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Complete Asymptotic Expansion of the Additive Mertens Sum $S_k(x)$

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Pith's one-line read The additive sum of reciprocal prime sums admits a complete asymptotic expansion whose coefficients are multiple logarithmic integrals over the unit cube.

desk verdict Solid, carefully executed complete asymptotic expansion for a new additive Mertens sum; the math holds and the limitations are stated honestly. read the letter →

arxiv 2607.04366 v2 pith:Z444PZL4 submitted 2026-07-05 math.NT

Daoyi Peng , Hao Liu This is my paper · ORCID
classification math.NT MSC 11N0511M0633B3041A60
keywords additiveMertenssumasymptoticexpansioncompletehomogeneoussymmetricpolynomialmultiplelogarithmicintegralsprimenumbertheoremDirichletetafunctionpolylogarithms
topics The Twin Prime Conjecture
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

Classical Mertens theorems describe sums of reciprocal primes; their multiplicative generalisations grow like powers of log log x. This paper studies the additive counterpart: the sum of 1/(p1+…+pk) over all primes ≤x. That sum grows like x^{k-1}/(log x)^k. The authors prove that it possesses a full asymptotic series in descending powers of log x, with every coefficient given by an absolutely convergent integral of a complete homogeneous symmetric polynomial against 1/(t1+…+tk) on the unit cube. They evaluate the first two coefficients in closed form for every k, the third coefficient completely for k≤4, and the entire infinite sequence of coefficients when k=2 (as explicit rational combinations of log 2 and zeta values). The result therefore supplies precise leading constants and arbitrarily many secondary terms for an elementary but previously unstudied prime sum.

What carries the argument

The coefficient integrals Ek,n themselves: after the prime-number theorem converts the original Stieltjes integral into a smooth main term, a linear rescaling uj=xtj and a multivariate Taylor expansion of the product of (1+wj)^{-1} factors produce exactly these integrals as the successive coefficients.

What would settle it

Compute the exact double or triple sum Sk(x) by prime convolution for several large x (e.g., 10^6–10^7) and check whether the successive partial sums of the claimed series reduce the relative error by roughly a factor of three each time, matching the explicit numerical tables already given for k=2 and k=3.

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Extended reading notes

Core claim

For every fixed k≥2 and every N≥0 the additive Mertens sum admits the complete expansion Sk(x)=(x^{k-1}/log^k x)∑_{n=0}^N Ek,n/log^n x + O(x^{k-1}/log^{k+N+1}x), where the coefficients are the absolutely convergent integrals Ek,n=(-1)^n ∫_{(0,1]^k} hn(log t1,…,log tk)/(t1+…+tk) dt and hn is the complete homogeneous symmetric polynomial of degree n.

Load-bearing premise

The classical prime-number theorem error is small enough that every term containing at least one residual factor is smaller than any negative power of log x; a weaker error would stop the expansion at finite order.

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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

0 major / 5 minor

Summary. The paper introduces the additive Mertens sum Sk(x)=∑_{p1,...,pk≤x} 1/(p1+⋯+pk) for fixed k≥2 and proves a complete asymptotic expansion Sk(x)=(x^{k-1}/log^k x)∑_{n=0}^N Ek,n/log^n x + O(x^{k-1}/log^{k+N+1}x) for every N≥0. The coefficients are the absolutely convergent integrals Ek,n=(-1)^n ∫_{(0,1]^k} hn(log t)/(∑t) dt with hn the complete homogeneous symmetric polynomial of degree n. Closed forms are obtained for Ek,0 and Ek,1 for all k, for the diagonal part of Ek,2 for all k (and fully for Ek,2 when k≤4), and for the entire sequence {E2,n} as explicit Q-linear combinations of log 2 and zeta values. The argument proceeds from the Stieltjes representation of Sk, the classical PNT error, isolation of the pure main term Mk, rescaling, and multivariate Taylor expansion with controlled remainder; numerical checks for k=2,3 corroborate the first few terms.

Significance. The additive Mertens sum appears not to have been studied previously; the paper supplies a complete asymptotic expansion of arbitrary order together with explicit arithmetic expressions for the first coefficients and a closed formula for the full sequence when k=2. The derivation is elementary (real-variable PNT, Laplace transforms, finite differences, Taylor remainder) and free of fitted parameters. The numerical tables for S2 and S3 provide independent verification of the closed forms. The work therefore fills a natural gap left by the multiplicative theory of Tenenbaum, Popa, Qi–Hu and supplies concrete, computable expansions that can be used in further analytic-number-theoretic estimates.

minor comments (5)
  1. In the abstract and Theorem 1 the O-term is written without the subscript k,N that appears later; a uniform notation would avoid any ambiguity about dependence of the implied constant.
  2. Section 4.3 and the proof of Theorem 2(ii) invoke symbolic verification for k=2….7; a short remark on the computer-algebra system used (or a reference to a supplementary notebook) would strengthen reproducibility.
  3. Proposition 2 for B4 lists a lengthy expression inside the Q-algebra generated by L3; a brief note that the representation is not unique (already mentioned in a footnote) could be moved into the main text for clarity.
  4. Table 1 and the numerical Tables 2–3 are clear, but the caption of Figure 1 could explicitly state that the curves are relative errors |approx-exact|/exact rather than absolute errors.
  5. A few minor typographical inconsistencies appear (e.g., “coeüients” vs. “coefficients”, occasional missing spaces around “=”); a light copy-edit would remove them.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: asymptotic expansion derived from PNT + Taylor remainder; coefficients defined by independent integrals later evaluated in closed form.

