Periodic minimum in the count of binomial coefficients not divisible by a prime

TL;DR

This work resolves, for a broad range of odd primes, the problem of locating the minimum of the periodic component governing the asymptotics of the binomial non-divisible counts $F_p(n)$. By deriving a $p$-adic recurrence and a precise periodic representation $F_p(n)=n^{\varrho_p}\mathcal{P}_p(\log_p n)$, the authors reduce the minimization to studying a one-dimensional function $G(s)$ on $[1/p,1]$. They introduce a magnifying mapping and a monotonic majorization framework to prove that the minimum occurs at points $\hat{s}_p(\xi,\eta)=(2\xi+1)/(2p)-\eta/p^2$, yielding explicit minima $\beta_p=B_{\xi,\eta}$ for many primes; this confirms Wilson's conjectures up to $p=113$ and extends the method through $p\le 2221$, with large-$p$ asymptotics suggesting systematic growth of the optimal digits. The results provide a detailed, constructive description of the minimal behavior of $F_p(n)$ modulo $p$ and illuminate the self-similar structure of the corresponding periodic function $\mathcal{P}_p$. Overall, the paper advances understanding of Pascal's triangle modulo $p$ and the extremal values of the associated counting function.

Abstract

The summatory function of the number of binomial coefficients not divisible by a prime is known to exhibit regular periodic oscillations, yet identifying the less regularly behaved minimum of the underlying periodic functions has been open for almost all cases. We propose an approach to identify such minimum in some generality, solving particularly a previous conjecture of B. Wilson [Asymptotic behavior of Pascal's triangle modulo a prime, Acta Arith. 83 (1998), pp. 105-116].

Figures (7)

• Figure 1: Fluctuations of $G(s)$, $s\in[p^{-1},1]$, for $p=3,5,7$.
• Figure 2: Graphical rendering of $\mathcal{Q}_{s_1}(t)$ (in blue) and $E_{s_1,1}(t)$ (in red) for $p=3,5,7$.
• Figure 3: A closer look at the fluctuations of $G$ in the smaller interval $\bigl[\frac{1}{p}, \frac{2}{p}\bigr]$ for $p=3,5,7$.
• Figure 4: $p=11$: $\mathcal{Q}_{s_i}(t)$ (in blue) and $E_{s_i,0}$ (in red) for $i=1,2$.
• Figure 5: $p=11$: $G(x)$ for $x\in\left[ \frac{1}{11},1\right]$ (left) and $\left[ \frac{14}{11^2}, \frac{17}{11^2}\right]$ (right), respectively.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Remark 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof