Thinned Wallis-type prime products in residue classes modulo $2^m$

Mike Winkler

arxiv: 2602.13371 · v3 · submitted 2026-02-13 · 🧮 math.GM

Thinned Wallis-type prime products in residue classes modulo 2^m

Mike Winkler This is my paper

Pith reviewed 2026-05-15 22:42 UTC · model grok-4.3

classification 🧮 math.GM

keywords Wallis productprime productsresidue classesDirichlet L-functionsMertens constantsarithmetic progressionsthinned productsquadratic character

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The pith

Thinned products of factors A(p) over primes in residue classes mod 2^m converge to a finite nonzero limit precisely when the classes satisfy a balance condition.

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

The paper studies infinite products over odd primes p of the factor A(p) = (p - χ4(p))/(p + χ4(p)), where χ4(p) equals 1 or -1 according to whether p is 1 or 3 mod 4, but only including primes that lie in a fixed union of residue classes modulo 2^m. It supplies an explicit criterion on those classes that guarantees the restricted product tends to a positive finite constant rather than zero or infinity. In the remaining cases it establishes that the partial products up to x behave asymptotically like C times a power of log x, and it writes the prefactor C in terms of Mertens-type products taken inside the chosen arithmetic progressions, which in turn are expressed using values of Dirichlet L-functions at s=1. A reader cares because the construction isolates how the quadratic character χ4 interacts with the distribution of primes in dyadic residue classes, turning a classical Wallis-type product into a family of thinner products whose convergence is completely controlled by modular arithmetic.

Core claim

For odd primes p define A(p) = (p − χ4(p))/(p + χ4(p)) with χ4(p) = 1 if p ≡ 1 mod 4 and −1 if p ≡ 3 mod 4. When the product of A(p) is taken only over primes belonging to a union of residue classes modulo 2^m, the infinite product converges to a finite nonzero value if and only if the selected classes obey a simple balance condition; otherwise the partial product up to primes ≤ x satisfies a logarithmic asymptotic whose leading term is determined by the density of the classes, and the multiplicative constant in that asymptotic is given explicitly by a ratio of Mertens constants in the relevant arithmetic progressions, equivalently by a ratio of Dirichlet L-values at s=1.

What carries the argument

The factor A(p) built from the non-principal character χ4 mod 4, restricted to primes in a union of residue classes mod 2^m, whose partial products are analyzed via the prime-number theorem in arithmetic progressions and the analytic continuation of the associated Dirichlet L-functions.

If this is right

The product converges to a nonzero constant precisely when the number of selected classes congruent to 1 mod 4 equals the number congruent to 3 mod 4, up to an explicit adjustment for the power of 2.
In all other cases the logarithm of the partial product grows like α log log x for an explicit density constant α depending only on the chosen classes.
The limiting constant (when it exists) equals a finite ratio of products of the form ∏ (1−1/p) taken inside each arithmetic progression, which reduces to a ratio of L(1,χ) values.
All such constants are therefore computable to arbitrary precision from known values of Dirichlet L-functions at s=1.

Where Pith is reading between the lines

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

The same thinning technique could be applied to other quadratic characters to produce families of products whose limits are controlled by different L-functions.
Numerical verification of the asymptotic for moderate moduli such as 2^6 or 2^7 would give a direct check on the size of the implied constants.
The criterion may extend to products taken over primes in unions of classes modulo other fixed integers, provided the corresponding Dirichlet density exists.

Load-bearing premise

The primes are asymptotically distributed among the residue classes modulo 2^m according to the densities predicted by Dirichlet's theorem.

What would settle it

Compute the partial product of A(p) over primes up to 10^12 in a specific unbalanced union of classes mod 32 and check whether its logarithm deviates from the predicted linear growth in log log x by more than a few percent.

read the original abstract

For odd primes $p$ we consider the factors \[ A(p)=\frac{p-\chi_4(p)}{p+\chi_4(p)}, \qquad \chi_4(p)= \begin{cases} 1,&p\equiv 1\pmod 4, \\ -1,&p\equiv 3\pmod 4, \end{cases} \] and study products of $A(p)$ restricted to unions of residue classes modulo $2^m$. We give a simple criterion for the existence of a finite nonzero limit, prove a logarithmic asymptotic in the general case, and express the limiting constant in terms of Mertens-type constants in arithmetic progressions (hence in terms of Dirichlet $L$-values).

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.

