Hari R. Iyer’s research while affiliated with Harvard University and other places

We prove an explicit Chebotarev variant of the Brun–Titchmarsh theorem. This leads to explicit versions of the best known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that limx→∞#{1≤n≤x∣τ(n)≠0}x>1-1.15×10-12,$\begin{aligned} \lim _{x \rightarrow \infty } \frac{\#\{1 \le n \le x \mid \tau (n) \ne 0\}}{x} > 1-1.15 \times 10^{-12}, \end{aligned}$where τ(n)$\tau (n)$ is Ramanujan’s tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers n such that τ(n)≠0$\tau (n) \ne 0$.

We prove an explicit Chebotarev variant of the Brun--Titchmarsh theorem. This leads to explicit versions of the best-known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that $\lim_{x \to \infty} \frac{\#\{1 \leq n \leq x \mid \tau(n) \neq 0\}}{x} > 1-1.15 \times 10^{-12},$ where $\tau(n)$ is Ramanujan's tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers n such that $\tau(n) \neq 0$.

Semi-regular sequences over F2 are sequences of homogeneous elements of the algebra B(n)=F2[X1,...,Xn]/(X12,...,Xn2), which have a given Hilbert series and can be thought of as having as few relations between them as possible. It is believed that most such systems are semi-regular and this property has important consequences for understanding the complexity of Gröbner basis algorithms such as F4 and F5 for solving such systems. We investigate the case where the sequence has length two and give an almost complete description of the number of semi-regular sequences for each n.