On the distribution of the strongly multiplicative function $2^{\omega(n)}$ on the set of natural numbers

A. Sankaranarayanan; K. Venkatasubbareddy

REVIEW 3 major objections 5 minor 11 references

On the distribution of the strongly multiplicative function $2^{\omega(n)}$ on the set of natural numbers

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper establishes explicit asymptotic formulas for counting integers by distinct prime factors, with an unconditional error term of order $x^{8/29+\varepsilon}$ and an $O(x^\varepsilon)$ error under the strong Riemann hypothesis.

desk verdict Theorem 2 is a solid unconditional improvement, but Theorem 1's proof has a clear T=x^10 error that needs fixing before the paper is publishable. read the letter →

arxiv 2502.02598 v1 pith:QEHPKLEM submitted 2025-01-21 math.NT

K. Venkatasubbareddy , A. Sankaranarayanan This is my paper · ORCID

classification math.NT MSC 11M0611M26

keywords stronglymultiplicativefunction2^{\omega(n)}RiemannzetaDirichletseriesPerronformulahypothesisexplicitzeros

topics The Riemann Hypothesis

open problems The Riemann Hypothesis

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

The paper studies the summatory function of $2^{\omega(n)}$, where $\omega(n)$ counts the distinct prime factors of $n$; on squarefree integers this function agrees with the divisor function $d(n)$. Its central object is the Dirichlet series $F(s)=\zeta(s)^2/\zeta(2s)$, and the paper proves that the partial sums of $2^{\omega(n)}$ are recovered by collecting the residues of $F(s)x^s/s$ at its poles. The unconditional theorem expresses the sum as $A_1x\log x+A_2x$ plus an explicit sum over the zeros of $\zeta(2s)$ in a strip, with an error term $O(x^{8/29+\varepsilon})$, improving the previous error term of shape $x^{1/2}\exp\{-C(\log x)^{3/5}(\log\log x)^{-1/5}\}$. Under the strong Riemann hypothesis, the zero sum reduces to zeros on the line $\Re(s)=1/4$ and the error is $O(x^\varepsilon)$. A sympathetic reader would care because the proof turns a classical counting problem into an explicit-formula problem whose error is governed by the zeros of the zeta function.

What carries the argument

The central object is the Dirichlet series $F(s)=\sum 2^{\omega(n)}n^{-s}=\zeta(s)^2/\zeta(2s)$. The paper applies Perron's formula to recover the partial sum as a contour integral of $F(s)x^s/s$, then shifts the contour to $\Re(s)=-1/2$ in a rectangle and collects residues at $s=1$, $s=0$, and the zeros of $\zeta(2s)$ that remain after cancellations against zeros of $\zeta(s)^2$. The unconditional improvement comes from choosing the horizontal sides at a special height $T^*\in[T,T+T^{1/3}]$, where both $\zeta$ and $1/\zeta$ are $O_\varepsilon(T^\varepsilon)$, and from bounding the integrand with a subconvexity estimate for $\zeta$; balancing $x^{1/4}T^{-29/84}=x^{1/2}T^{-29/42}$ gives $T=x^{21/29}$ and error $x^{8/29+\varepsilon}$.

What would settle it

Search a range of large $T$ for a height $T^*\in[T,T+T^{1/3}]$ where both $\zeta(\sigma+iT^*)$ and its reciprocal are bounded by $T^\varepsilon$ for $1/2\le\sigma\le2$; if such a height never exists for some large $T$, the horizontal-line estimate behind the unconditional error term collapses.

Watch

Extended reading notes

Core claim

The paper's central claim is that the distribution of $2^{\omega(n)}$ is fully encoded, up to a small remainder, by the quotient $F(s)=\zeta(s)^2/\zeta(2s)$. Applying Perron's formula and shifting the contour to $\Re(s)=-1/2$ gives, via Cauchy's residue theorem, the leading terms $A_1x\log x+A_2x$ plus a sum of residues at the poles of $F(s)x^s/s$ lying in the rectangle; unconditionally the surviving poles are the zeros $\rho=\beta+i\gamma$ of $\zeta(2s)$ with $0\le\beta\le1/2$ and $0<|\gamma|\le x^{21/29}$, and the remainder is $O_\varepsilon(x^{8/29+\varepsilon})$. Assuming the strong Riemann hypothesis, all zeros of $\zeta(2s)$ lie on $\Re(s)=1/4$ and are simple, so the pole sum runs over that line up to a height controlled by $x$, and the error becomes $O(x^\varepsilon)$.

