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Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...

I am interested in an upper bound on the following incomplete Kloosterman sum $$ \sum_{\substack{x=1 \\ x+_{_{\bf Z}}x^{-1}>p}}^{p-1}e\left(\frac{x+x^{-1}}{p}\right).$$ Using the Weil's bound it ...

Consider $$S_N:=\sum_{n=1}^N \exp\left(2\pi i\left(\frac12n^2x+\alpha n\right)\right)$$ where $\alpha$ is irrational. For certain $x$ (say integer) we can get that this is bounded for all $N$. I am ...

I am working through the problem presented in Chapter 8 of Iwaniec and Kowalski’s Analytic Number Theory (specifically inequalities (8.8) and (8.9)) and I am struggling with the transition between ...

Let $a,q,N$ be integers such that $N/2 \leq q \leq N$ and $a/q \notin \mathbb{Z}$. Is the following estimate true, and, if so, how can it be proved? \[\left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \...