Additivity gives \(f(1)=0\), and we set \(f(0)=0\) only for the telescoping

notation used by Mangerel.

Write

This is Mangerel's \(B_f(X)^2\); I use \(V_f\) to avoid confusing it with a

descent set.

Proposition 3.1 — an exact sufficient criterion

Claim (b). If

then \(f(n)=c\log n\) for every \(n\), for some real \(c\ge0\).

Proof. Equation (51) of Mangerel, specialized to \(Y=X\), gives

For completeness, its mechanism is transparent:

After summation, the first term telescopes. Mangerel's Lemma 3.5 bounds

the endpoint by \(V_f(X)\log X/\sqrt X\), while Cauchy–Schwarz and the

Turán–Kubilius inequality bound the sum supported on \(\mathcal D\) by

\(V_f(X)\sqrt{\rho(X)}\). Condition (C) makes the left side of (1)

\(o(1)\). Mangerel's Theorem 3.7 (attributed there to Kátai and Wirsing)

then gives \(f=c\log\) identically. Finally \(c<0\) would make every

consecutive step a descent, so the original density-zero hypothesis forces

\(c\ge0\). ∎

This criterion is not the conjecture: \(\rho=o(1)\) alone gives no useful

control over the product \(V_f\sqrt{\rho}\).

Corollary 3.2 — bounded variance is settled

Claim (b). If \(V_f(X)=O(1)\) and the original hypothesis

\(\rho(X)=o(1)\) holds, then \(f\equiv0\).

Indeed, (C) holds, so Proposition 3.1 gives \(f=c\log\). For \(c\ne0\),

the prime terms make

\(V_{c\log}(X)\asymp |c|\log X\) (the standard prime-number-theorem

estimate also recorded in Mangerel), contradicting boundedness. Hence

\(c=0\).

This special case is consistent with Erdős's Theorem X and is included here

because it cleanly identifies where a counterexample would have to live:

its Turán–Kubilius scale \(V_f(X)\) must diverge.

4. An implicit extension of Mangerel's stated corollary

Define the higher-prime-power part

and

In the published paper \(f\in\mathcal A_s\) means precisely

Theorem 4.1 — ordinary additive \(\mathcal A_s\) functions

Claim (b). Let \(f:\mathbb N\to\mathbb R\) be additive, not necessarily

completely additive. Assume (2), and suppose that for some fixed

Then \(f(n)=c\log n\) identically, with \(c\ge0\).

Derivation checked line by line.

1. Theorem 1.8 applies to every additive \(f\in\mathcal A_s\), so it

supplies \(\lambda(X)\) with

\(V_{f-\lambda(X)\log}(X)=o(V_f(X))\) and

\(|\lambda(X)|\ll V_f(X)/\log X\).

2. Lemma 8.7 therefore gives, for every fixed \(\varepsilon>0\),

\[ V_f(X)\ll_\varepsilon(\log X)^{1+\varepsilon}. \tag{4} \]

3. Take \(\varepsilon=\delta/3\) in (4). Combining (3) with (1) gives

\[ \frac1X\sum_{n\le X}|f(n)-f(n-1)| \ll (\log X)^{1+\delta/3-(2+\delta)/2} +\frac{(\log X)^{2+\delta/3}}{\sqrt X}. \]

The first exponent is exactly \(-\delta/6\); the standalone checker

also verifies this arithmetic over rational numbers. Both terms tend

to zero.

4. The Kátai–Wirsing theorem again gives \(f=c\log\).

Mangerel states Corollary 1.7 for completely additive \(f\), because

complete additivity plus \(V_f\to\infty\) is an easy sufficient condition

for membership in \(\mathcal A\) (Lemma 3.6). Once membership in

\(\mathcal A_s\) is assumed explicitly, complete additivity is not used in

the proof. Theorem 4.1 is therefore an explicit, verified corollary of the

published proof for a broader class of ordinary additive functions. I did

not find it separately advertised in the paper, and I do not claim that

this observation closes the original problem.

