Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.761126 
Title:  Zeros of quadratic forms and the delta method  
Author:  Viswanathan, Vinay 
ISNI:
0000 0004 7432 7874


Awarding Body:  University of Bristol  
Current Institution:  University of Bristol  
Date of Award:  2018  
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Abstract:  
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.


Supervisor:  Browning, Tim  Sponsor:  Not available  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.761126  DOI:  Not available  
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