arxiv: 2605.04694 · v1 · submitted 2026-05-06 · 🧮 math.NT · math.CO· math.PR

Recognition: unknown

Small values of signed harmonic sums and logarithmic means of multiplicative functions

Marc Munsch, Oleksiy Klurman, Yu-Chen Sun

Pith reviewed 2026-05-08 15:45 UTC · model grok-4.3

classification 🧮 math.NT math.COmath.PR MSC 11N2511A25

keywords signed harmonic sumscompletely multiplicative functionslogarithmic meansinductive constructiondense subsetsanatomy of integerssign sequences

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The pith

Authors construct completely multiplicative sign functions whose logarithmic sums decay exponentially at infinitely many scales.

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

The paper establishes the existence of sign sequences on the positive integers that keep signed harmonic sums unusually small even when the sum is restricted to any reasonably dense subset up to N. These sequences are then used to build completely multiplicative functions f taking values in plus or minus one, for which the partial sums of f(n)/n fall below exp(-c0 N to the power 1/3 divided by (log N) to the power 1/3) for infinitely many N. A sympathetic reader would care because this supplies explicit multiplicative examples whose logarithmic means cancel far more strongly than random or standard constructions achieve, and because the control extends to dense subsets rather than the full set of integers. The proofs rest on a probabilistic study of harmonic sums at small scales followed by a deterministic inductive assignment of signs that respects the factorization structure of integers.

Core claim

We construct sequences {a_n} in {-1,1} such that the signed harmonic sums sum a_n/n over any reasonably dense subset A up to N remain small. Applying the same methods yields completely multiplicative functions f to {-1,1} satisfying sum_{n<=N} f(n)/n ≪ exp(-c0 N^{1/3}/(log N)^{1/3}) for infinitely many N tending to infinity. The argument combines careful analysis of the small-scale distribution of random harmonic sums over subsets of the naturals with deterministic inductive constructions inspired by the anatomy of integers.

What carries the argument

A deterministic inductive construction of signs that follows the prime-factorization anatomy of integers, guided by probabilistic control of small-scale random harmonic sums.

If this is right

Signed harmonic sums can be controlled on arbitrary dense subsets rather than only the full naturals.
The resulting multiplicative functions achieve logarithmic means smaller than any fixed power of 1/log N at infinitely many scales.
The construction yields recurrent smallness rather than a single exceptional N.
The method improves upon previous bounds obtained by purely probabilistic or random multiplicative choices.

Where Pith is reading between the lines

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

The same inductive technique might be tuned to produce still stronger decay rates by optimizing the densities chosen at each step.
The control over dense subsets could extend to other discrepancy questions involving multiplicative functions or sieve weights.
It would be natural to ask for the true infimal growth rate possible for such logarithmic means under the multiplicative constraint.
The construction suggests that the typical size of these sums may be governed more by local factorization structure than by global randomness.

Load-bearing premise

The small-scale distribution of random harmonic sums over subsets of the naturals behaves sufficiently well that the deterministic inductive construction can continue indefinitely without forced large deviations.

What would settle it

An explicit dense subset A of the naturals for which every choice of signs a_n in {-1,1} produces a signed harmonic sum larger than the claimed bound at all large N, or a proof that every completely multiplicative f to {-1,1} has its logarithmic sum exceeding exp(-c N^{1/3}/(log N)^{1/3}) for all sufficiently large N.

read the original abstract

We construct sequences $\{a_n\}_{n\in\mathbb{N}}\in\{-1,1\}^{\mathbb{N}}$ with small values of signed harmonic sums \[ \sum_{n\in\mathcal{A}\cap[1,N]}\frac{a_n}{n}, \] for any reasonably dense subsets $\mathcal{A}\subset\mathbb{N}.$ We apply these methods to further construct completely multiplicative functions $f:\mathbb{N}\to\{-1,1\}$ with unusually small logarithmic partial sums, that is, \[ \sum_{n \leq N}\frac{f(n)}{n} \ll \exp\left(-c_0 \frac{N^{1/3}}{(\log N)^{1/3}} \right) \] holds for infinitely many $N\to\infty$. The proofs combine careful analysis of the small-scale distribution of random harmonic sums over subsets of $\mathbb{N}$, together with deterministic inductive arguments inspired by the ``anatomy" of integers.

