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Assume throughout that $k \neq 0 $ since otherwise the probability is $1$ and not $0$ as required. A special case of Corollary 1.9 of https://arxiv.org/pdf/2001.10970.pdf says that if we have two inte …

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_ …

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 …

I would be thankful if someone had references to provide...

Reference request: Davenport proved that for every fixed $N>1 $ one has $$ \sup_{\alpha \in \mathbb R } \left | \sum_{1\leq n \leq x } \mu(n) \exp(2 \pi i \alpha n )\right | = O_N\left( \frac{x}{(\log …

The primes are equidistributed in the residue classes $1(\!\!\!\mod{4})$ and $3(\!\!\!\mod 4)$. We also know (for example, by Rubinstein-Sarnak) that the patterns cannot be eventually alternating, i.e …

The Siegel-Walfisz principle for a function $f(n)$ states that for all $A>0$ fixed then whenever $a$ modulo $q$ is a residue class with $a$ and $q$ coprime then one has $$\sum_{\substack{n\leq x \\ n …

This is a reference request. I know that Hardy and Littlewood gave a proof of the ternary Goldbach for sufficiently large odd integers under the assumption of GRH. Is there a modern account of th …

Sums of the form $S_0=\sum_{n,m}f(n)g(m)$ where $f,g$ have some multiplicative property should be doable by a double Euler formula. However it is more useful to have a multidimensional Euler formula f …

A few more ideas: using the Chebyshev upper bound, by partial summation we have $\sum_{p>y}p^{-2}=O(\frac{1}{y \log y})$ and therefore we see that $s(n)=\sum_{p \leq \frac{n}{\log n}}\frac{(p-1,n)}{ …