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arxiv: 2604.17832 · v1 · submitted 2026-04-20 · 🧮 math.NT

Duality Between Prime Factors and The Prime Number Theorem For Arithmetic Progressions -- Higher Order Dualities

Krishnaswami Alladi , Sroyon Sengupta This is my paper

Pith reviewed 2026-05-10 04:15 UTC · model grok-4.3

classification 🧮 math.NT
keywords Möbius functionomega functionprime factor dualityarithmetic progressionsPrime Number Theoremdensity theoremshigher-order identitiesquantitative forms
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The pith

Higher-order dualities between prime factors imply that sums of the Möbius function times powers of the number of distinct prime factors equal zero.

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

The paper extends duality results between the largest and smallest prime factors of integers to all higher orders k. It proves that for each k at least 2 the sum from n equals 2 to infinity of mu of n times omega of n to the k divided by n equals zero. Using the general duality identity and the prime number theorem for arithmetic progressions it further shows that sums restricted to numbers whose smallest prime factor lies in a given arithmetic progression also vanish when multiplied by omega to the k minus 1. These are presented in quantitative form and recast as density statements equaling one over the Euler totient function. A reader would care because the results connect the distribution of prime factors directly to classical theorems on primes in arithmetic progressions.

Core claim

Using the k-th order duality identity from Alladi's 1977 paper, which involves the Möbius function and the (k-1)th power of omega(n), the authors show that the sum over all n of mu(n) omega(n)^k / n equals zero for k greater than or equal to 2. They further establish that for coprime j and l the sum over n with smallest prime factor congruent to j modulo l of mu(n) omega(n)^{k-1} / n equals zero for k at least 3, and equivalently that the sum with the binomial coefficient choose k-1 of omega minus one equals (-1)^k over phi(l). All statements are proved with explicit error terms.

What carries the argument

The k-th order duality identity relating the k-th largest and k-th smallest prime factors via the Möbius function and omega(n) to the power k-1.

If this is right

  • For any k greater than or equal to 2 the series sum from n equals 2 to infinity of mu(n) omega(n)^k over n equals zero.
  • For every k greater than or equal to 3 and coprime integers j and l the restricted sum over n with smallest prime factor congruent to j modulo l of mu(n) omega(n)^{k-1} over n equals zero.
  • The restricted sum satisfies (-1)^k times the sum of mu(n) times binomial(omega(n)-1, k-1) over n equals one over phi(l).
  • All identities hold with explicit quantitative error terms derived from the prime number theorem in arithmetic progressions.

Where Pith is reading between the lines

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

  • The vanishing sums provide a way to express the average of powers of omega(n) under the Möbius weight as zero, which may connect to other weighted averages in analytic number theory.
  • The residue-class formulation could be tested computationally for small k to verify the density equals one over phi(l) within the error bounds.
  • Similar duality constructions might apply to sums involving other arithmetic functions of the prime factors.

Load-bearing premise

The higher-order duality identities established in Alladi (1977) hold without modification for the quantitative forms, and the Prime Number Theorem for Arithmetic Progressions applies directly to the residue-class sums involving ω(n)^{k-1}.

What would settle it

Numerical computation of partial sums up to a large bound such as 10^7 of mu(n) omega(n)^3 / n to check whether the value is within the explicit error term of zero predicted by the quantitative form.

read the original abstract

In 1977, the first author observed a duality between the largest and smallest prime factors of integers, and established as a consequence some new results on the M\"obius function $\mu(n)$ using the Prime Number Theorem for Arithmetic Progressions. In that 1977 paper, higher order dualities were observed involving the $k$-th largest and $k$-th smallest prime factors, facilitated by the M\"obius function and $\omega(n)^{k-1}$, where $\omega(n)$ is the number of distinct prime factors on $n$. In 2024, the first author and Jason Johnson proved new results involving $\mu(n)$ and $\omega(n)$, by exploiting the second order duality identity of Alladi (1977). We establish here extensions to all higher orders $k$, the results of Alladi (1977) and of Alladi-Johnson (2024), by utilizing the $k$-th order duality in Alladi's 1977 paper. First, we show that for each $k\geq 2$, $$ \sum_{n=2}^{\infty} \frac{\mu(n)\omega(n)^{k}}{n} =0, $$ where $\mu(n)$ is the M\"obius Function and $\omega(n)$ counts the number of distinct prime factors of $n$. Further, using the General Duality Identity and the Prime Number Theorem of Arithmetic Progressions, we prove that for integers $j,\ell$ satisfying $1 \leq j \leq \ell$ and $(j,\ell)=1$ $$ \sum_{\substack{n=2 \\ p_1(n) \equiv j\;(mod\;\ell)}}^{\infty} \frac{\mu(n)\omega(n)^{k-1}}{n}=0, \nonumber $$ for every $k \geq 3$; this result for $k=1$ is due to Alladi (1977) and for $k=2$ due to Alladi-Johnson (2024). We also recast this result in the following manner as a density-type theorem: for integers $j,\ell$ satisfying $1 \leq j \leq \ell$ and $(j,\ell)=1$ $$ (-1)^k\sum_{\substack{n=2 \\ p_1(n) \equiv j\;(mod\;\ell)}}^{\infty} \frac{\mu(n){\omega(n)-1 \choose k-1}}{n}=\frac{1}{\varphi(\ell)}, \nonumber $$ for every $k \geq 3$. All results are established here in quantitative form.

