Erdős problem #1144: live audit, exact square-kernel reduction, and exhaustive finite data

Access/search date: 2026-07-28 UTC.

Result in one paragraph

The mandatory collision gate did not fire: the live page says OPEN, has 0 claimed proofs, and lists Currently working on this problem None. I did not prove the requested almost-sure positive limsup. I obtained three fully checked pieces of progress:

1. an exact reduction

\[ \frac{1}{\sqrt N}\sum_{n\le N}F(n) = \sum_{a\le N}\frac{g(a)}{\sqrt a} - \frac1{\sqrt N}\sum_{a\le N}g(a) \left\{\sqrt{\frac Na}\right\}, \tag{1} \] where \(g\) is the squarefree-supported Rademacher function formed from the same prime signs;

1. a rigorous grid lemma showing exactly what positive large-fluctuation

theorem for the first term in (1) would suffice, while the fractional-part term costs only \(L^2\)-norm at most \(1\) at each deterministic \(N\);

1. exhaustive, non-Monte-Carlo data for every one of the \(2^{18}=262144\)

prime-sign assignments through \(N=64\), plus an exact variance formula and its asymptotic \[ \operatorname{Var}\!\left(\sum_{n\le N}F(n)\right) =\frac{6}{\pi^2}N\log N+O(N). \]

The exact missing ingredient is directional and pathwise: current theorems give sign changes and upper bounds, but not growing positive fluctuations of the critical weighted squarefree sum on a grid where (1)'s error can be controlled. Nothing below claims that the finite data proves an asymptotic trend.

Claim labels used throughout:

• [A] elementary-rigorous;
• [B] rigorous modulo an explicitly named theorem;
• [C] plausible/structural-unverified;
• [D] computational-only.

0. Mandatory live-page gate

I fetched the live page and its LaTeX view with a Bright Data residential browser, including a full-page screenshot and the rendered DOM. This was necessary because the site is Cloudflare-walled to ordinary datacenter access.

The rendered live state was:

• status OPEN;
• 0 comments on this problem;
• 0 claimed proofs for this problem;
• Interested in collaborating None;
• Currently working on this problem None;
• This problem looks difficult None;
• This problem looks tractable None;
• last edited 26 January 2026.

Thus neither the claimed-proof nor the current-worker stop condition applied.

Verbatim live LaTeX statement

Let $f$ be a random completely multiplicative function, where for each prime $p$ we independently choose $f(p)\in \{-1,1\}$ uniformly at random. Is it true that\[\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N}}=\infty\]with probability $1$?

Everything mathematical listed on the live page

The page says that this is sometimes called a Rademacher function, while warning that “Rademacher function” is also used for the different model which is zero on non-squarefree integers. It points to problem #520 for that model and also contrasts it with the completely multiplicative Steinhaus model, whose prime values are uniform on the unit circle.

The page's one cited result is:

Atherfold \([{\rm At25}]\) has proved that, almost surely, \[ > \sum_{m\leq N}f(m)\ll N^{1/2}(\log N)^{1+o(1)}. > \]

Its bibliography identifies [At25] as Christopher Atherfold, Almost sure bounds for weighted sums of Rademacher random multiplicative functions, arXiv:2501.11076 (2025). The marker and reference transcription in this section are [A] as statements about what the authoritative page displayed.

1. Primary-source literature audit

Original source

The source marker [Va99,1.11] is real. A scan of the 1999 conference booklet Some of Paul's Favorite Problems contains this question as problem 1.11. The scan defines the independent prime signs, extends them completely multiplicatively, and asks for the same almost-sure infinite limsup after division by \(\sqrt N\). The scan is poor, so the live page, as required by the task, remains the transcription used as ground truth. [A]

Aymone: unweighted sign changes, but no amplitude

Marco Aymone, Sign changes of the partial sums of a random multiplicative function II, Comptes Rendus Mathématique 362 (2024), 895–901, DOI 10.5802/crmath.615, proves in Theorem 1.2 that for a Rademacher random completely multiplicative \(F\),

changes sign infinitely often almost surely for every \(0\le\alpha<1/2\). At \(\alpha=0\), this applies to the exact unweighted sum in problem #1144. It does not give a lower bound for the size of either sign, so it does not imply the requested normalized limsup. [B]

Atherfold: current upper bound and the exact obstruction

The current PDF of Atherfold, arXiv:2501.11076v4 is dated 3 February 2026. In that paper “Rademacher random multiplicative function” means the squarefree-supported model \(g\), not the completely multiplicative target \(F\). Its Theorem 1 proves, for every \(\varepsilon>0\), almost surely

Theorem 3 supplies only an absolute lower bound of order \((\log\log x)^{-1/2}\) at arbitrarily large \(x\); that bound tends to zero and is not a growing positive fluctuation. [B]

Atherfold's equations (1.7)–(1.8) explicitly pass between \(g\) and the completely multiplicative \(F\). Corollary 1 gives

almost surely, and partial summation gives the live page's unweighted \(\sqrt x\) times essentially one logarithm upper bound. Immediately after equation (1.7), the paper says that the fractional-part sum is difficult to control because its weight breaks the multiplicative structure. Equation (1) below is the normalized form of precisely that issue. [B]

