All Questions

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)\...

Let $S(\alpha) = \sum_{n\leq N} w(n) e^{2\pi i \alpha n}$ for some function $w$ defined on $\mathbb{R}$. Suppose $\alpha_1, \ldots, \alpha_R$ are real numbers that are $\delta$-spaced modulo $1$, for ...