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 / 0 minor

Summary. The paper extends Alladi's 1977 duality results and the 2024 Alladi-Johnson work to arbitrary order k by invoking the k-th order duality identity from the 1977 paper. It claims three main results, all in quantitative form: (i) ∑_{n≥2} μ(n) ω(n)^k / n = 0 for each k≥2; (ii) for coprime j,ℓ the restricted sum ∑_{n: p_1(n)≡j mod ℓ} μ(n) ω(n)^{k-1}/n =0 for k≥3; and (iii) the binomial recasting (-1)^k ∑_{n: p_1(n)≡j mod ℓ} μ(n) binom(ω(n)-1,k-1)/n = 1/φ(ℓ) for k≥3. The proofs are said to combine the General Duality Identity with the Prime Number Theorem for arithmetic progressions.

Significance. If the quantitative derivations are complete and the error terms are controlled, the work supplies a uniform higher-order generalization of the known duality identities, yielding new vanishing and density statements for Möbius sums weighted by powers or binomial coefficients in ω(n) and restricted to residue classes of the smallest prime factor. This would strengthen the link between prime-factor dualities and the PNT in AP, but the significance remains conditional on the missing quantitative details.

major comments (2)
  1. Abstract and the statements of the main theorems: the quantitative versions of the k-th order duality identities are applied directly to the conditionally convergent series ∑ μ(n) ω(n)^{k-1}/n summed over arithmetic progressions on p_1(n). The manuscript must supply the explicit error-term estimates or uniform bounds that justify passing the PNT for AP through these weighted identities, because the 1977 identities were originally derived formally and the standard PNT for AP does not automatically furnish the required uniformity over the residue classes when polynomial weights in ω(n) are present.
  2. The claim that all results hold 'in quantitative form' (abstract) is load-bearing for the extension beyond the k=1,2 cases already in the literature. Without the detailed derivation of the remainder terms when the General Duality Identity is inserted into the residue-class sums, it is impossible to verify that the error is o(1) uniformly in j,ℓ and k.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments correctly identify that our claim of quantitative forms for the higher-order results requires explicit justification of error terms when combining the General Duality Identity with the Prime Number Theorem in arithmetic progressions. We address each point below and will revise the manuscript accordingly to supply the missing uniform estimates.

read point-by-point responses

  1. Referee: Abstract and the statements of the main theorems: the quantitative versions of the k-th order duality identities are applied directly to the conditionally convergent series ∑ μ(n) ω(n)^{k-1}/n summed over arithmetic progressions on p_1(n). The manuscript must supply the explicit error-term estimates or uniform bounds that justify passing the PNT for AP through these weighted identities, because the 1977 identities were originally derived formally and the standard PNT for AP does not automatically furnish the required uniformity over the residue classes when polynomial weights in ω(n) are present.

    Authors: We agree that the 1977 identities were formal and that direct substitution into conditionally convergent series over residue classes of p_1(n) requires care. The General Duality Identity is an exact finite-sum relation for each n; when summed over the indicated arithmetic progressions it produces an expression whose main term is controlled by the PNT in AP and whose error is the sum of the PNT error weighted by the polynomial in ω(n). Because the PNT in AP holds with error o(x/φ(ℓ)) uniformly for ℓ fixed (or growing slowly) and the weight ω(n)^{k-1} grows slower than any positive power of log n on average, the contribution of the error term remains o(1) after summation. We will add an explicit lemma deriving the o(1) remainder, using the known effective error in the PNT for AP together with a truncation argument that bounds the tail where ω(n) exceeds a fixed multiple of log log x. This establishes uniformity in j for each fixed coprime pair (j,ℓ) and each fixed k. revision: yes

  2. Referee: The claim that all results hold 'in quantitative form' (abstract) is load-bearing for the extension beyond the k=1,2 cases already in the literature. Without the detailed derivation of the remainder terms when the General Duality Identity is inserted into the residue-class sums, it is impossible to verify that the error is o(1) uniformly in j,ℓ and k.

    Authors: The manuscript asserts quantitative forms but presents the combination of the General Duality Identity with the PNT in AP at a high level. For each fixed k the polynomial degree is fixed, so the required uniformity in j,ℓ follows from the standard uniformity of the PNT in AP for moduli up to any fixed bound (or slowly growing with the truncation level). We will insert a new subsection that carries out the remainder analysis in full, showing that the total error after inserting the duality identity is bounded by the PNT error term plus a negligible tail controlled by the density of integers with large ω(n). This will make the o(1) statement verifiable and will also clarify that the uniformity holds for each fixed k (with the implied constant depending on k). revision: yes

Circularity Check

0 steps flagged

Minor self-citation to 1977 duality identities with independent derivation

full rationale

The paper derives the higher-order sums and density theorems by invoking the k-th order duality identities from Alladi (1977) together with the external Prime Number Theorem for Arithmetic Progressions. These inputs are prior independent results rather than redefined or fitted within the present manuscript; the new claims for k ≥ 3 follow as logical consequences without internal reduction to tautology. The self-citation is to foundational prior work by the first author and is not load-bearing in the sense that the quantitative extensions retain separate content from the cited identities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on previously published duality identities and the standard Prime Number Theorem for Arithmetic Progressions rather than introducing new fitted constants or postulated entities.

axioms (2)
  • domain assumption Prime Number Theorem for Arithmetic Progressions
    Invoked to evaluate the residue-class sums involving the smallest prime factor p1(n).
  • domain assumption k-th order duality identities from Alladi (1977)
    Used as the foundation to extend the Möbius-function sums to arbitrary k.

pith-pipeline@v0.9.0 · 5801 in / 1420 out tokens · 45484 ms · 2026-05-10T04:15:48.467116+00:00 · methodology

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Reference graph

Works this paper leans on

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