This thesis presents solutions to three problems. First, we show that the optimal covering exponent for the $3$sphere is $frac{4}{3},$ and this is joint work with T. D. Browning and R. S. Steiner. Next, we prove a result involving $h(n)$, the class number of an imaginary quadratic field with fundamental discriminant $n$. We give an asymptotic formula for correlations involving $h(n)$ and $h(nl)$ over fundamental discriminants that avoid the congruence class $1 pmod{8}$. The result is uniform in the shift $l$, and along the way we also derive an asymptotic formula for correlations between $r_Q(n)$, the number of representations of an integer by a positive definite quadratic form $Q$. Finally, we study sums of normalised Hecke eigenvalues $lambda(n)$ of holomorphic cusp forms over thin sequences. Let $F(bs{x})$ be a diagonal quadratic form in $4$ variables, we give an upper bound for the problem of counting integer solutions of bounded height to $F(bs{x})=0$ weighted by $lambda(x_1)$, and as a consequence we derive upper bounds for certain generalised cubic divisor sums. All three problems are solved by counting integer zeros of quadratic forms using the $delta$method.
Date of Award  25 Sep 2018 

Original language  English 

Awarding Institution   The University of Bristol


Supervisor  Tim Browning (Supervisor) 

Zeros of quadratic forms and the delta method
Viswanathan, V. (Author). 25 Sep 2018
Student thesis: Doctoral Thesis › Doctor of Philosophy (PhD)