Desk Editor's Note private letter to a colleague

Winkler gives a clean arithmetic criterion for when thinned Wallis products over primes in residue classes mod 2^m converge to a nonzero finite limit, expressed via standard Mertens constants in APs.

read the letter

The paper works out the convergence behavior of products of (p - chi4(p))/(p + chi4(p)) when the primes are restricted to unions of residue classes modulo 2^m. It supplies a simple condition on the classes that makes the weighted sum of chi4 over those classes vanish, which cancels the divergent log-log term and leaves a finite nonzero limit. When the condition fails, it gives the logarithmic asymptotic. The limit is written explicitly in terms of the Mertens constants C(a, 2^m) for the individual classes, which reduce to products involving Dirichlet L-values at s=1. All of this rests on the prime number theorem in arithmetic progressions and the known non-vanishing of L(1, chi) for the characters modulo 2^m; no extra hypotheses are needed. The argument is direct and uses only established machinery, so the derivations look routine once the setup is in place. The main limitation is that the result is a targeted extension rather than a new method; the criterion itself is basically the vanishing of a finite sum over the residue classes, which is easy to check but does not open broad new territory. The citation pattern is appropriate and points back to the classical sources for Mertens theorems in APs. This is useful for anyone who needs explicit constants for such restricted prime products, for instance in work on generalizations of the Wallis product or on Euler products with character twists. A reader who already knows the standard results on primes in APs will find the explicit form and the convergence test immediately applicable. The paper deserves a serious referee because the claims are stated precisely and rest on unconditionally proven tools; any gaps would be in the details of the bookkeeping rather than in the overall logic.

Referee Report

0 major / 3 minor

Summary. The manuscript examines products ∏ A(p) over odd primes p in unions of residue classes modulo 2^m, where A(p)=(p−χ4(p))/(p+χ4(p)) and χ4 is the non-principal character mod 4. It states a simple arithmetic criterion for convergence to a finite nonzero limit, proves a logarithmic asymptotic when the product diverges, and expresses the limiting constant explicitly via Mertens-type constants C(a,2^m) in arithmetic progressions (hence via Dirichlet L-values at s=1).

Significance. If the derivations hold, the work extends the classical Wallis and Mertens products to thinned prime sets in residue classes mod 2^m, with the limit expressed in terms of standard, independently defined constants from the literature. The unconditional reliance on the prime-number theorem in arithmetic progressions and the non-vanishing of L(1,χ) for non-principal characters mod 2^m is a strength, as is the explicit arithmetic criterion for the vanishing of the divergent log-log coefficient.

minor comments (3)

[§2] §2: the precise definition of the admissible unions of residue classes modulo 2^m should be stated as a numbered condition or lemma before the main theorem, to make the criterion immediately checkable.
[Eq. (3.2)] Eq. (3.2): the logarithmic asymptotic is stated with an error term O(1); a brief indication of how the error is controlled (via the prime-number theorem in APs) would improve readability.
[Table 1] Table 1: the numerical values of the limiting constants for small m should include a column comparing against direct partial products to illustrate convergence rate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment of its significance in extending classical Wallis and Mertens products to thinned prime sets in residue classes modulo 2^m. The unconditional use of the prime-number theorem in arithmetic progressions and the explicit arithmetic criterion for convergence are strengths we appreciate being highlighted. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation applies the prime-number theorem in arithmetic progressions modulo 2^m and the analytic continuation/non-vanishing of Dirichlet L-functions (standard, externally proven results) to obtain a convergence criterion for the thinned product and an explicit expression for the limit via pre-existing Mertens constants C(a,2^m) and L(1,χ) values. These constants are defined independently in the literature by convergent Euler products; the paper does not redefine them or fit parameters to the target product. No self-definitional steps, fitted-input predictions, load-bearing self-citations, or smuggled ansatzes appear in the given derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard analytic number theory tools without introducing new free parameters or postulated entities.

axioms (1)

standard math Prime number theorem in arithmetic progressions and analytic properties of Dirichlet L-functions
Invoked to establish the logarithmic asymptotic and to express the limiting constant in terms of known quantities.

pith-pipeline@v0.9.0 · 5407 in / 1383 out tokens · 39849 ms · 2026-05-15T22:42:23.730487+00:00 · methodology

discussion (0)

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

log A(p) = −2 χ4(p)/p + O(1/p^3); log PS(x) = −2 μ(S)/φ(q) log log x + C(q,S) + o(1)

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