Load-bearing premise

The unconditional error term rests on an imported lemma, not proved in the paper, that just above any large height there is a line where both the zeta function and its reciprocal are bounded by a small power of the height; if that lemma is false, the $O(x^{8/29+\varepsilon})$ bound does not follow.

Editorial extensions

If this is right

The new unconditional error exponent $8/29$ is smaller than the exponent $131/416$ appearing in the best-known Dirichlet divisor problem, so this sum is now known more accurately than the analogous divisor sum.
The main fluctuation in the sum is displayed explicitly as a sum of residues at zeros of $\zeta(2s)$ up to height $x^{21/29}$, instead of being hidden inside an opaque error term.
Under the strong Riemann hypothesis the zero sum collapses to the line $\Re(s)=1/4$ and the error becomes $O(x^\varepsilon)$, with the remaining question being the size of the oscillatory zero sum, which the authors conjecture is $O(x^{1/4+\varepsilon})$.
The contour procedure is designed to transfer to other zeta-quotient Dirichlet series, including those for $\varphi(n)/n$, $d(n^2)$, $d(n)^2$, and the squareful indicator function.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One consequence the paper leaves implicit is that the final exponent $8/29$ is a function of the input subconvexity exponent: a stronger bound than the $13/42$ estimate for $\zeta$ would mechanically lower the error term through the same balancing step.
A natural extension would treat $2^{k\omega(n)}$ for fixed $k\ge2$, whose Dirichlet series is $\zeta(s)^k/\zeta(ks)$; the analogous zero sum would be governed by zeros of $\zeta(ks)$ with a height cutoff determined by the same balancing identities.
A numerical probe of the explicit zero sum could test whether the true error is closer to $O(x^{1/4+\varepsilon})$ than to $O(x^{8/29+\varepsilon})$: averaging the residue sum over dyadic $x$ and checking whether its mean square grows like $x^{1/2+\varepsilon}$ would support the plausible conjecture in the paper's Remark 2.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the paper has one solid result, Theorem 2, and one conditional theorem whose stated error term is not supported by the proof as written. Theorem 1 chooses T=x^10 and then claims an O(x^ε) error, but the vertical integral at Re(s)=-1/2 is O(x^{-1/2}T^{1/2}) = O(x^{9/2+ε}) with that choice. The fix is to take T=x, which makes the vertical term O(1) and the remaining error O(x^ε), but it also changes the zero cutoff from x^10 to x. That is a substantive change to the main term, not a typo, so the theorem statement needs revising.

What is genuinely new: Theorem 2 gives, unconditionally, an asymptotic formula with an explicit sum over poles of ζ(s)^2/ζ(2s) and error O(x^{8/29+ε}). That improves the previous O(x^{1/2} exp{-C (log x)^{3/5}/(log log x)^{1/5}}) error term from the literature. The method is standard Perron contour plus existing zeta estimates, but the decomposition is explicit and the exponent 8/29 is competitive (smaller than 131/416 in the divisor problem). No fitted parameters; the residues are derived. The paper is clearly written and the unconditional argument appears coherent.

Soft spots, in proportion: aside from the Theorem 1 error, the proof of Theorem 2 says "we make a special choice T such that 2T = T*" of Lemma 2.4. That is confusing. Lemma 2.4 gives a good height T* near T; you would normally pick the contour height to be exactly T*, not half of it. If the authors mean the chosen contour height is T*, then the notation is just awkward. If they mean half, the horizontal bounds at height T are not justified because the 1/ζ bound only holds at T*. Needs clarification. The unconditional proof depends on Lemma 2.4 (an external result from [8]) to control 1/ζ(2s) on horizontal lines; this is cited without proof, which is acceptable, but it is load-bearing. The constant ζ(0) is omitted from the main terms but absorbed into the error; minor.

Who this is for: analytic number theorists working on multiplicative functions and zeta zeros. Theorem 2 deserves publication. Theorem 1, once corrected, is a nice conditional explicit formula. I would send this to a serious referee with a request to fix the T choice in Theorem 1 and clarify the height selection in Theorem 2.

