Problem 377: separating a trivial uniform bound from digit-count information
1) OBJECTIVE AND SOURCE REVIEW
Attempt dated 2026-09-23. The target is boundedness of $f(n)=\sum_{p\le n,\ p\nmid\binom{2n}{n}}1/p$. I read all of
377.tex and checked the 1975 primary paper. The source's bound $\log(m+1)+o(1)$ above $n^{1/(m+1)}$ already follows by dropping the nondivisibility restriction and applying the prime reciprocal asymptotic. Its digit-window lemma is valid, but its stated leading constant is not an improvement over that trivial estimate. Here I give a self-contained averaged consequence of the digit condition, without presenting it as a new literature result.
2) WORK
For an odd prime $p$, put $r_p=(p+1)/2$ and $L=\lfloor\log X/\log p\rfloor$. Nondivisibility means that every base-$p$ digit of $n$ belongs to $\{0,\ldots,(p-1)/2\}$. Hence the number $G_p(X)$ of such integers $0\le n\le X$ is at most $r_p^{L+1}$. Since $X\ge p^L$, \[
\frac{G_p(X)}{pX}\le\left(\frac{r_p}{p}\right)^{L+1}.
\] Fix $0<c<\sqrt{\log2}$ and set $y=\exp(c\sqrt{\log X})$. Choose a constant $c_0$ with $c^2<c_0<\log2$. For all sufficiently large fixed $P$, every prime $p\ge P$ has $\log(p/r_p)\ge c_0$. Since $L+1>\log X/\log p$, we obtain \[
\sum_{P\le p\le y}\frac{G_p(X)}{pX}
\le y\exp\left(-\frac{c_0\log X}{\log y}\right)
=\exp\left((c-c_0/c)\sqrt{\log X}\right)\to0.
\] For each of the finitely many primes below $P$, $G_p(X)/(pX)\to0$ directly from the digit bound; prime two contributes nothing for positive $n$. Interchanging two finite sums therefore proves \[
\frac1{\lfloor X\rfloor}\sum_{1\le n\le X}
\sum_{\substack{p\le\min(n,y)\\p\nmid\binom{2n}{n}}}\frac1p\longrightarrow0.
\] By Markov's inequality, the contribution from this growing small-prime range is $o(1)$ outside a set of $o(X)$ integers, in the usual convergence-in-density sense.
3) ADVERSARIAL VERIFICATION
An averaged estimate permits exceptional integers with large contributions. It therefore does not conflict with the source's observation that any unboundedness must arise from small primes relative to $n$. The cutoff uses $X$, the averaging range, rather than silently replacing it by each individual $n$. The primary paper already establishes stronger average behavior for the full function; this proof is an elementary verification of one mechanism, not an improvement on that theorem.
4) FINAL
UNRESOLVED. Typical digit behavior is controlled, while boundedness at every exceptional integer remains open in this attempt. No new verified worst-case bound for $f$ is obtained.
5) SOURCES CHECKED
The complete supplied version; P. Erdős, R. L. Graham, I. Z. Ruzsa and E. G. Straus,
On the Prime Factors of $\binom{2n}{n}$, digit criterion and Theorem 2 checked at
https://renyi.hu/~p_erdos/1975-27.pdf, 2026-09-23.
6) COMPLETION ESTIMATE
Subjective progress toward uniform boundedness or an unbounded sequence.
COMPLETION: 10%