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The key fact here is that the integer pairs $(x,y) \in (0,p)^2$ such that $xy \equiv 1 \bmod p$ are asymptotically equidistributed as $p \rightarrow \infty$. This is a consequence of Weil's 1948 boun …

This is true, and follows from one-dimensional Szemeredi. Fix $\delta > 0$ such that $|S\cap A_n| \geq \delta |A_n|$ infinitely often. Let $r = \lfloor \sqrt n \rfloor$, so $A_n$ is contained in the …

You're dividing the square $S = \{ (x,y) \colon 0 < x < 1, 0 < y < 1\}$ into two regions according to the parity of $\lfloor 1/(xy) \rfloor$, separated by the segments of the hyperbolas $xy = 1/n$ ($n …

"Rational to transcendental power is irrational" is easy given the transcendence of numbers like $\log_2 3$, e.g. $2^{\frac12 \log _{\,2}\! 3} = \sqrt 3$. For a transcendental example, once numbers l …

Here's an elementary proof of the inequality $$ (1) \qquad\qquad \sum_{k=1}^{N-1} \left(1-\frac{k}{N}\right)B_2(\{kx\}) \ge \frac1{12N} - \frac1{12}. \qquad\qquad\phantom{(1)} $$ This is nearly the …

Not necessarily. The first counterexample might be $q=14$ and $f(n)=1, -1, -1, -1, -1, 1, 0, -1, 1, 1, 1, 1, -1, 0$ for $n=1,2,3,\ldots,14$.

No, it is not true that $M(D) = o(|D|^{3/2})$. For example, if $D = -4abcd$ where $a,b,c,d$ are odd, pairwise coprime and of roughly equal size, then the forms $ab\,x^2+cd\,y^2, ac\,x^2+bd\,y^2, ad\, …

The number $S_q$ depends on the choice of additive character $\psi$: changing $\psi$ multiplies each $g(\psi,\chi)$ by some $(q-1)$-th root of unity depending also on $\chi$; this is often harmless, b …

It seems unlikely that one can prove anything nontrivial, but it's still interesting to consider what ought to be true, and to experimentally compute for small $k$. Let $$ \delta_k = \min_{\ell_1<\ …

Actually $\zeta_{K_n}(\sigma)$ is bounded for any fixed $\sigma > 1$. Let $N = 2^n = [K_n : {\bf Q}]$. Then all the local factors of $\zeta(\sigma)$, other than the factor $(1-2^{-\sigma})^{-1}$ fo …

If $\Phi(0) = 1 + \epsilon$ then $\,\widehat{\!\Phi\!}\,(t) \gg \epsilon^{-1/2}$, even under the weaker assumption that $\,\widehat{\!\Phi\!}\,(t) \geq 0$ for all $t$; and this is best possible up to …

You are right to question this. The product $\prod_p \left(1 - \chi(p)/p\right)^{-1}$ (where $\chi = (-1/\cdot)$ is the Dirichlet character mod $4$) does converge, and the limit is $L(1,\chi) = \pi/4 …

Let $s_n = 2^n$ and choose for $a$ any real number that's normal in base $2$.

Once you know that the coefficients are all positive (see postscript), it's easy to get an effective upper bound that grows as $\exp(4\pi \sqrt{n})$, which is within a factor $O(\sqrt n)$ of the corre …

You're surely right that there cannot be a "closed form" for such a series; but it can still be approximated to any desired precision. The defining sum $$ x = x(a) = \sum_{i=0}^\infty \sum_{j=0}^\in …