5. Exact finite-prefix computation

This section gives a sharp concrete result, with rational primal and dual

certificates.

For \(N\ge1\), assign one unrestricted real variable

\(x_{p^k}=f(p^k)\) to every prime power \(p^k\le N+1\). Unique

factorization and additivity give

Thus every prefix condition \(f(n+1)\ge f(n)\) is a rational linear

inequality. Conversely, any assignment to these finitely many variables

extends to a global additive function by assigning arbitrary values to all

remaining prime powers. Normalize \(f(2)=1\) and optimize \(f(3)\).

Claim (a). This finite LP model is exactly equivalent to the stated

prefix problem: (5) proves necessity, and assigning the remaining

prime-power values proves sufficiency.

The checker asks HiGHS only to locate a candidate. It converts the full

primal vector and all dual multipliers to Fraction, then independently

checks:

• every prefix inequality exactly;
• \(f(2)=1\) exactly;
• the required sign of every dual multiplier;
• exact dual stationarity in every prime-power coordinate; and
• exact equality of primal and dual objectives.

Consequently the displayed endpoints are sharp exact computer-assisted

theorems, not rounded LP output.

No exceptional position

Claim (d). Among all additive \(f\) with \(f(2)=1\) and

\(f(n+1)\ge f(n)\) for every \(1\le n\le N\), the exact range endpoints

for \(f(3)\) are:

| \(N\) | exact lower bound | exact upper bound | interval width |

|---:|---:|---:|---:|

| 20 | \(4/3\) | \(2\) | 0.666666666667 |

| 50 | \(10/7\) | \(7/4\) | 0.321428571429 |

| 100 | \(61/40\) | \(33/20\) | 0.125000000000 |

| 200 | \(242/155\) | \(129/80\) | 0.051209677419 |

| 500 | \(134/85\) | \(1087/682\) | 0.017371053993 |

| 1000 | \(574/363\) | \(251/158\) | 0.007340377306 |

| 2000 | \(94221/59509\) | \(18630/11743\) | 0.003170323444 |

Claim (a). A logarithmic solution normalized by \(f(2)=1\) has

\(f(3)=\log3/\log2\). Numerically,

lies strictly inside the \(N=2000\) certified interval. The table proves

only these finite statements; it does not prove that the endpoints converge.

At most one exceptional position

For each possible exceptional \(e\in[1,N]\), I removed only the inequality

at \(e\), solved the two LPs, and exactly certified them. Taking the best

endpoint over all \(e\) therefore covers precisely all prefixes with at

most one descent.

Claim (d).

| \(N\) | exact lower bound | exact upper bound | lower endpoint descends at | upper endpoint descends at |

|---:|---:|---:|---:|---:|

| 50 | \(11/8\) | \(9/5\) | 14 | 17 |

| 100 | \(3/2\) | \(5/3\) | 50 | 57 |

| 200 | \(115/74\) | \(97/60\) | 122 | 89 |

Here is an explicit lower-endpoint construction for \(N=50\). Give the

prime powers \(q\le51\) the following values, and extend arbitrarily on all

other prime powers:

q:       2    3     4     5     7     8     9     11    13    16
f(q):    1   11/8  15/8  17/8  21/8  11/4  23/8  25/8   7/2  15/4

q:      17    19    23    25    27   29   31   32    37   41  43   47    49
f(q):  15/4  31/8  33/8  33/8  9/2  9/2  9/2  9/2  39/8  5   5   41/8  41/8

Equation (5) defines the resulting additive function. Direct exact

evaluation gives \(f(n+1)\ge f(n)\) for \(1\le n\le50\), except

and \(f(3)=11/8\). The dual certificates for all 50 choices of the

exceptional position prove that no smaller \(f(3)\) is possible in this

normalized regime.

6. Standalone re-verifier

The checker is

erdos1122_wave8g_verify.py, SHA-256

2945be633a2d428dec8251730aca96fba5e75a922bb75dcf0071e9f258952ad4.

