Uniform bounds for lattice point counting and partial sums of zeta functions

We prove uniform versions of two classical results in analytic number theory. The first is an asymptotic for the points a complete lattice $$\Lambda \subseteq \mathbb {R}^d$$ inside d-sphere radius R. In contrast to previous works, we obtain error terms with implied constants depending only on d. Secondly, let $$\phi (s) = \sum _n a(n) n^{-s}$$ be ‘well behaved’ zeta function. A method Landau yields asymptotics partial sums $$\sum _{n < X} a(n)$$ , power saving terms. Following exposition due Chandrasekharan and Narasimhan, version where term will depend ‘shape functional equation’, implying families functions same equation.