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arxiv: 2605.01412 · v1 · submitted 2026-05-02 · 🧮 math.NT

A logarithmic structure theorem for multiplicative functions with small partial sums

Dimitris Koukoulopoulos This is my paper

Pith reviewed 2026-05-09 17:44 UTC · model grok-4.3

classification 🧮 math.NT
keywords multiplicative functionssmall partial sumsstructure theoremDirichlet serieszeroslogarithmic intervalsvon Mangoldt convolutionprime averages
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The pith

Multiplicative functions whose partial sums are small must have their prime values averaging close to negative integers on a chain of logarithmic intervals.

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

The paper aims to classify the possible behaviors of multiplicative functions that stay small when summed up to x. Under a bound on the partial sums that improves with higher powers of the logarithm and a control on the convolution with the von Mangoldt function limiting the complexity to D, it establishes the existence of transition points Q_j. At these scales, the real part of f at primes settles near -j for stretches of the log scale. This matters because it pins down what such small functions can look like at primes, potentially aiding in their explicit construction or exclusion in number theoretic problems. The result also ties these scales to the zeros of the generating Dirichlet series when the sum bound is sharper.

Core claim

For multiplicative functions f with |sum_{n≤x} f(n)| ≤ x (log Q)^{A-D-1} / (log x)^A for x≥Q and |Λ_f|≤DΛ, there exist m in 0 to D and Q_D ≤ ⋯ ≤ Q_m <∞ such that for j=m to D and compact I subset [Q_{j+1}, Q_j), the sum_{p in I} Re(f(p)+j)/p is O_{A,D}(1). When the sum bound is x^{1-1/log Q}/(log x)^{D+1}, the parameters m and Q_j are related to the zeros of sum f(n) n^{-s} inside the ball of radius 1/log Q centered at 1.

What carries the argument

The sequence of scales Q_j and the index m, which divide the positive reals into intervals on which the average Re(f(p) + j) for primes p is bounded by a constant.

If this is right

  • Such functions exhibit at most D changes in their average behavior at primes as one moves to larger scales.
  • The averages of Re(f(p)) over primes in logarithmic intervals are controlled, implying limited oscillation.
  • Stronger partial sum bounds allow the scales Q_j to be located using the zeros of the associated Dirichlet series near s=1.
  • The theorem extends the known structure for D=1 to higher D, broadening the class of functions to which it applies.

Where Pith is reading between the lines

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

  • This structure may allow one to deduce that f is pretentious to a specific character or power in each interval.
  • The result could be applied to bound the number of such functions or to study their Dirichlet series more precisely.
  • One testable extension is to verify the relation between Q_j and zeros for explicit examples satisfying the bounds for small D.

Load-bearing premise

The key premise is that the multiplicative function satisfies |Λ_f| ≤ D times the von Mangoldt function, which is necessary to ensure the prime values are controlled enough for the logarithmic structure to emerge.

What would settle it

Observing a multiplicative function that meets the partial sum bound but for which Re(f(p) + j) summed over primes in some large interval I within one of the supposed ranges grows larger than any constant would falsify the theorem.

read the original abstract

Let $D\in\mathbb{N}$, let $A>D+1$, and let $Q\geqslant3$. Consider the class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ such that $|\sum_{n\leqslant x}f(n)|\le x(\log Q)^{A-D-1}/(\log x)^A$ for all $x\geqslant Q$, and such that $|\Lambda_f|\leqslant D\Lambda$, where $\Lambda_f$ is defined via the Dirichlet convolution identity $f\log=\Lambda_f*f$ and $\Lambda$ denotes von Mangoldt's function. We prove there exist parameters $m\in\{0,1,\dots,D\}$ and $Q=Q_D\leqslant Q_{D-1}\le \cdots\leqslant Q_m<Q_{m+1}=\infty$ such that $\sum_{p\in I} \mathrm{Re}(f(p)+j)/p=O_{A,D}(1)$ for all $j=m,m+1,\dots,D$ and all compact intervals $I\subset[Q_{j+1},Q_j)$. Moreover, when $|\sum_{n\leqslant x}f(n)|\le x^{1-1/\log Q}/(\log x)^{D+1}$ for all $x\geqslant Q$, we relate the parameters $m$ and $Q_j$ to the location of zeroes of the Dirichlet series $\sum_{n\geqslant1} f(n)/n^s$ in the ball $B(1,1/\log Q)$. These results generalize work of the author when $D=1$. Their proof builds on earlier work of the author with Soundararajan, and of Sachpazis.