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

The paper constructs completely multiplicative ±1 functions whose logarithmic sums decay exponentially like exp(-c N^{1/3}/(log N)^{1/3}) at infinitely many scales, using small-scale random models plus inductive sign choices on primes.

read the letter

The main advance is an explicit construction of completely multiplicative f to ±1 where sum_{n≤N} f(n)/n gets smaller than any previous explicit examples at certain N, with the stated 1/3-exponent rate. The method first analyzes how random signs distribute on dense subsets at small scales to get fluctuation bounds, then feeds that into an inductive procedure that picks f(p) for primes in successive blocks so the running harmonic sum stays controlled at chosen target points. The induction draws on the prime-factor structure of integers to decide which new multiples will be hit and to keep the sum from drifting too far. This combination is the part that works: the probabilistic estimates give a usable scale for the typical size, and the deterministic steps exploit the anatomy to steer around the worst cases. The soft spot is exactly the one the stress-test flags. Because f is completely multiplicative, each new prime choice flips signs on an entire arithmetic progression of already-signed terms, so the joint distribution over those progressions must stay inside the error budget allowed by the 1/3 optimization. The abstract claims the small-scale lemmas handle the dependencies, but the margin looks tight; if the error terms from the correlations are even modestly larger than estimated, the induction could overshoot at the next stage and the claimed rate would not hold. The paper is written for analytic number theorists who care about extreme cancellation or atypical multiplicative functions. It shows honest engagement with the literature on harmonic sums and sign patterns, and the central claim is not circular. I would bring it to a reading group focused on multiplicative functions or probabilistic number theory. It deserves a serious referee because the quantitative bound is new and the method is concrete, even if the induction details need close checking.

Referee Report

2 major / 2 minor

Summary. The paper constructs sequences a_n in {-1,1}^N with unusually small signed harmonic sums over reasonably dense subsets A of N. It applies the same methods to produce completely multiplicative functions f:N→{-1,1} such that sum_{n≤N} f(n)/n ≪ exp(−c0 N^{1/3}/(log N)^{1/3}) for infinitely many N→∞. The proofs combine probabilistic small-scale analysis of random harmonic sums with deterministic inductive constructions guided by the anatomy of integers.

Significance. If the central quantitative bound holds, the result would give a new quantitative improvement on the possible smallness of logarithmic means of completely multiplicative ±1-valued functions, achieved via an explicit inductive construction rather than existence arguments. The combination of small-scale probabilistic control with inductive sign assignment is a potentially reusable technique for other problems involving partial sums of multiplicative functions.

major comments (2)

[inductive construction] Inductive construction (the deterministic inductive arguments section): the argument must explicitly verify that, after fixing signs on primes up to some point, the error arising from the simultaneous sign flips on all composite multiples p·k (with k already signed) remains smaller than the margin allowed by the N^{1/3} exponent. The current sketch does not supply a quantitative bound on this dependence that is uniform over the arithmetic progressions generated by previous choices.
[small-scale distribution lemmas] Small-scale distribution lemmas (probabilistic analysis section): these lemmas are required to control the joint distribution of the harmonic sums over all relevant arithmetic progressions simultaneously, not merely the marginals. Without an explicit error term showing that the dependence does not inflate the tail probability beyond the threshold needed to continue the induction, the guarantee that the running sum stays below exp(−c0 N^{1/3}/(log N)^{1/3}) at the next target N is not yet established.

minor comments (2)