The critical weighted completely multiplicative sum

Rodrigo Angelo and Max Wenqiang Xu, Oscillations of random multiplicative functions under initial bias, arXiv:2411.14447v2 (24 March 2026), prove that

changes sign infinitely often almost surely. This closes the weighted sign-change question that Aymone had left open, but again provides no growing one-sided amplitude for the unweighted sum in #1144. [B]

I searched the exact limsup expression, the phrases “random completely multiplicative”, “unweighted sums of completely multiplicative Rademacher functions”, and recent 2025–2026 large-fluctuation papers. I verified the primary sources above, including their current versions, but found no paper claiming the positive infinite normalized limsup in #1144. This is an honest search result, not a proof of bibliographic completeness. [C]

2. Exact square-kernel decomposition

To avoid the terminology collision, use the following notation throughout:

• \((\varepsilon_p)_p\) are independent uniform signs;
• \(F\) is the target completely multiplicative function,

\(F(p)=\varepsilon_p\);

• \(g\) is the squarefree-supported function

\[ g(a)=\mu^2(a)\prod_{p\mid a}\varepsilon_p. \]

Every positive integer \(n\) has a unique representation

The parity of every prime exponent is carried by \(a\), so

The sum in (3) has exactly one nonzero term. Therefore, for \(S_F(N):=\sum_{n\le N}F(n)\),

These identities are [A].

Writing \(\lfloor y\rfloor=y-\{y\}\) in (4) gives the central exact reduction

where

No approximation has occurred in (5). [A]

Orthogonality and the inexpensive fixed-\(N\) error

For squarefree \(a,b\),

Indeed, if \(a\ne b\), some prime occurs in exactly one of them and its independent sign has mean zero. Consequently,

This is [A]. Notice carefully that (8) is a fixed-\(N\) statement; it does not by itself control a supremum over many or randomly selected \(N\).

A precise sufficient grid lemma

Let \(\mathcal T_k\) be deterministic finite sets of positive integers and let \(u_k\to\infty\). If

then (8), Markov's inequality, and the union bound give

The first Borel–Cantelli lemma therefore implies, almost surely,

No independence between different \(N\)'s is needed. [A]

It follows immediately from (5) that #1144 would have a positive answer if one could construct \((\mathcal T_k,u_k)\) satisfying (9) such that, almost surely for infinitely many \(k\),

At such \(k\), (10) gives some \(N\in\mathcal T_k\) with

Thus (LF) is an explicit sufficient missing lemma, not a reformulation with the fractional error swept aside. [A]

Current lower bounds quoted above do not establish (LF): they are absolute rather than positive, do not grow, and do not provide the deterministic-grid uniformity needed for (9). [B]

3. Exact mean, variance, and fourth-moment object

Equation (4) is a Walsh expansion with nonnegative integer coefficients. Its constant character is \(a=1\). From (7),

because \(F(n)\) has mean \(1\) exactly when \(n\) is a square. Also,

Both formulas are exact and [A].

The standard elementary estimate

follows, for example, by inserting \(\mu^2(a)=\sum_{d^2\mid a}\mu(d)\) and using the harmonic-sum estimate. Moreover,

Summing this error for \(a\le N\), using \(\sum_{a\le N}a^{-1/2}=O(\sqrt N)\), and then applying (13) to (12) gives

This is [A].

Equation (14) says that the centered variable has a growing \(L^2\) scale \(\sqrt{N\log N}\), but it does not give a lower-tail probability: a second moment may be carried by rare sign configurations. For reference, if \(X_N=S_F(N)-\lfloor\sqrt N\rfloor\), then its exact fourth moment is

Indeed, a product of the four Walsh characters has nonzero expectation exactly when every prime occurs with even total exponent. Formula (15) is [A]. Controlling (15), or suitable low moments of a maximum across scales, strongly enough to prove a positive tail is part of the analytic wall.

4. Exact exhaustive computation through \(N=64\)

For \(n\le64\), \(F(n)\) depends only on the 18 prime signs for

The checker represents the parity-of-exponent squarefree kernel of \(n\) as an 18-bit mask. For an assignment mask \(e\),

It exhausts all \(2^{18}=262144\) assignments; no pseudorandom numbers or floating-point comparisons enter the crossing counts. To test \(S_e(n)/\sqrt n\ge T\), it uses the exact integer condition

The method and the resulting table are [D].

Define

The exact exhaustive values are:

These are finite-horizon facts only. In particular, the unique path reaching ratio \(8\) by time \(64\) is the all-\(+1\) assignment, and its probability \(2^{-18}\) says nothing by itself about almost-sure recurrence. [D]

The same exhaustive pass gives the following exact endpoint data. Here \(\mu=\mathbb E S_F(N)\), \(m_j=\mathbb E(S_F(N)-\mu)^j\), and the last column is \(m_4/m_2^2\).

At \(N=64\), the endpoint distribution has

The increase in the four displayed finite kurtoses is suggestive of a heavy-tail issue but is not asserted to continue asymptotically. The exact numbers are [D] and the extrapolation is deliberately withheld.

