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;
2. 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 \(N\);
3. 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*](https://numdam.org/articles/10.5802/crmath.615/),
Comptes Rendus Mathématique 362 (2024), 895–901,
DOI 10.5802/crmath.615, proves in Theorem 1.2 that for a Rademacher random
completely multiplicative \(F\),
\[ \sum_{n\le x}\frac{F(n)}{n^\alpha} \]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
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
\[ M_g(x):=\sum_{n\le x}\frac{g(n)}{\sqrt n} \ll(\log\log x)^{3/4+\varepsilon}. \tag{2} \]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
\[ \sum_{n\le x}\frac{F(n)}{\sqrt n} \ll \log x\,(\log\log x)^{3/4+\varepsilon} \]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*](https://arxiv.org/abs/2411.14447),
arXiv:2411.14447v2 (24 March 2026), prove that
\[ \sum_{n\le x}\frac{F(n)}{\sqrt n} \]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
\[ n=ab^2,\qquad a\ \text{squarefree}. \]The parity of every prime exponent is carried by \(a\), so
\[ F(n)=g(a) =\sum_{b^2\mid n}g(n/b^2). \tag{3} \]The sum in (3) has exactly one nonzero term. Therefore, for
\(S_F(N):=\sum_{n\le N}F(n)\),
\[ \begin{aligned} S_F(N) &=\sum_{\substack{ab^2\le N\\a\ {\rm squarefree}}}g(a)\\ &=\sum_{a\le N}g(a)\left\lfloor\sqrt{\frac Na}\right\rfloor. \tag{4} \end{aligned} \]These identities are [A].
Writing \(\lfloor y\rfloor=y-\{y\}\) in (4) gives the central exact
reduction
\[ \boxed{\quad \frac{S_F(N)}{\sqrt N}=M_g(N)-R_g(N), \quad} \tag{5} \]where
\[ M_g(N)=\sum_{a\le N}\frac{g(a)}{\sqrt a},\qquad R_g(N)=\frac1{\sqrt N}\sum_{a\le N}g(a) \left\{\sqrt{\frac Na}\right\}. \tag{6} \]No approximation has occurred in (5). [A]
Orthogonality and the inexpensive fixed-\(N\) error
For squarefree \(a,b\),
\[ \mathbb E[g(a)g(b)]= \begin{cases} 1,&a=b,\\ 0,&a\ne b. \end{cases} \tag{7} \]Indeed, if \(a\ne b\), some prime occurs in exactly one of them and its
independent sign has mean zero. Consequently,
\[ \mathbb E|R_g(N)|^2 =\frac1N\sum_{\substack{a\le N\\a\ {\rm squarefree}}} \left\{\sqrt{\frac Na}\right\}^{\!2} \le \frac1N\sum_{a\le N}\mu^2(a) \le1. \tag{8} \]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
\[ \sum_k\frac{|\mathcal T_k|}{u_k^2}<\infty, \tag{9} \]then (8), Markov's inequality, and the union bound give
\[ \sum_k \mathbb P\!\left(\max_{N\in\mathcal T_k}|R_g(N)|>u_k\right) \le \sum_k\frac{|\mathcal T_k|}{u_k^2}<\infty. \]The first Borel–Cantelli lemma therefore implies, almost surely,
\[ \max_{N\in\mathcal T_k}|R_g(N)|\le u_k \quad\text{for all sufficiently large }k. \tag{10} \]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 \(k\),
\[ \max_{N\in\mathcal T_k}M_g(N)\ge2u_k. \tag{LF} \]At such \(k\), (10) gives some \(N\in\mathcal T_k\) with
\[ \frac{S_F(N)}{\sqrt N}\ge u_k\longrightarrow\infty. \]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),
\[ \mathbb E S_F(N)=\left\lfloor\sqrt N\right\rfloor, \tag{11} \]because \(F(n)\) has mean \(1\) exactly when \(n\) is a square. Also,
\[ \boxed{\quad \operatorname{Var}(S_F(N)) =\sum_{\substack{2\le a\le N\\a\ {\rm squarefree}}} \left\lfloor\sqrt{\frac Na}\right\rfloor^{\!2}. \quad} \tag{12} \]Both formulas are exact and [A].
The standard elementary estimate
\[ \sum_{a\le N}\frac{\mu^2(a)}a =\frac{6}{\pi^2}\log N+O(1) \tag{13} \]follows, for example, by inserting
\(\mu^2(a)=\sum_{d^2\mid a}\mu(d)\) and using the harmonic-sum estimate.
Moreover,
\[ \left| \left\lfloor\sqrt{\frac Na}\right\rfloor^{\!2}-\frac Na \right| \le 2\sqrt{\frac Na}. \]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
\[ \boxed{\quad \operatorname{Var}(S_F(N)) =\frac{6}{\pi^2}N\log N+O(N). \quad} \tag{14} \]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
\[ \mathbb E X_N^4 = \sum_{\substack{a_1,a_2,a_3,a_4\in[2,N]\ {\rm squarefree}\\ a_1a_2a_3a_4\ {\rm a\ square}}} \prod_{i=1}^4 \left\lfloor\sqrt{\frac N{a_i}}\right\rfloor. \tag{15} \]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
\[ 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61. \]The checker represents the parity-of-exponent squarefree kernel of \(n\)
as an 18-bit mask. For an assignment mask \(e\),
\[ F_e(n)=(-1)^{\operatorname{popcount}(e\mathbin{\&}\kappa(n))}. \]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
\[ S_e(n)>0,\qquad S_e(n)^2\ge T^2n. \tag{16} \]The method and the resulting table are [D].
Define
\[ H_X(T)= \mathbb P\!\left(\max_{1\le n\le X}\frac{S_F(n)}{\sqrt n}\ge T\right). \]The exact exhaustive values are:
| \(X\) | \(H_X(2)\) | \(H_X(3)\) | \(H_X(4)\) | \(H_X(5)\) | \(H_X(6)\) | \(H_X(7)\) | \(H_X(8)\) |
|---:|---:|---:|---:|---:|---:|---:|---:|
| 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 | \(83033/262144\) | \(7927/65536\) | \(15635/262144\) | \(6633/262144\) | \(959/131072\) | \(219/262144\) | \(1/262144\) |
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 \(m_4/m_2^2\).
| \(N\) | \(\mu\) | \(m_2\) | \(m_3\) | \(m_4\) | 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 | \(4796/841\) |
At \(N=64\), the endpoint distribution has
\[ \mathbb P(S_F(64)=-20)=2^{-18},\quad \mathbb P(S_F(64)=0)=\frac{28529}{262144},\quad \mathbb P(S_F(64)=64)=2^{-18}. \]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:
| \(N\) | exact \(\operatorname{Var}(S_F(N))\) | \(\operatorname{Var}/(N\log N)\) |
|---:|---:|---:|
| \(10\) | 9 | 0.390865033713 |
| \(10^2\) | 210 | 0.456009205998 |
| \(10^3\) | 3383 | 0.489739410760 |
| \(10^4\) | 47296 | 0.513509795402 |
| \(10^5\) | 614180 | 0.533469969791 |
| \(10^6\) | 7540120 | 0.545772084815 |
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
\(mn\le64\);
3. checks the unique \(n=ab^2\) decomposition and every coefficient in
(4) through \(N=64\);
4. exhausts the full 18-dimensional sign cube;
5. checks the finite crossing counts and centered moments against explicit
integers; and
6. 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;
2. 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.