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

0 major / 2 minor

Summary. The paper proves a structure theorem for multiplicative functions f:ℕ→ℂ satisfying |∑_{n≤x} f(n)| ≤ x (log Q)^{A-D-1}/(log x)^A for all x≥Q (with A>D+1, Q≥3) and |Λ_f|≤DΛ, where Λ_f arises from the convolution f log = Λ_f * f and Λ is the von Mangoldt function. It establishes the existence of m∈{0,…,D} and a non-increasing sequence Q=Q_D≤⋯≤Q_m<∞=Q_{m+1} such that ∑_{p∈I} Re(f(p)+j)/p = O_{A,D}(1) for each j=m,…,D and every compact interval I⊂[Q_{j+1},Q_j). Under the stronger hypothesis |∑_{n≤x} f(n)|≤x^{1-1/log Q}/(log x)^{D+1}, the parameters m and Q_j are related to the location of zeros of the Dirichlet series ∑ f(n)n^{-s} inside the disk B(1,1/log Q). The result generalizes the author's D=1 case and builds on prior joint work with Soundararajan and on Sachpazis.

Significance. If correct, the theorem supplies a precise logarithmic-scale decomposition of the prime values of such f, controlled by the order D of the Λ_f bound. This refines the understanding of multiplicative functions whose partial sums are smaller than the trivial bound, with direct implications for the analytic properties of their Dirichlet series. The explicit linkage between the structural parameters (m,Q_j) and zero locations under the stronger hypothesis is a notable strengthening. The inductive construction on D appears to be a natural and technically substantive extension of the D=1 theory.

minor comments (2)
  1. [§1] §1 (Introduction): the statement of the main theorem could usefully include a brief reminder of the precise definition of Λ_f via the convolution identity, even though it is standard, to improve readability for readers outside the immediate subfield.
  2. [Abstract / final theorem] The transition from the partial-sum hypothesis to the zero-location statement in the final paragraph of the abstract (and presumably in the corresponding theorem) relies on standard growth estimates; a short sentence indicating which lemma or standard reference supplies the necessary bound on |∑ f(n)n^{-s}| would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our work, as well as the recommendation for minor revision. The report correctly identifies the main theorem, its generalization from the D=1 case, and the connection to zero locations under the stronger hypothesis. No specific major comments or requests for changes are provided in the report.

Circularity Check

0 steps flagged

Minor self-citation of prior D=1 case but derivation remains independent

full rationale

The central theorem establishes the existence of m and the sequence Q_j directly from the partial-sum bound and the hypothesis |Λ_f| ≤ DΛ via an inductive argument on D. This does not reduce any claimed prediction to a fitted input or self-definition. The paper notes that the result generalizes the author's own D=1 work and builds on joint work with Soundararajan and on Sachpazis, but these citations supply supporting lemmas rather than load-bearing uniqueness theorems or ansatzes that would force the conclusion. The zero-location statement in the ball B(1,1/log Q) follows from standard Dirichlet-series growth estimates controlled by the same Λ_f bound. No step equates a derived quantity to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the standard definition of multiplicative functions, the Dirichlet convolution identity defining Λ_f, and the uniform partial-sum hypothesis; no free parameters are fitted, no new entities are postulated, and the Q_j are existential outputs of the theorem rather than inputs.

axioms (3)
  • domain assumption f is multiplicative
    The class of functions under consideration is defined to be multiplicative.
  • domain assumption The partial-sum bound holds for all x ≥ Q
    This is the primary hypothesis that triggers the structure theorem.
  • domain assumption |Λ_f| ≤ D Λ
    The bound on the von Mangoldt-like function derived from the convolution identity f log = Λ_f * f.

pith-pipeline@v0.9.0 · 5603 in / 1661 out tokens · 29747 ms · 2026-05-09T17:44:16.940855+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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    S. Sachpazis,On multiplicative functions with small partial sums.Int. Math. Res. Not. IMRN 2024, no. 4, 2937– 2964. D ´EPARTEMENT DE MATH ´EMATIQUES ET DE STATISTIQUE, UNIVERSIT ´E DEMONTR ´EAL, CP 6128SUCC. CENTRE-VILLE, MONTR ´EAL, QC H3C 3J7, CANADA Email address:dimitris.koukoulopoulos@dms.umontreal.ca

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