Referee Report

3 major / 5 minor

Summary. The paper studies the summatory function of the strongly multiplicative function 2^{\omega(n)}. The authors write its Dirichlet series as F(s)=\zeta(s)^2/\zeta(2s) and apply Perron's formula with a contour shift to \Re(s)=-1/2. Under a strong Riemann hypothesis they claim, in Theorem 1, an asymptotic formula with leading terms A_1 x\log x + A_2 x, a sum over residues at zeros of \zeta(2s) with \Re(s)=1/4 and |\Im(2s)|<x^{10}, and error O(x^\varepsilon). In Theorem 2 they claim an unconditional formula with the same leading terms, a sum of residues over zeros of \zeta(2s) with 0\le\Re(s)\le 1/2 and |\Im(s)|\le x^{21/29}, and error O(x^{8/29+\varepsilon}). The proofs use the functional equation of \zeta, subconvexity bounds, and a good-height lemma from the literature.

Significance. If the conditional result is repaired, it gives a natural analogue of Dirichlet-type expansions for 2^{\omega(n)} under strong RH with very small error O(x^\varepsilon). The unconditional Theorem 2 is the more substantial contribution: after correcting the cutoff issue noted below, it would provide an explicit residue-sum formula with error O(x^{8/29+\varepsilon}), improving on the cited unconditional error of size x^{1/2}\exp\{-C(\log x)^{3/5}/(\log\log x)^{1/5}\}. A genuine strength is that the constants A_1 and A_2 are computed explicitly from residues rather than fitted, and there is no circular dependence: the proof cites Lemma 2.4 from a published paper, and no ad-hoc axioms or invented entities are introduced.

major comments (3)

[Section 3, Eq. (3.1) and subsequent estimates] The proof of Theorem 1 is internally inconsistent at the final choice T=x^{10}. The vertical integral is bounded by |I_1|\ll x^{-1/2+\varepsilon}T^{1/2}, and substituting T=x^{10} gives O(x^{9/2+\varepsilon}), not O(x^\varepsilon). The horizontal terms are O(x^{1/4}T^{-1/2+3\varepsilon})=O(x^{-19/4+\varepsilon}) and O(x^{1+\varepsilon}T^{-1+3\varepsilon})=O(x^{-9+\varepsilon}), and the Perron error is O(x^{-9+\varepsilon}), so the vertical term completely dominates and the claimed O(x^\varepsilon) error in Theorem 1 is not established. The argument can be repaired by taking T=x, which gives |I_1|=O(1) and the other displayed terms O(x^\varepsilon) after renaming \varepsilon, but this changes the zero-sum cutoff in Theorem 1 from x^{10} to x; the statement of Theorem 1 must be corrected accordingly.
[Section 3, estimate of I_2+I_3] In the horizontal estimate for the segment \sigma\in[1/4,1/2], the denominator \zeta(2\sigma+2iT) is controlled by T^{3\varepsilon} using Lemma 2.3. As stated, Lemma 2.3 gives 1/\zeta(\sigma+it)\ll T^\varepsilon only for \sigma>1/2, but for \sigma=1/4 the argument of \zeta(2s) has real part exactly 1/2, so the lemma does not apply uniformly on the closed segment. The authors should either justify this boundary behavior explicitly or choose the contour height via a good-height lemma such as Lemma 2.4, as is done in the proof of Theorem 2.
[Section 4, paragraph 'we make a special choice T such that 2T=T^*'] The statement of Theorem 2 fixes the residue-sum cutoff at |\gamma|\le x^{21/29}, but the proof uses a height T=T^*/2 where T^* is the good height supplied by Lemma 2.4. Lemma 2.4 only guarantees that T^* lies in an interval [T_0,T_0+T_0^{1/3}], not that T^* equals 2x^{21/29} exactly. Since the number and size of the residues with ordinates between the chosen good height and x^{21/29} are not controlled, replacing the actual contour height by x^{21/29} is not automatic. The theorem should either state the cutoff as the chosen good height or give an additional argument showing that the difference is absorbed in O(x^{8/29+\varepsilon}).

minor comments (5)

