Let $E_1$ and $E_2$ be $overline{mathbb{Q}}$-nonisogenous, semistable elliptic curves over $mathbb{Q}$, having respective conductors $N_{E_1}$ and $N_{E_2}$ and both without complex multiplication. For each prime $p$, denote by $a_{E_i}(p) := p+1-#E_
i(mathbb{F}_p)$ the trace of Frobenius. Under the assumption of the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power $L$-functions $L(s, mathrm{Sym}^i E_1otimesmathrm{Sym}^j E_2)$ where $i,jin{0,1,2}$, we prove an explicit result that can be stated succinctly as follows: there exists a prime $p mid N_{E_1}N_{E_2}$ such that $a_{E_1}(p)a_{E_2}(p)<0$ and [ p < big( (32+o(1))cdot log N_{E_1} N_{E_2}big)^2. ] This improves and makes explicit a result of Bucur and Kedlaya. Now, if $Isubset[-1,1]$ is a subinterval with Sato-Tate measure $mu$ and if the symmetric power $L$-functions $L(s, mathrm{Sym}^k E_1)$ are functorial and satisfy GRH for all $k le 8/mu$, we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime $p mid N_{E_1}$ such that $a_{E_1}(p)/(2sqrt{p})in I$ and [ p < left((21+o(1)) cdot mu^{-2}log (N_{E_1}/mu)right)^2. ]
Let $mathcal{I} subset mathbb{N}$ be an infinite subset, and let ${a_i}_{i in mathcal{I}}$ be a sequence of nonzero real numbers indexed by $mathcal{I}$ such that there exist positive constants $m, C_1$ for which $|a_i| leq C_1 cdot i^m$ for all $i i
n mathcal{I}$. Furthermore, let $c_i in [-1,1]$ be defined by $c_i = frac{a_i}{C_1 cdot i^m}$ for each $i in mathcal{I}$, and suppose the $c_i$s are equidistributed in $[-1,1]$ with respect to a continuous, symmetric probability measure $mu$. In this paper, we show that if $mathcal{I} subset mathbb{N}$ is not too sparse, then the sequence ${a_i}_{i in mathcal{I}}$ fails to obey Benfords Law with respect to arithmetic density in any sufficiently large base, and in fact in any base when $mu([0,t])$ is a strictly convex function of $t in (0,1)$. Nonetheless, we also provide conditions on the density of $mathcal{I} subset mathbb{N}$ under which the sequence ${a_i}_{i in mathcal{I}}$ satisfies Benfords Law with respect to logarithmic density in every base. As an application, we apply our general result to study Benfords Law-type behavior in the leading digits of Frobenius traces of newforms of positive, even weight. Our methods of proof build on the work of Jameson, Thorner, and Ye, who studied the particular case of newforms without complex multiplication.
