We compute upper and lower bounds for the approximation of hyperbolic functions at points $1/s$ $(s=1,2,\cdots )$ by rationals $x/y$, such that $x,y$ satisfy a quadratic equation. For instance, all positive integers $x,y$ with $y\equiv 0(mod2)$ solving the Pythagorean equation $x^2+y^2=z^2$ satisfy $$|ysinh(1/s)-x|\gg \frac{loglogy}{logy}.$$ Conversely, for every $s=1,2,\cdots$ there are infinitely many coprime integers $x,y$, such that $$|ysinh(1/s)-x|\ll \frac{loglogy}{logy}$$ and $x^2+y^2=z^2$ hold simultaneously for some integer $z$. A generalization to the approximation of $h\left(e^{1/s}\right)$ for rational...

We provide upper bounds for the mean square integral $${\int }_X^{2X}{\left({𝔻}_k(x+h)-{𝔻}_k\left(x\right)\right)}^2\mathrm{d}x,$$ where $h=h\left(X\right)\gg 1,h=o\left(x\right)\mathrm{as}X\to \infty$ and $h$ lies in a suitable range. For $k\ge 2$ a fixed integer, ${𝔻}_k\left(x\right)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_k\left(n\right)$, generated by ${\zeta }^k\left(s\right)$.

For $\epsilon \>0$ and any sufficiently large odd $n$ we show that for almost all $k\le R:=n^{1/5-\epsilon }$ there exists a representation $n=p_1+p_2+p_3$ with primes $p_i\equiv b_i$ mod $k$ for almost all admissible triplets $b_1,b_2,b_3$ of reduced residues mod $k$.

Let ${𝔖}_n$ denote the symmetric group with $n$ letters, and $g\left(n\right)$ the maximal order of an element of ${𝔖}_n$. If the standard factorization of $M$ into primes is $M=q_1^{{\alpha }_1}{q}_2^{{\alpha }_2}...q_k^{{\alpha }_k}$, we define $\ell \left(M\right)$ to be $q_1^{{\alpha }_1}+q_2^{{\alpha }_2}+...+q_k^{{\alpha }_k}$; one century ago, E. Landau proved that $g\left(n\right)={max}_{\ell \left(M\right)\le n}M$ and that, when $n$ goes to infinity, $logg\left(n\right)\sim \sqrt{nlog\left(n\right)}$. There exists a basic algorithm to compute $g\left(n\right)$ for $1\le n\le N$; its running time is $𝒪\left(N^{3/2}/\sqrt{logN}\right)$ and the needed memory is $𝒪\left(N\right)$; it allows computing $g\left(n\right)$ up to, say, one million. We describe an algorithm to calculate $g\left(n\right)$ for $n$ up to ${10}^{15}$. The main idea is to use the...

Given an integer base $q\ge 2$ and a completely $q$-additive arithmetic function $f$ taking integer values, we deduce an asymptotic expression for the counting function $$N_f\left(x\right)=\#\left\{0\le n\<x\left|f\right(n)\mid n\right\}$$ under a mild restriction on the values of $f$. When $f=s_q$, the base $q$ sum of digits function, the integers counted by $N_f$ are the so-called base $q$ Niven numbers, and our result provides a generalization of the asymptotic known in that case.