REVIEW
On a series related to the M\"obius function
Number Theory · Submitted 16 Oct 2026 · 7 references
Pith's verdict: A clean asymptotic refinement with an explicit constant, using standard methods carefully applied.
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The paper
The paper claims
This paper establishes an asymptotic formula with an explicit constant for a weighted Möbius sum over integers whose smallest prime divisor lies in a fixed arithmetic progression, and under GRH shows that this constant is almost always…
The reading
What carries the argument
The key technical ingredient is Proposition 2, which gives an asymptotic for $V_p(X)=\sum_{n\leq X,\,P^-(n)=p}\mu(n)\omega(n)/n$, namely $V_p(X)=\zeta(1,p)/(p\log X)+O((\log p)^2/(p(\log X)^2))$ uniformly for $p\leq\exp(\sqrt{\log X})$. This is proved using the Selberg–Delange method: the Dirichlet series $F_p(s)$ is expressed in terms of $\zeta(s+1,p)/\zeta(s+1)$ and a Hankel contour integration is performed using the classical zero-free region of the Riemann zeta function. From $V_p(X)$, Theorem 1 follows by orthogonality of Dirichlet characters, splitting the sum over primes into small and large contributions. The GRH analysis of the constants $C(k,l)$ uses Lemma 4 (an identity from the Euler product) and estimates for character sums over primes derived from GRH and an upper bound sieve.
Extended reading notes
Core claim
The paper's main result (Theorem 1) is that for integers $l,k$ with $(l,k)=1$ and large $X$, $$\sum_{\substack{n\leq X\\ P^-(n)\equiv l(\mod k)}} \frac{\mu(n)\omega(n)}{n} = \frac{C(k,l)}{\log X} + O\left(\frac{(\log\log X)^4}{(\log X)^2}\right),$$ where $$C(k,l) = -\frac{1}{\varphi(k)} + \sum_p \left(1_{p\equiv l(\mod k)} - \frac{1}{\varphi(k)}\right)\frac{\zeta(1,p)}{p},$$ and $\zeta(1,p)=\prod_{q\leq p}(1-1/q)^{-1}$. This improves on Tenenbaum's $O$-bound and gives an explicit main term. Under the assumption of GRH (Theorem 5), the paper shows that as $k\to\infty$, the number of reduced residue classes $l$ mod $k$ with $C(k,l)\geq 0$ is $O(\varphi(k)/\log k)$, so that $C(k,l)$ is negative for almost all classes. Thus the leading term is typically negative, consistent with the earlier result that the full series sums to zero.
Load-bearing premise
For Theorem 1, the proof assumes that the Riemann zeta function has no zeros in the region $\sigma \geq 1 - c_0/\log(|t|+2)$ for some absolute constant $c_0$ (the classical zero-free region). For Theorem 5, it assumes the Generalized Riemann Hypothesis for Dirichlet $L$-functions, so that character sums over primes are small enough to control the constant's deviation from its average.
Editorial extensions
If this is right
- The asymptotic formula pinpoints the leading term of the partial sums, showing that they decay like $1/\log X$ with a constant that can be positive, negative, or zero depending on $l$ and $k$.
- The explicit form of $C(k,l)$ allows numerical computation for small $k$, and the paper reports that all computed values for $3\leq k\leq 300$ are non-zero, suggesting that none of these partial sums is identically zero for $X$ large enough.
- Under GRH, the overwhelming negativity of $C(k,l)$ implies that for most moduli and most residue classes, the partial sums are eventually negative, mirroring the sign of the complete series.
- The methods used (Selberg–Delange and character sums) can likely be adapted to similar sums involving the second-largest prime divisor, as hinted in the introduction.
Reading between the lines
- The error term $O((\log\log X)^4/(\log X)^2)$ is likely improvable: the exponent 4 on the double logarithm may be reduced to 1 or less with a refined treatment of the tail sum over primes.
- The explicit constant $C(k,l)$ can be expressed in terms of Mertens' theorem and the partial Euler products; its sign distribution without GRH would require new techniques, as the current proof heavily relies on the uniform character sum bounds that GRH provides.
- Similar weighted sums with $\omega(n)$ replaced by $\omega(n)^r$ or other additive functions may admit analogous asymptotics, possibly with constants that reflect higher-order Selberg–Delange residues.
- The numerical observation that $C(k,l)\neq 0$ for $k\leq 300$ suggests a deeper conjecture: $C(k,l)$ never vanishes for any coprime $l,k$. This would imply that the partial sums are never asymptotically smaller than $1/\log X$.
Original abstract
For a natural number $n>1$ let $P^-(n)$ denote its smallest prime divisor. Alladi and Johnson proved that $\sum_{P^{-}(n)\equiv l \Mod{k}} \frac{\mu(n)\omega(n)}{n} = 0$ for $(l,k)=1$, and Tenenbaum showed that the partial sums of this series up to $X$ are $O_\varepsilon\bigl((\log\log X)^{5/3+\varepsilon}/\log X\bigr)$. In this article, we prove that these partial sums are in fact equal to $C(k,l)/\log X + O\bigl((\log\log X)^4/(\log X)^2\bigr)$, with an explicit constant $C(k,l)$. We also investigate the behaviour of these constants. In particular, we show under GRH that the number of reduced classes $l\Mod{k}$ for which $C(k,l)<0$ is $\varphi(k)(1- o(1))$ as $k\to\infty$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
For a natural number $n>1$ let $P^-(n)$ denote its smallest prime divisor. Alladi and Johnson proved that $\sum_{P^{-}(n)\equiv l \Mod{k}} \frac{\mu(n)\omega(n)}{n} = 0$ for $(l,k)=1$, and Tenenbaum showed that the partial sums of this series up to $X$ are $O_\varepsilon\bigl((\log\log X)^{5/3+\varepsilon}/\log X\bigr)$. In this article, we prove that these partial sums are in fact equal to $C(k,l)/\log X + O\bigl((\log\log X)^4/(\log X)^2\bigr)$, with an explicit constant $C(k,l)$. We also investigate the behaviour of these constants. In particular, we show under GRH that the number of reduced classes $l\Mod{k}$ for which $C(k,l)<0$ is $\varphi(k)(1- o(1))$ as $k\to\infty$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For a natural number $n>1$ let $P^-(n)$ denote its smallest prime divisor. | background 1.00 | 0.12 | 0.01 | asserted 0.86 |
| s1 | Alladi and Johnson proved that $\sum_{P^{-}(n)\equiv l \Mod{k}} \frac{\mu(n)\omega(n)}{n} = 0$ for $(l,k)=1$, and Tenenbaum showed that the partial sums of this series up to $X$ are $O_\varepsilon\bigl((\log\log X)^{5/3+\varepsilon}/\log X\bigr)$. | background 1.00 | 0.02 | 0.03 | asserted 1.00 |
| s2 | In this article, we prove that these partial sums are in fact equal to $C(k,l)/\log X + O\bigl((\log\log X)^4/(\log X)^2\bigr)$, with an explicit constant $C(k,l)$. | result 0.95 | 0.97 | 0.91 | asserted 1.00 |
| s3 | We also investigate the behaviour of these constants. | contribution 1.00 | 0.90 | 0.03 | asserted 0.62 |
| s4 | In particular, we show under GRH that the number of reduced classes $l\Mod{k}$ for which $C(k,l)<0$ is $\varphi(k)(1- o(1))$ as $k\to\infty$. | result 0.99 | 0.95 | 0.13 | asserted 0.99 |
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