Run from the repository root:

python runs/erdos1122_wave8g_verify.py

It completed in 5.4 seconds on this VM and printed

ALL EXACT CERTIFICATES VERIFIED. In addition to all LP certificates, it

recomputes factorizations from a home-built smallest-prime-factor sieve,

checks the exponent \(-\delta/6\), and checks on three finite

prime-power models the coefficient-free involution

That last check is a sanity test for the symmetry behind Erdős's Theorem X,

not a replacement for the theorem.

7. Exact wall

Diagnosis (b). The work above does not solve #1122. The checked

theorems and inequalities identify two precise analytic gaps.

7.1 The density-rate barrier even inside \(\mathcal A_s\)

For \(f\in\mathcal A_s\), Mangerel obtains

The best directly available absolute-gap estimate is still (1):

The original assumption \(\rho(X)=o(1)\) does not imply that the first

term is \(o(1)\). The exact missing lemma sufficient to finish this route

is

under only \(\rho(X)=o(1)\), or any replacement strong enough to invoke

Kátai–Wirsing. Mangerel explicitly notes after Corollary 1.7 that obtaining

(M) under weaker decay of \(|\mathcal D(X)|/X\) is the unresolved issue.

The Cauchy–Schwarz route loses the square root and naturally creates the

exponent \(2\) in (3).

7.2 Large prime spikes outside \(\mathcal A_s\)

Theorem 1.8 needs \(F_f(\varepsilon)\to0\). Without it, a sparse set of

primes with \(|f(p)|\gg V_f(X)\) can dominate the second moment exactly on

their sparse sets of multiples, defeating the short-interval

\(\ell^2\) estimate. A full solution must either:

1. prove from almost-everywhere monotonicity itself that these large-prime

contributions cannot occur; or

2. develop a replacement for the \(\ell^2\)/Cauchy–Schwarz step that is

insensitive to such spikes.

Mangerel's unconditional Theorem 1.9 gives scale-dependent

\(\lambda(X),\eta(X)\) such that

for all but \(o(X)\) integers \(n\le X\), with slow variation on scales

\(X\mapsto X^u\). This does not stabilize \(\lambda\) to a constant or

remove \(\eta\): the exceptional set need not simultaneously avoid

\(n,m,nm\), so additivity cannot yet be applied to all three values. This

is the other exact uniformity failure.

7.3 Why larger finite computation is not the missing step

The LP table is useful as an exact local benchmark, but every finite prefix

admits non-logarithmic perturbations around \(\log n\); a finite rational

polyhedron cannot supply the needed asymptotic stabilization. The current

dense implementation at \(N=10^5\) would use roughly \(N\pi(N)\) doubles,

about 7.7 GB before solver overhead, and exceed the permitted few-minute

budget. (c, resource forecast only) A sparse implementation could

probably produce a larger finite table with a budget of roughly 1–5

core-hours, but it would not prove (M) or control large prime spikes. I

therefore did not run heavier computation.

8. Verified outcome

• (b) A clean sufficient criterion is

\(V_f(X)(\sqrt{\rho(X)}+\log X/\sqrt X)=o(1)\).

• (b) The original conjecture holds in the bounded-\(V_f\) regime,

necessarily with \(f\equiv0\).

• (b) Mangerel's quantitative conclusion extends verbatim from

completely additive functions to all ordinary additive

\(f\in\mathcal A_s\).

• (d) Exact sharp normalized prefix intervals were certified through

\(N=2000\) with no descents and through \(N=200\) with one allowed

descent.

• The full density-zero question remains open; the exact missing work is

an average-gap lemma such as (M), together with control of the

large-prime-spike regime.

PARTIAL: Certified sharp finite prefix bounds and an ordinary-additive extension of Mangerel's sparse-descent theorem; #1122 remains open at the average-gap and large-prime-spike barriers.