As a separate arithmetic check on (12), the script recomputed:

The limiting coefficient in (14) is \(6/\pi^2=0.607927101854\ldots\). The table is [D]; equation (14) is the rigorous reason for its limiting value.

5. Reproducibility

The standalone checker is runs/erdos1144_wave8g_reverify.py. It uses only the Python standard library. It independently:

1. rebuilds all prime-exponent parity masks;
2. checks \(\kappa(mn)=\kappa(m)\mathbin{\rm XOR}\kappa(n)\) whenever

\(mn\le64\);

1. checks the unique \(n=ab^2\) decomposition and every coefficient in

(4) through \(N=64\);

1. exhausts the full 18-dimensional sign cube;
2. checks the finite crossing counts and centered moments against explicit

integers; and

1. recomputes (12) with a new squarefree sieve through \(N=10^6\).

Run:

python runs/erdos1144_wave8g_reverify.py

Observed complete output:

Structural identities: PASS
Exact cube: 262144 = 2^18 assignments

Finite-horizon counts for max_{n<=X} S(n)/sqrt(n) >= T:
X  T=2  T=3  T=4  T=5  T=6  T=7  T=8
8  1/4  0  0  0  0  0  0
16 9/32  5/64  1/64  0  0  0  0
32 305/1024  111/1024  41/1024  5/1024  0  0  0
64 83033/262144  7927/65536  15635/262144  6633/262144  959/131072  219/262144  1/262144

Exact endpoint centered moments (mean, m2, m3, m4, kurtosis):
8  (2, 8, 12, 152, 19/8)
16 (4, 16, 66, 952, 119/32)
32 (5, 51, 450, 10797, 3599/867)
64 (8, 116, 1980, 76736, 4796/841)

S(64) distribution spot checks:
P[S(64)=-20]=1/262144, P[S(64)=0]=28529/262144, P[S(64)=64]=1/262144

Exact variance values and Var(S(N))/(N log N):
     10:        9  ratio=0.390865033713
    100:      210  ratio=0.456009205998
   1000:     3383  ratio=0.489739410760
  10000:    47296  ratio=0.513509795402
 100000:   614180  ratio=0.533469969791
1000000:  7540120  ratio=0.545772084815
Limit coefficient 6/pi^2 = 0.607927101854

ALL CHECKS PASSED in 5.223 seconds

The source SHA-256 is

b33f1fff6395bd0f8904a630951c334fb01a8e0ec13123a91df64980785cd9a4

The observed computation is [D].

6. Why the standard machinery stalls

The reduction (5) separates two genuinely different requirements:

1. produce growing positive fluctuations of \(M_g(N)\), not merely large

absolute values or sign changes;

1. obtain them on deterministic test sets sparse enough for (9), or prove

a stronger maximal estimate for \(R_g\).

Atherfold's theorem supplies the relevant almost-sure upper bound for \(M_g\), but its proved lower bound does not grow. Aymone and Angelo–Xu supply sign changes for adjacent sums, but a sign change can occur at arbitrarily small normalized amplitude. None of those named theorems implies (LF). [B]

The exact variance (14) cannot replace (LF). A Paley–Zygmund argument would need a suitably uniform fourth-moment or low-moment estimate, and then a dependence argument across scales. Multiplicative Walsh characters have many relations \(a_1a_2a_3a_4=\square\), exactly the configurations counted in (15). The finite kurtosis table illustrates, but does not prove asymptotically, why a Gaussian bounded-kurtosis shortcut is unsafe. [A] for the logical requirement and [D] for the illustration.

There is also no useful brute-force route to the infinite statement. Exact path enumeration through \(X\) has \(2^{\pi(X)}\) assignments and roughly \(X2^{\pi(X)}\) value updates. At \(X=128\), this is \(2^{31}\) paths and \(2.75\times10^{11}\) updates, roughly \(1\)–\(5\) core-hours in an optimized C++ implementation at a realistic \(3\times10^7\)–\(10^8\) updates/second. At \(X=256\), it becomes \(2^{54}\) paths and \(4.61\times10^{18}\) updates, about 1460 core-years even at \(10^8\) updates/second. I did not run either computation. These cost estimates are [D], and no finite extension would prove the almost-sure limsup anyway.

7. Honest final state

The target remains open. The useful reduction is (5), strengthened by the grid lemma (9)–(LF): the fractional part obstruction can be paid for on sufficiently sparse deterministic grids, so the central missing theorem is a positive, growing, almost-sure large-fluctuation result for the critical weighted squarefree Rademacher sum \(M_g\) with compatible grid uniformity. The exact finite cube and moment computations are reproducible checks of the structure, not evidence claimed to close the required uniformity step.

PARTIAL: Reduced #1144 exactly to a weighted squarefree positive-fluctuation lemma plus an \(L^2\)-controlled fractional error, proved the exact variance \((6/\pi^2)N\log N+O(N)\), and exhaustively verified all paths through \(N=64\); the required growing positive grid fluctuation remains unproved.