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arXiv2026

An Asymptotic for Non-Wieferich Numbers

A cyclic number is a positive integer $n$ such that $\gcd(n, φ(n)) = 1$. Erdős showed that the number of cyclic numbers up to $x$, $C(x)$, has $C(x) \sim e^{-γ}x/\log_3(x)$ asymptotically as $x$ tends to infinity. We define Wieferich numbers to be odd, positive integers $n$ such that $(n, (2^{\ell(n)}-1)/n) > 1$, where $\ell(n)$ is the smallest positive integer with $2^{\ell(n)} \equiv 1 \mod n$. Franco and Pomerance showed that non-Wieferich numbers have density zero among the odd integers. We show that the counting function of $n \leq x$ in the symmetric difference of non-Wieferich numbers and cyclic numbers is $O(x/(\log_3(x))^2)$, under a mild assumption on the counting function of Wieferich primes.

math.NT↗

arXiv2023

Tamagawa Products for Elliptic Curves Over Number Fields

In recent work, Griffin, Ono, and Tsai constructs an $L-$series to prove that the proportion of short Weierstrass elliptic curves over $\mathbb{Q}$ with trivial Tamagawa product is $0.5054\dots$ and that the average Tamagawa product is $1.8183\dots$. Following their work, we generalize their $L-$series over arbitrary number fields $K$ to be \[L_{\mathrm{Tam}}(K; s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(K; m)}{m^s},\] where $P_{\mathrm{Tam}}(K;m)$ is the proportion of short Weierstrass elliptic curves over $K$ with Tamagawa product $m$. We then construct Markov chains to compute the exact values of $P_{\mathrm{Tam}}(K;m)$ for all number fields $K$ and positive integers $m$. As a corollary, we also compute the average Tamagawa product $L_{\mathrm{Tam}}(K;-1)$. We then use these results to uniformly bound $P_{\mathrm{Tam}}(K;1)$ and $L_{\mathrm{Tam}}(K,-1)$ in terms of the degree of $K$. Finally, we show that there exist sequences of $K$ for which $P_{\mathrm{Tam}}(K;1)$ tends to $0$ and $L_{\mathrm{Tam}}(K;-1)$ to $\infty$, as well as sequences of $K$ for which $P_{\mathrm{Tam}}(K;1)$ and $L_{\mathrm{Tam}}(K;-1)$ tend to $1$.

math.NT↗