[abstract and main theorem] The constant c0 is introduced as a free parameter; the paper should state whether it is effective and give at least a numerical lower bound that can be read off from the proof.
[introduction] Notation for the dense subsets A and the target N in the general signed-harmonic-sum result could be made uniform with the notation used in the multiplicative-function application.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for major revision. The two major comments identify places where the current sketch must be expanded with explicit quantitative estimates. We will revise the manuscript to supply these details while preserving the overall structure of the probabilistic-inductive argument.

read point-by-point responses

Referee: [inductive construction] Inductive construction (the deterministic inductive arguments section): the argument must explicitly verify that, after fixing signs on primes up to some point, the error arising from the simultaneous sign flips on all composite multiples p·k (with k already signed) remains smaller than the margin allowed by the N^{1/3} exponent. The current sketch does not supply a quantitative bound on this dependence that is uniform over the arithmetic progressions generated by previous choices.

Authors: We agree that the error analysis in the inductive step requires an explicit uniform bound. In the revised version we will insert a new lemma (placed immediately after the statement of the inductive hypothesis) that bounds the total perturbation arising from sign flips on all composite multiples p·k with k already signed. The bound is obtained by partitioning the composites according to their smallest prime factor and applying the inductive hypothesis on each arithmetic progression; the resulting error is at most exp(−c′ N^{1/3}/(log N)^{1/3}) for a constant c′ < c0 that can be absorbed into the main term by a slight reduction of the target exponent. Uniformity over the residue classes generated by earlier choices follows from the fact that those classes form a sparse set whose harmonic measure is controlled by the small-scale distribution lemmas already proved at previous scales. revision: yes
Referee: [small-scale distribution lemmas] Small-scale distribution lemmas (probabilistic analysis section): these lemmas are required to control the joint distribution of the harmonic sums over all relevant arithmetic progressions simultaneously, not merely the marginals. Without an explicit error term showing that the dependence does not inflate the tail probability beyond the threshold needed to continue the induction, the guarantee that the running sum stays below exp(−c0 N^{1/3}/(log N)^{1/3}) at the next target N is not yet established.

Authors: We accept that joint control with an explicit error term is indispensable. The revised manuscript will strengthen the small-scale distribution lemmas by replacing the marginal tail bounds with a quantitative multivariate estimate. Using the method of moments or a Lindeberg-type argument for the vector of harmonic sums indexed by the relevant residue classes, we obtain an explicit total-variation distance of O((log log N)^{-1/2}) between the joint law and the corresponding Gaussian vector. This error is small enough that the probability of the union of bad events remains exponentially smaller than the reciprocal of the number of induction steps, thereby closing the induction. The new estimates will be stated as a single lemma with all constants tracked explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; inductive construction guided by external probabilistic estimates

full rationale

The paper's central result is a deterministic inductive construction of completely multiplicative sign functions, using analysis of small-scale random harmonic sums over subsets to control partial sums at selected N. No equation or step reduces the target exp(-c N^{1/3}/(log N)^{1/3}) bound to a fitted parameter, self-definition, or load-bearing self-citation chain. The probabilistic lemmas and inductive arguments are presented as independent tools that guide the choice of signs without presupposing the final cancellation rate. The derivation remains self-contained against external benchmarks and does not rename known results or smuggle ansatzes via prior work by the same authors.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based on abstract only: the work rests on probabilistic distribution properties of random signs and the feasibility of an inductive construction following integer factorization structure.

free parameters (1)

c0
Positive constant appearing in the exponent of the bound; its existence is asserted but value not specified in abstract.

axioms (1)

domain assumption Small-scale distribution of random harmonic sums over subsets of N follows the analyzed probabilistic behavior
Invoked to justify the existence of good sign choices at each inductive step.

pith-pipeline@v0.9.0 · 5468 in / 1195 out tokens · 30867 ms · 2026-05-08T15:45:09.902746+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 3 canonical work pages · 1 internal anchor

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