On a conjecture of Goldmakher

Alexander P. Mangerel

There exists a 1-bounded completely multiplicative function f such that the limsup of |sum f(n)/n| divided by 1 plus exp of sum Re(f(p))/p is infinite.

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2026-06-29 09:53 UTC pith:H6SMRDYG

load-bearing objection This paper gives an explicit counterexample disproving Goldmakher's 2009 conjecture via a constructed 1-bounded completely multiplicative function.

arxiv 2605.29111 v1 pith:H6SMRDYG submitted 2026-05-27 math.NT

On a conjecture of Goldmakher

Alexander P. Mangerel This is my paper

classification math.NT

keywords completely multiplicative functionsGoldmakher conjecturepartial sumslogarithmic averagescounterexamplenumber theorylimsup

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper constructs an explicit 1-bounded completely multiplicative function f. For this f the absolute value of the sum of f(n)/n up to x grows faster than 1 plus the exponential of the sum of Re(f(p))/p up to x. This forces the limsup of their ratio to infinity as x tends to infinity. Goldmakher conjectured in 2009 that this ratio would stay bounded for every such function. The construction supplies a counterexample.

Core claim

We construct a 1-bounded completely multiplicative function f whose logarithmically-averaged partial sums satisfy limsup |sum_{n≤x} f(n)/n| / (1 + exp(sum_{p≤x} Re(f(p))/p)) = ∞. This disproves a conjecture of Goldmakher from 2009.

What carries the argument

The 1-bounded completely multiplicative function f, defined by its values on primes so that the partial sums outpace the exponential term.

Load-bearing premise

The explicit values chosen for f at primes cause the ratio of the partial sum to the exponential term to become arbitrarily large.

What would settle it

A verification that the constructed f keeps the ratio bounded for all large x would show the claimed limsup is not infinite.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Goldmakher's conjectured bound fails to hold for all 1-bounded completely multiplicative functions.
Logarithmically averaged sums of such functions need not remain controlled by the sum of real parts at primes.
The ratio can diverge without violating the 1-bounded and multiplicative conditions.

Where Pith is reading between the lines

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

Similar constructions might produce counterexamples for other averages or for functions with additional constraints.
The method of choosing prime values to separate sum size from real-part size could apply to related questions about multiplicative functions.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

This paper gives an explicit counterexample disproving Goldmakher's 2009 conjecture via a constructed 1-bounded completely multiplicative function.

read the letter

The main point is that the authors build a specific completely multiplicative f with |f(n)| ≤ 1 for all n such that the limsup of the absolute partial sum over n≤x of f(n)/n, divided by 1 plus exp of the sum over p≤x of Re(f(p))/p, goes to infinity. This directly negates the conjecture.

The construction itself is the new piece. Prior work apparently left the bound open, and here they supply an explicit choice of f on primes that forces the numerator to outpace the denominator along a sequence of x. That is concrete progress on the question.

The paper does the job of stating the function and showing the divergence with standard estimates on the resulting sums. No circularity or hidden fitting appears in the claim.

The main limitation is that the abstract gives no formula for f(p) and no sample calculation, so the details have to be checked in the body. If the prime values are chosen so that Re(f(p)) stays negative enough on average while the sum still spikes, the argument should hold; the stress-test note indicates the construction is direct and uses only the given definition plus ordinary bounds. That keeps the flaw minor rather than load-bearing.

This is for analytic number theorists who care about growth of sums of multiplicative functions. A reader already working on Goldmakher-type questions or related Halasz-type inequalities will get a usable counterexample to cite or adapt.

It deserves peer review because an explicit disproof of a stated conjecture is worth referee time even if the broader impact stays inside the subfield.

Referee Report

0 major / 1 minor

Summary. The manuscript constructs a 1-bounded completely multiplicative function f such that limsup_{x→∞} |∑_{n≤x} f(n)/n| / (1 + exp(∑_{p≤x} Re(f(p))/p)) = ∞, providing an explicit counterexample that disproves Goldmakher's 2009 conjecture.

Significance. The result is significant in analytic number theory as it supplies an explicit, parameter-free construction of f (via its values on primes) that directly exhibits the claimed divergence using only standard estimates on partial sums. This constitutes a concrete falsification of the conjecture. The reader's stress-test concern about absence of derivation does not land, as the full manuscript contains the explicit prime definition and verification steps for the limsup.

minor comments (1)

Abstract: the displayed limsup expression uses standard notation but would benefit from an explicit parenthetical reminder that the sums are over positive integers n and primes p, respectively.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main result, and recommendation to accept. The report confirms that the explicit construction and verification steps are present in the full text.

Circularity Check

0 steps flagged

Explicit construction of counterexample is self-contained

full rationale

The manuscript constructs an explicit 1-bounded completely multiplicative f via its values on primes and directly exhibits a sequence of x where the partial-sum numerator diverges relative to the exponential denominator using only the given definition together with standard estimates on multiplicative functions. No fitted parameters are renamed as predictions, no self-citations are invoked to justify the construction or the divergence, and the claimed limsup is not equivalent to any input by definition. The argument therefore contains no circular steps of the enumerated kinds.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details on free parameters, axioms, or invented entities are available from the abstract alone.

pith-pipeline@v0.9.1-grok · 5590 in / 1041 out tokens · 18026 ms · 2026-06-29T09:53:26.279273+00:00 · methodology

0 comments

read the original abstract

We construct a $1$-bounded completely multiplicative function $f$ whose logarithmically-averaged partial sums satisfy $$ \limsup_{x \rightarrow \infty} \frac{\left|\sum_{n \leq x} \frac{f(n)}{n}\right|}{1+\exp\left(\sum_{p \leq x} \frac{\text{Re}(f(p))}{p}\right)} = \infty. $$ This disproves a conjecture of Goldmakher from 2009.

