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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 …

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 …

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 …

I am reading about Dirichlet polynomials in the book Analytic Number Theory by the said authors. Can anyone justify the following inequality? Assume that $a(n),b(m)$ are sequences of non-negative numb …

Let $n $ be even and define $$ Q(n)=\sum_{\substack{ p,q \ \textrm{ primes} \\p+q=n }}\left(\frac{p}{q} \right),$$ where $\left(\frac{p}{q} \right)$ is the quadratic Legendre symbol. Has this sum been …

For each positive integer n we may define the convergent sum $$ s(n)=\sum_{p}\frac{(n,p-1)}{p^2} $$ where the summation is over primes p and $(a,b)$ denotes the greatest common divisor of a,b. It is …

Let $f:\mathbb N \to \mathbb C$ be an arithmetic function. There are various ways to define what the Siegel–Walfisz (S–W) property is for $f(n)$. One simple way is that there exists some function $g …

Let $(\frac{m}{n})$ be the Jacobi quadratic symbol defined for positive squarefree odd integers $n,m$. Does the following sum go to infinity? $$ \sum_{1\leq n \leq (\log x)^{100} } \mu^2(2n) \sum_{(\l …

Let $b:\mathbb N\to \{0,1\}$ be the indicator function of integers that are a sum of two non-zero integer squares. Let $f(t)\in \mathbb Z[t]$ be an irreducible polynomial of degree $2$ with positive l …