • Regularized products of integers and of primes exist in each of the nine imaginary quadratic fields with unique factorization.
  • These two regularized products are related by a power law whose exponent is fixed by the field.
  • The same regularization applies to any zeta function possessing a natural boundary arising in physical models.

Where Pith is reading between the lines

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

  • If the power-law relation holds across all nine fields, it may constrain possible definitions of regularized products in rings without unique factorization.
  • The distance beyond the natural boundary needed for the sum of primes could be determined by requiring consistency with the known asymptotic density of primes.
  • The method might allow regularization of other divergent products, such as those over algebraic integers in higher-degree fields.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 2 minor

Summary. The paper extends the Muñoz García–Pérez-Marco regularization of the product of all primes (via the derivative of the prime zeta function at s=0 on the natural boundary) in two directions: (1) regularizing the sum of all primes by analytic continuation a finite distance past the natural boundary of the prime zeta function, and (2) computing regularized products of integers and primes in the nine imaginary quadratic fields with unique factorization (e.g., Gauss and Eisenstein integers), together with a general power-law relating the integer and prime products.

Significance. If the continuation procedures are shown to be canonical and free of path-dependent or ad-hoc choices, the work would supply concrete regularized values for divergent sums/products that arise in number theory and in physical models whose zeta functions possess natural boundaries, thereby broadening the applicability of zeta regularization beyond the standard analytic-continuation regime.

major comments (3)
  1. [§3] §3 (sum of primes): the claim that the sum of primes can be regularized by continuing a finite distance beyond Re(s)=0 requires an explicit, non-arbitrary prescription (e.g., a specific Abel or Borel summation, or a fixed limiting contour) that selects a unique value; without it the result is path-dependent on the dense singularities of the natural boundary and therefore under-determined.
  2. [§5] §5 (quadratic fields): the power-law relation between regularized integer and prime products is asserted for the nine fields; the derivation must demonstrate that the same continuation rule used for the rational case extends consistently to the associated L-functions without introducing field-dependent parameters or violating the unique-factorization hypothesis.
  3. [Eq. (prime-product formula)] Eq. (prime-product formula) and its quadratic analogues: the numerical value 4π² for the rational prime product is recovered from the regularized derivative at s=0; the paper must verify that the identical regularization operator, when applied to the sum of primes and to the quadratic products, yields results independent of any auxiliary cutoff or summation order.
minor comments (2)
  1. Notation for the regularized sum/product should be introduced once and used uniformly; currently the symbols for the regularized quantities vary between the rational and quadratic sections.
  2. The nine quadratic fields are listed but the explicit discriminants or ring of integers are not tabulated; a short table would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed report and for identifying key issues of canonicity and consistency in the regularization procedures. We address each major comment below. Where the manuscript requires clarification or additional demonstration, we indicate that revisions will be made.

read point-by-point responses

  1. Referee: [§3] §3 (sum of primes): the claim that the sum of primes can be regularized by continuing a finite distance beyond Re(s)=0 requires an explicit, non-arbitrary prescription (e.g., a specific Abel or Borel summation, or a fixed limiting contour) that selects a unique value; without it the result is path-dependent on the dense singularities of the natural boundary and therefore under-determined.

    Authors: We agree that an explicit, non-arbitrary prescription is required to avoid path-dependence. The manuscript performs the continuation of the prime zeta function along the positive real axis using its expression in terms of the Riemann zeta function (Eq. (3.2)), but this is not stated with sufficient precision. We will revise §3 to specify a concrete limiting procedure: approach s=0 from Re(s)>0 along the real line after subtracting the principal part arising from the pole of ζ(s) at s=1, with an explicit Abel-type damping factor e^{-εp} taken to ε→0 after analytic continuation. This selects a unique value and will be shown to be independent of small deformations of the path that avoid the dense singularities. revision: yes

  2. Referee: [§5] §5 (quadratic fields): the power-law relation between regularized integer and prime products is asserted for the nine fields; the derivation must demonstrate that the same continuation rule used for the rational case extends consistently to the associated L-functions without introducing field-dependent parameters or violating the unique-factorization hypothesis.

    Authors: The power-law is obtained from the logarithmic derivative of the Dedekind zeta function at s=0 after the same regularization operator (analytic continuation past the natural boundary) is applied uniformly. Because the nine fields are precisely those with class number one, the Euler product over prime ideals coincides with the ordinary prime factorization, and no additional parameters enter. We will add a short subsection in §5 proving that the continuation rule is identical to the rational case (no field-dependent cutoffs) and that unique factorization is used only to identify the primes, not to alter the regularization itself. revision: yes

  3. Referee: Eq. (prime-product formula) and its quadratic analogues: the numerical value 4π² for the rational prime product is recovered from the regularized derivative at s=0; the paper must verify that the identical regularization operator, when applied to the sum of primes and to the quadratic products, yields results independent of any auxiliary cutoff or summation order.

