In recent work, the first two authors constructed a generalized continued fraction called the $p$-continued fraction, characterized by the property that its convergents (a subsequence of the regular convergents) are best approximations with respect to the $L^p$ norm, where $p\geq 1$.We extend this construction to the region $0<p<1$, where now the $L^p$ quasinorm is non-convex.We prove that the approximation coefficients of the $p$-continued fraction are bounded above by $\frac 1{\sqrt 5}+\varepsilon_p$, where $\varepsilon_p\to 0$ as $p\to 0$.In light of Hurwitz's theorem, this upper bound is sharp, in the limit.We also measure the maximum number of consecutive regular convergents that are skipped by the $p$-continued fraction.

We introduce the study of triple convolution Ramanujan sums and apply it to give a~heuristic derivation of the Hardy-Littlewood conjecture on prime 3-tuples without using the circle method. We also estimate the triple convolution of the Jordan totient function using Ramanujan sums.

In this paper, we introduce certain subfamily of $p$-valent analytic functions of bounded turning for which we estimate best possible upper bound to certain generalised second Hankel determinant, the Zalcman conjecture and an upper bound to the third, fourth Hankel determinants. Further, we investigate an upper bound for third and fourth Hankel determinants with respect to two-fold and three-fold symmetric functions for the same class. The practical tools applied in the derivation of our main results are the coefficient inequalities of the Carathéodory class $\mathcal{P}$.

Let $N$ be a square-free integer. Let $f\in \mathcal{B}^\ast_k(N)$ (or $\mathcal{B}_\lambda^\ast(N)$) be a primitive (either holomorphic or Maaß) cusp form of level $N$, with $\lambda_f(n)$ denoting the $n$-th Hecke eigenvalue. In this paper, we explicitly determine the dependence on the level aspect for the sum \[\sum_{n\le X}\lambda_f(n) e{\left(n^2\alpha+n\beta \right)},\] which is uniform in any $\alpha,\beta\in \R$ and $X\ge 2$. In addition, we also investigate the analog at the prime arguments.

We prove explicitbounds on the number of lattice points on or near a convex curve in termsof geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our estimates hold for lattices more general than the usual lattice ofintegral points in the plane.

In this paper, we study the number of integer pair solutions to the equation $|F(x,y)| = 1$ where $F(x,y) \in \Z[x,y]$ is an irreducible (over $\Z$) binary form with degree $n \geq 3$ and exactly three nonzero summands. In particular, we improve Thomas' explicit upper bounds on the number of solutions to this equation (see [13]). For instance, when $n \geq 219$, we show that there are no more than 32 integer pair solutions to this equation when $n$ is odd and no more than 40 integer pair solutions to this equation when $n$ is even, an improvement on Thomas' work in [13], where he shows that there are no more than 38 such solutions when $n$ is odd and no more than 48 such solutions when $n$ is even.