[Section 3, Eq. (3.1)] The intermediate formula contains the term \zeta(0) from the residue at s=0; this is absorbed into O(x^\varepsilon) in Theorem 1, but the absorption is not stated explicitly and may confuse readers.
[Section 2, Lemma 2.4] Since Theorem 2 depends on Lemma 2.4 for the reciprocal of \zeta on a horizontal line, the authors should give the exact statement and proof location in [8], or include a proof sketch, rather than simply citing the result.
[Theorem 1 and Theorem 2] The notation \gamma_{1/4} in the zero-sum of Theorem 1 is not defined precisely; it should be stated that \gamma_{1/4}=\Im(2\rho) for a zero \rho of \zeta(2s).
[Front matter] There are several typographical issues in the author affiliation and abstract (for example, "Hyderaba d" and "T elangana"), which should be corrected in production.
[Remark 5] Remark 5 lists several further asymptotics that might follow by the same method, but no proofs or precise statements are provided; it should be labeled as speculative or removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the asymptotic formulas are obtained from the Euler product/Perron/residue calculation; the cited Lemma 2.4 is an external zeta-function estimate, not an input-equivalent prediction.

full rationale

The derivation chain is a standard Perron-formula contour integration. F(s)=ζ^2(s)/ζ(2s) is proved directly from the Euler product (Lemma 2.1), and the constants A1, A2 and the coefficients Aγ are computed as residues of F(s)x^s/s (Lemma 2.2), not fitted to the target sum. Theorem 1's zero sum is the residue contribution at zeros of ζ(2s) under the stated strong-Riemann-hypothesis assumption; Theorem 2's residue sum is the same mechanism without the hypothesis, with horizontal integrals controlled by standard zeta bounds. The only overlapping-author citation is Lemma 2.4, taken from [8] (Ramachandra–Sankaranarayanan 1991), used to bound 1/|ζ(2s)| on a horizontal line. That is a published, external, parameter-free estimate about the zeta function; it does not assume or encode the asymptotic being proved and is not derived from it, so under the rules it is real evidence and does not make the argument circular. An apparent arithmetic inconsistency exists in Theorem 1's proof (setting T=x^{10} makes the vertical integral O(x^{9/2+ε}), not O(x^ε)), but that is a correctness defect, not a reduction of the claimed result to its inputs. No fitted-input-as-prediction, no definitional circularity, and no renaming of a known pattern occur.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central formulas are derived by contour integration; no free parameters are fitted to data, and no new entities are postulated. Strong RH is an explicit external assumption, not a derived conclusion. The two cited external lemmas (Bourgain; Ramachandra and Sankaranarayanan) are load-bearing but standard.

assumptions (6)

standard math Euler product and meromorphic continuation of the Riemann zeta function, including the functional equation zeta(s)=chi(s)zeta(1-s) and |chi(s)| ~ |t|^{1/2-sigma}
Invoked in Lemma 2.1 and throughout Sections 3 and 4 to shift the contour and bound F(s).
standard math Perron's formula
Used at the start of the proofs of Theorems 1 and 2 to express the summatory function as a contour integral.
domain assumption Strong Riemann hypothesis: all nontrivial zeros of zeta(s) and zeta(2s) lie on their critical lines and are simple
This is the explicit conditional assumption of Theorem 1; the location and simplicity of poles in the residue sum follow from it.
standard math Lemma 2.4 of Ramachandra and Sankaranarayanan [8]: existence of a horizontal line T* on which 1/zeta(sigma+iT*) is bounded by T^epsilon uniformly for 1/2<=sigma<=2
Cited, not proved in this paper; it supplies the reciprocal-zeta bound needed for the unconditional horizontal integral estimates in Theorem 2.
standard math Bourgain's subconvexity bound for zeta (Lemma 2.5): zeta(sigma+it) << (|t|+10)^{max{13/42(1-sigma),0}+epsilon}
Cited, not proved; used in Theorem 2 to bound the numerator zeta(s)^2 on the horizontal segments.
standard math Residue theorem and Cauchy's theorem for rectangles
Used to collect residues at poles inside the contour.

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Pith. "Pith review of On the distribution of the strongly multiplicative function $2^{\omega(n)}$ on the set of natural numbers." pith.science (2026). https://pith.science/paper/QEHPKLEM

@misc{pith2026250202598,
  author       = {Pith},
  title        = {Pith review of: On the distribution of the strongly multiplicative function $2^\omega(n)$ on the set of natural numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QEHPKLEM}},
  note         = {Machine review of arXiv:2502.02598}
}

abstract

In this paper, we study the distribution of the sequence of integers $2^{\omega(n)}$ under the assumption of the strong Riemann hypothesis, where $\omega(n)$ denotes the number of distinct prime divisors of $n$. We provide an asymptotic formula for the sum $\displaystyle\sum_{n\leq x}2^{\omega(n)}$ under this assumption. We study the sum $\displaystyle\sum_{n\leq x}2^{\omega(n)}$ unconditionally too.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages

