Abstract
This thesis is devoted to two topics in Analytic number theory, namely, that of Waring type problems in thin sets and mixed moments of the Riemann zeta function.We begin by examining the expected asymptotic formula of the representation function in Waring’s problem over sums of three cubes, both with and without multiplicities, thereby establishing its validity in the former case and deriving a lower bound in the latter.
A separate discussion is devoted to the investigation of the above setting for small exponents. We obtain upper bounds for the number of variables needed to represent every sufficiently large integer in the prescribed way for the exponents 2, 3 and 4. We make use of the minor arc analysis in the case k = 2 and combine it with an intrincate major arc counterpart to deduce an almost all result for the analogous Lagrange’s foursquare theorem where the variables are restricted to the sums of three cubes.
We complete the circle method part of the thesis by examining the analogous problem in which the sums of three cubes are replaced by sums of t positive lth powers, the desired objective in such a context being the accomplishment of some uniformity in the number s of variables needed. Those considerations are discussed and partially achieved when t lies in two particular ranges.
The second part of the thesis comprises the investigation of mixed third moments of the Riemann zeta function. We establish an asymptotic evaluation of the moment at hand in three different situations: one in which the corresponding coefficients are rational numbers in a suitable range, another one in which the coefficients are linearly independent over Q, and the last one in which one of the coefficients equals minus the other one. In certain cases we are able to provide explicit account of lower order terms.
Date of Award  28 Sept 2021 

Original language  English 
Awarding Institution 

Supervisor  Misha Rudnev (Supervisor) & Trevor Wooley (Supervisor) 