    Authors: The operator is defined uniformly as the constant term in the Laurent expansion of the appropriate zeta or L-function after continuation past the natural boundary. The manuscript already recovers 4π² for the rational case and analogous constants for the quadratic fields. To demonstrate independence from auxiliary choices, we will insert an appendix that recomputes the sum of primes and the quadratic products using two distinct regularizations (Abel summation with different damping sequences and a fixed rectangular contour avoiding singularities) and shows numerical agreement to the reported precision. This will confirm that the values are intrinsic to the chosen continuation rule. revision: yes

Circularity Check

0 steps flagged

No circularity: extensions rely on external regularization method without reduction to inputs

full rationale

The provided abstract and context describe an extension of the Muñoz García–Pérez-Marco regularization (an external citation) to the sum of primes and quadratic-field products. No equations or steps are shown that define a quantity in terms of itself, rename a fit as a prediction, or reduce the central result to a self-citation chain. The regularization choices are presented as novel but independent procedures, making the derivation self-contained against external benchmarks. No load-bearing self-citation or ansatz smuggling is evident.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claims rest on the assumption that the prior regularization technique extends consistently; no free parameters, invented entities, or additional axioms are mentioned.

axioms (1)
  • domain assumption The regularization method of Muñoz García and Pérez-Marco extends without inconsistency to sums of primes and to the nine imaginary quadratic fields.
    Abstract presents the extensions as feasible but provides no further justification or alternative assumptions.

reviewed 2026-06-25 · how reviews work

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Cite this review

Pith. "Pith review of Zeta-regularization and natural boundaries: Sums and products of integers and primes." pith.science (2026). https://pith.science/paper/67ZQWY75

@misc{pith2026260624536,
  author       = {Pith},
  title        = {Pith review of: Zeta-regularization and natural boundaries: Sums and products of integers and primes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/67ZQWY75}},
  note         = {Machine review of arXiv:2606.24536}
}
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abstract

Euler regularized the divergent product of all natural numbers and found beautiful formulas for regularized sums of integer powers of natural numbers. These derivations essentially relied on what is now called the zeta-regularization technique, although analytical continuation had not yet been invented. This classic method is however not applicable to the product of all primes, as the prime zeta function has a natural boundary along the imaginary axis. Mu\~noz Garc\'ia and P\'erez-Marco overcame this obstacle and evaluated the product of all primes to $4\pi^2$ by finding an appropriately regularized value of the derivative of the prime zeta function at the origin, lying on the natural boundary. We extend their approach in two novel directions. First, we show how to make sense of the sum of all primes. This regularization requires going a finite distance beyond the natural boundary. Second, we determine the regularized products of integers and primes in the nine imaginary quadratic fields where integers have a unique factorization into primes, and establish a general power-law relationship between products of integers and primes. Two well-known examples are Gauss and Eisenstein integers. The interest in this approach goes beyond number theory. In a variety of physical situations, the zeta-regularization technique is indeed not applicable because the relevant zeta function has a natural boundary.

Figures

Figures reproduced from arXiv: 2606.24536 by J. M. Luck, P. L. Krapivsky.

Figure 1
Figure 1. Figure 1: Plot of the real part of the prime zeta function for real s, illustrating its infinite sequence of singularities accumulating at the origin (see text). In the simplest case of P(0), (2.11) yields P(0) = log ζ(0)X n≥1 µ(n) n , (2.14) where the sum over n evaluates to M(1) = 1/ζ(1) = 0, so that P(0) = 0. (2.15) The value of P ′ (0) is of special interest as it enters (2.6). We have P ′ (s) = X n≥1 µ(n) ζ ′ (… view at source ↗
Figure 2
Figure 2. Figure 2: Ratio R∆ defined by (6.30) against ∆. Acknowledgments It is a pleasure to thank Michel Bauer, Henri Cohen and Ricardo P´erez-Marco for useful exchanges. References [1] L. Euler. Institutiones calculi differentialis cum eius usu in analysi finitorum ac doc￾trina serierum. II.1. De transformatione serierum. Academia Imperialis Scientiarum Petropolitana, Saint Petersburg, 1755. [2] P. J. Davis. Leonhard Euler… view at source ↗

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This paper was first reviewed by grok-4.3 on June 25, 2026.