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Richert proved in https://link.springer.com/article/10.1007/BF01399533 that $$ \zeta(s) =O\left( |\Im(s)|^{100(1-\Re(s))^{3/2}} (\log |\Im(s)|)^{2/3}\right)$$ uniformly in the region $\Re(s)\in [1/2,1 …

It is well know that the density of positive square-free integers up to $x$ is asymptotically $x/\zeta(2)$. The error term $$ E(x)=\sum_{1\leq n \leq x } \mu(n)^2 -\frac{x}{\zeta(2)} $$ can easily p …

Let $r(n)$ be the number of ways of writing $n$ as the sum of two integer squares. Asymptotics for the shifted convolution problem $$ \sum_{n\in \mathbb N\cap[1,x]}r(n) r(n+1)$$ are quite classical; a …

There are formulas for counting the number of representations of a positive integer $N$ as a sum of three integer squares. What is a reference for $$ \#\{(x,y,z)\in \mathbf{N}^3: 5^4 x^2+y^2+z^2=N\} ? …

Assume that $F(x_1,\ldots, x_n)$ is a polynomial with integer coefficients that is "square-free" over $\mathbb Q$, i.e. it does not have repeated polynomial factors whose coefficients are in $\mathbb …

This is a reference question: Let $\psi(x)$ be the psi-Chebyshev function. Is there any unconditional result in the literature that proves that there exists $0<a<2$ such that $$ \int_2^x (\psi(y)-y)^2 …

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 $f:\mathbb N \to \mathbb C$ 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 …

One can prove that in fact the function mpe has bounded average which clearly improves $\log \log$. If a positive integer n has $mpe(n)=m$ then there exists a prime power p^m that divides n hence the …

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 …

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 $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 …

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 …

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 …

One can do a bit better. For simpler presentation assume that we instead consider the function $b'$ that is the indicator function of integers all of whose prime divisors are $1 \mod 4$. We have $b'\l …