full rationale

The central claim (Theorem 1) is obtained by writing Sk as a Stieltjes integral, expanding dπ = du/log u + dR via the classical PNT error, isolating the pure main term Mk, proving every term containing at least one dR is O(x^{k-1} exp(-c'√ log x)) (Lemma 2), rescaling uj = x tj, and expanding the product of 1/(L + log tj) by the multivariate Taylor formula with integral remainder whose remainder is controlled by the absolute convergence of the coefficient integrals (Lemmas 1 and 3). The coefficients Ek,n are defined by those integrals (Definition 1) and are subsequently evaluated by Laplace transforms, generating functions satisfying higher-order ODEs, and finite-difference extraction (Propositions 1–2, Lemmas 4–5, Theorem 3); none of these evaluations is fitted to Sk data or presupposes the expansion. Numerical checks use independent exact convolutions of the prime indicator and are not used to derive any coefficient. There is no self-definitional loop, no fitted parameter renamed as prediction, and no load-bearing self-citation of an unverified uniqueness claim. The derivation is self-contained against the classical PNT error term.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

Pure analytic-number-theory derivation. No free parameters are fitted. The only external inputs are standard theorems (PNT with classical error, Frullani, properties of polylogarithms and finite differences). The objects Sk and Ek,n are definitions, not postulated physical entities. Absolute convergence of the coefficient integrals is proved inside the paper.

assumptions (4)
  • standard math Prime-number theorem with classical error: π(u)-Li(u)≪ u exp(-c√ log u) for some c>0.
    Invoked in §3.1–3.2 to control all error integrals containing at least one dR factor; without it the complete expansion to arbitrary N fails.
  • standard math Frullani integral and the vanishing of the Stirling-number sum that cancels the singularity at zero.
    Used in the evaluation of Ek,0 (Proposition 1).
  • standard math Integral representation of forward differences and the Laplace transforms of U and w that produce G0 and G1.
    Core of the generating-function extraction for Ek,1 and the diagonal of Ek,2 (Lemmas 4–5).
  • domain assumption Absolute convergence of the multiple integrals defining Ek,n for all k≥2, n≥0.
    Proved in Lemma 1 by largest-coordinate decomposition and elementary one-dimensional integrals; required for the remainder estimates.

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Pith. "Pith review of Complete Asymptotic Expansion of the Additive Mertens Sum $S_k(x)$." pith.science (2026). https://pith.science/paper/Z444PZL4

@misc{pith2026260704366,
  author       = {Pith},
  title        = {Pith review of: Complete Asymptotic Expansion of the Additive Mertens Sum $S_k(x)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z444PZL4}},
  note         = {Machine review of arXiv:2607.04366}
}
abstract

Let $p_1, \dotsc, p_k$ be primes not exceeding $x$ ($k \geqslant 2$), and define the additive Mertens sum \[ S_k(x) = \sum_{p_1 \leqslant x} \cdots \sum_{p_k \leqslant x} \frac{1}{p_1 + \dotsm + p_k}. \] In contrast to Tenenbaum's generalized (multiplicative) Mertens sum, whose leading term has order $(\log \log x)^k$, the sum $S_k(x)$ has leading term of order $x^{k-1}/\log^k x$. We establish the complete asymptotic expansion \[ S_k(x) = \frac{x^{k-1}}{\log^k x} \sum_{n=0}^{N} \frac{E_{k,n}}{\log^n x} + O\left(\frac{x^{k-1}}{\log^{k+N+1} x}\right) \quad (\forall\, N \geqslant 0), \] where the coefficients are given by absolutely convergent multiple logarithmic integrals \[ E_{k,n} = (-1)^n \int_{(0,1]^k} \frac{h_n(\log t_1, \dotsc, \log t_k)}{t_1 + \dotsm + t_k}\, \mathrm{d}\mathbf{t}, \] with $h_n$ the complete homogeneous symmetric polynomial of degree $n$. We give closed-form expressions for the first two coefficients $E_{k,0}$ and $E_{k,1}$ for all $k$, and obtain the closed form for the diagonal part of the third coefficient $E_{k,2}$ (with $E_{k,2}$ fully explicit for $k \leqslant 4$); consequently, the first three terms of the expansions of $S_2(x)$ and $S_3(x)$ are fully explicit. For $k = 2$, we further obtain a closed-form expression for the entire sequence $\{E_{2,n}\}_{n \geqslant 0}$, whose values are explicit $\mathbb{Q}$-linear combinations of $\log 2$ and zeta values $\zeta(j)$. The proofs rely on the real-variable form of the prime number theorem, variable rescaling, and multivariate Taylor remainder estimates.

Figures

Figures reproduced from arXiv: 2607.04366 by the authors.

Figure 1
Figure 1. Relative errors of the m-term approximations of S2(x) for m = 1, . . . , 5 over 5 × 105 ⩽ x ⩽ 5 × 106 . Each additional coefficient E2,m−1 reduces the error by roughly one order of magnitude; at m = 5, the error has reached the 0.2% level. Numerical verification for k = 3. To confirm that the three-term expansion in Corollary 2 behaves as predicted, we performed an analogous computation for S3(x). The exact value is… view at source ↗
Figure 1
Figure 1. Relative errors of the m-term approximations of S2(x) for m = 1, . . . , 5 over 5 × 105 ⩽ x ⩽ 5 × 106 . Each additional coefficient E2,m−1 reduces the error by roughly a factor of three; at m = 5, the error has reached the 0.2% level [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗

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Works this paper leans on

10 extracted references · 1 linked inside Pith

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