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I will work with $f=u^2+v^2$ and the region $|u|,|v|\leq X$ but the argument easily generalises. The numbers $m$ go up to $X^2$ but by Dirichlet's divisor trick we can assume that they go up to $\sqrt …

Is there an example of a quasiprojective variety $X$ defined over ℚ such that $X$ is unirational over all finite fields, and $X$ is unirational over $\mathbb{R}$, and $X$ is not unirational over $\m …

I was wondering whether this result is in the bibliography and how one might go about proving it: $$ \lim_{x\to \infty} \frac{\log x }{x} \sum_{p \leq x \atop p\text{ prime }}\mu(p-1)=0.$$ The sum is …

We know that the average of $\phi(n)/n$ is approximated by a constant. Here $\phi $ is the Euler quotient function. One can furthermore show asymptotics with a secondary main term, at least for the sm …

Let $p $ be an odd prime. Assume that we have the following perfect pattern: all the primes below $p$ are successively quadratic residues and quadratic non-residues. What can we say about $p$? Is it p …

Let $p$ be an odd prime and define $n(p)$ be the smallest positive quadratic non-residue modulo $p$. By Ankeny and later effective work of Lamzouri, Li, and Soundararajan we know that under GRH one ha …

In the sieve book of Cojocaru and Murty they give a simple application of the square sieve of Heath-Brown, namely in Theorem 2.3.5 of their book they prove that $$\#\{1\leq n \leq x:f(n)=\square\}\ll_ …

Please ignore the previous version of this answer. Motivated by Lucia's comment, we use smooth numbers to show that the number in question is $o(x)$. First note that for $100\%$ of all integers one h …

Let $a$ be an integer which is neither a square nor $-1$. Artin's conjecture states that there are infinitely many primes $p$ for which $a$ is a primitive root modulo $p$. My question is whether there …

It is known that all smooth projective quartic hypersurfaces of suitably large dimension are unirational over $\overline{\mathbb{Q}}$. Are there any results regarding unirationality over $\mathbb{Q}$ …

If for every $q>1 $ the function $b(n)^q$ has average $O_q(1)$ then by H"{o}lder one can get $$ \sum_{n<x} \lambda^{\omega(n)} b(n) \ll_\epsilon x (\log x)^{\lambda-1+\epsilon}$$ for every fixed $\eps …

Let $d(n)$ be the number of positive integers that divide $n$. It is well known that $d(n)$ is on average $\log n$. However, it is also well known that for most $n$ the number $d(n)$ is rather close t …

As Ofir says this shouldn't be a problem to prove directly using Perron. But let me describe another way to get the same result: we shall use the well-known result of Davenport that for any $A>0$ one …

We use the bound given by my previous answer: $$\sum_{n<x}\mu(kn)\ll_A \frac{kx}{(\log (kx))^A}.$$ Now we improve this bound by using that the sum $$\sum_{d \in \mathbb N\atop d\mid n, d\mid k}\mu(d)$ …

The natural density is $0$. This is due to the function $\sigma(n)/n$ possessing a continuous distribution, i.e. there exists a continuous function $\Phi:\mathbb{R}\to \mathbb{R}$ such that for all $a …