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Suppose that $A\subseteq \mathbb{N}$ and suppose that you have an estimate of the form $$ \sum_{\substack{a\le x \\ a\in A}}f(a) \sim g(x). $$ With this information is it possible to get an asymptotic …

Let $e(x)=e^{2\pi i x}$ and consider the following functions defined for $x\in [0,1]$: $$ f_1(x)=\frac{e(10x)-e(x)}{e(x)-1}, \quad f_2(x)=\frac{e(110x)-e(11x)}{e(11x)-1}, $$ and $$ f_3(x)=\frac{e(1010 …

Let $k$ be an integer and consider the function $$ f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t). $$ I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to …

We know that the prime number theorem is equivalent to the statement $$ M(x)=\sum_{n\le x}\mu(n)=o(x). $$ By using Ramanujan sums, we can write $M(x)$ as $$ M(x)=\sum_{q\le x}\sum_{\substack{0\lt a\le …

Let $S_x(t)=\sum_{n\le x} a_n e(nt)$, where $e(x)=e^{2\pi i x}$. Then, the large sieve inequality tells us that $$ \sum_{q\le Q} \sum_{\substack{0\lt a \lt q \\ (a,q)=1}}|S_x(a/q)|^2 \le (Q^2+4\pi x)\ …