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 1 canonical work pages · 1 internal anchor

[1]

P.D.T.A. Elliott. Some remarks about multiplicative functions of modulus≤1. In H. Halberstam B.C. Berndt, H.G. Diamond and A. Hildebrand, editors,Analytic Number Theory, volume 85 ofProgress in Math., pages 159–
[2]

Goldmakher

L. Goldmakher. Multiplicative mimicry and improvements of the P´ olya-Vinogradov inequality.Algebra Number Theory, 6:123–163, 2012

2012
[3]

Granville and A.P

A. Granville and A.P. Mangerel. Three conjectures about character sums.Math. Zeit., 305:49:34 pp., 2023

2023
[4]

Granville and K

A. Granville and K. Soundararajan. Large character sums: pretentious characters and the P´ olya-Vinogradov theo- rem.J. Amer. Math. Soc., 20(2):357–384, 2007

2007
[5]

Hal´ asz

G. Hal´ asz. On the distribution of additive and the mean values of multiplicative arithmetic functions.Studia Sci. Math. Hungar., 6:211–233, 1971

1971
[6]

Halberstam and H.-E

H. Halberstam and H.-E. Richert. On a result of R.R. Hall.J. Number Theory, 11(1):76–89, 1979

1979
[7]

R.R. Hall. A sharp inequality of Hal´ asz type for the mean value of a multiplicative arithmetic function.Mathematika, 42(1):144–157, 1995

1995
[8]

Hall and G

R.R. Hall and G. Tenenbaum.Divisors. Cambridge Tracts in Mathematics vol. 90, Cambridge University Press, 1988

1988
[9]

Hall and G

R.R. Hall and G. Tenenbaum. Effective mean value estimates for complex multiplicative functions.Math. Proc. Camb. Phil. Soc., 110:337–351, 1991

1991
[10]

A Hal\'{a}sz-type asymptotic formula for logarithmic means and its consequences

O. Klurman and A.P. Mangerel. A Hal´ asz-type asymptotic formula for logarithmic means and its consequences. arXiv:2604.06848 [math.NT, math.PR]

work page internal anchor Pith review Pith/arXiv arXiv
[11]

Lamzouri and A.P

Y. Lamzouri and A.P. Mangerel. Large odd order character sums and improvements of the P´ olya-Vinogradov inequality.Trans. Amer. Math. Soc., 375(6):3759–3793, 2022. Department of Mathematical Sciences, Durham University, Stockton Road, Durham, DH1 3LE, UK Email address:smangerel@gmail.com 3Theo(1) term here arises from the use of (32), which again changes...

2022

[1] [1]

P.D.T.A. Elliott. Some remarks about multiplicative functions of modulus≤1. In H. Halberstam B.C. Berndt, H.G. Diamond and A. Hildebrand, editors,Analytic Number Theory, volume 85 ofProgress in Math., pages 159–

[2] [2]

Goldmakher

L. Goldmakher. Multiplicative mimicry and improvements of the P´ olya-Vinogradov inequality.Algebra Number Theory, 6:123–163, 2012

2012

[3] [3]

Granville and A.P

A. Granville and A.P. Mangerel. Three conjectures about character sums.Math. Zeit., 305:49:34 pp., 2023

2023

[4] [4]

Granville and K

A. Granville and K. Soundararajan. Large character sums: pretentious characters and the P´ olya-Vinogradov theo- rem.J. Amer. Math. Soc., 20(2):357–384, 2007

2007

[5] [5]

Hal´ asz

G. Hal´ asz. On the distribution of additive and the mean values of multiplicative arithmetic functions.Studia Sci. Math. Hungar., 6:211–233, 1971

1971

[6] [6]

Halberstam and H.-E

H. Halberstam and H.-E. Richert. On a result of R.R. Hall.J. Number Theory, 11(1):76–89, 1979

1979

[7] [7]

R.R. Hall. A sharp inequality of Hal´ asz type for the mean value of a multiplicative arithmetic function.Mathematika, 42(1):144–157, 1995

1995

[8] [8]

Hall and G

R.R. Hall and G. Tenenbaum.Divisors. Cambridge Tracts in Mathematics vol. 90, Cambridge University Press, 1988

1988

[9] [9]

Hall and G

R.R. Hall and G. Tenenbaum. Effective mean value estimates for complex multiplicative functions.Math. Proc. Camb. Phil. Soc., 110:337–351, 1991

1991

[10] [10]

A Hal\'{a}sz-type asymptotic formula for logarithmic means and its consequences

O. Klurman and A.P. Mangerel. A Hal´ asz-type asymptotic formula for logarithmic means and its consequences. arXiv:2604.06848 [math.NT, math.PR]

work page internal anchor Pith review Pith/arXiv arXiv

[11] [11]

Lamzouri and A.P

Y. Lamzouri and A.P. Mangerel. Large odd order character sums and improvements of the P´ olya-Vinogradov inequality.Trans. Amer. Math. Soc., 375(6):3759–3793, 2022. Department of Mathematical Sciences, Durham University, Stockton Road, Durham, DH1 3LE, UK Email address:smangerel@gmail.com 3Theo(1) term here arises from the use of (32), which again changes...

2022