[8]

Ramachandra and A

K. Ramachandra and A. Sankaranarayanan, Notes on the Riemann zeta- function, J. Indian Math. Soc. 57 (1991), no.1-4, 67–77

work page 1991
[1]

Ivi´ c,The Riemann Zeta-Function: Theory and Applications , Dover Publi- cations, Inc., Mineola, New Y ork, 2003

A. Ivi´ c,The Riemann Zeta-Function: Theory and Applications , Dover Publi- cations, Inc., Mineola, New Y ork, 2003

work page 2003
[2]

Bourgain, Decoupling, exponential sums and the Riemann zeta function , J

J. Bourgain, Decoupling, exponential sums and the Riemann zeta function , J. Amer. Math. Soc., 30 (2017) 205-224

work page 2017
[3]

Two classical lattice point problems

K. Corr´ adi, and I. K´ atai,A comment on K. S. Gangadharan’s paper entitled “Two classical lattice point problems” , Magyar Tud. Akad. Mat. Fiz. Oszt. K¨ ozl.17 (1967), 89–97

work page 1967
[4]

K. S. Gangadharan, Two classical lattice point problems , Proc. Cambridge Philos. Soc. 57 (1961), 699–721. 15

work page 1961
[5]

J. L. Hafner, New omega theorems for two classical lattice point problems , Invent. Math. 63 (1981), no. 2, 181–186

work page 1981
[6]

M. N. Huxley, Exponential sums and lattice points. III , Proc. London Math. Soc. (3) 87 (2003), no. 3, 591–609

2003
[7]

Jia and A

C. Jia and A. Sankaranarayanan, The mean square of the div isor function. Acta Arith. 164 (2014), no. 2, 181–208

work page 2014

Show all 11 references

[9]

Ramachandra and A

K. Ramachandra and A. Sankaranarayanan, On an asymptoti c formula of Srinivasa Ramanujan. Acta Arith. 109 (2003), no. 4, 349–357

2003
[10]

E. C. Titchmarsh and D. R. Heath-Brown, The Theory of the Riemann Zeta- function, second edition, Clarendon Press, Oxford, 1986

1986
[11]

V enkatasubbareddy and A

K. V enkatasubbareddy and A. Sankaranarayanan, On the distribution of the restricted sequence of integers 2ω(n), Integers 24 (2024), Paper No. A110, 15 pp. 16

2024

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [8]

Ramachandra and A

K. Ramachandra and A. Sankaranarayanan, Notes on the Riemann zeta- function, J. Indian Math. Soc. 57 (1991), no.1-4, 67–77

work page 1991

[2] [1]

Ivi´ c,The Riemann Zeta-Function: Theory and Applications , Dover Publi- cations, Inc., Mineola, New Y ork, 2003

A. Ivi´ c,The Riemann Zeta-Function: Theory and Applications , Dover Publi- cations, Inc., Mineola, New Y ork, 2003

work page 2003

[3] [2]

Bourgain, Decoupling, exponential sums and the Riemann zeta function , J

J. Bourgain, Decoupling, exponential sums and the Riemann zeta function , J. Amer. Math. Soc., 30 (2017) 205-224

work page 2017

[4] [3]

Two classical lattice point problems

K. Corr´ adi, and I. K´ atai,A comment on K. S. Gangadharan’s paper entitled “Two classical lattice point problems” , Magyar Tud. Akad. Mat. Fiz. Oszt. K¨ ozl.17 (1967), 89–97

work page 1967

[5] [4]

K. S. Gangadharan, Two classical lattice point problems , Proc. Cambridge Philos. Soc. 57 (1961), 699–721. 15

work page 1961

[6] [5]

J. L. Hafner, New omega theorems for two classical lattice point problems , Invent. Math. 63 (1981), no. 2, 181–186

work page 1981

[7] [6]

M. N. Huxley, Exponential sums and lattice points. III , Proc. London Math. Soc. (3) 87 (2003), no. 3, 591–609

2003

[8] [7]

Jia and A

C. Jia and A. Sankaranarayanan, The mean square of the div isor function. Acta Arith. 164 (2014), no. 2, 181–208

work page 2014