On a modular form of Zaremba's conjecture

Let m>=1 be an arbitrary fixed integer and let N_m(x) count the number of odd integers u<=x such that the order of 2 modulo u is not divisible by m. In case m is prime estimates for N_m(x) were given by H. Mueller that were subsequently sharpened into an asymptotic estimate by the present author. Mueller on his turn extended the author's result to the case where m is a prime power and gave bounds in the case m is not a prime power. Here an asymptotic for N_m(x) is derived tha...

It is shown that for some explicit constants $c>0, A>0$, the asymptotic for the number of positive non-square discriminants $D<x$ with fundamental solution $\varepsilon_D< x^{\frac 12+\alpha}, 0<\alpha <c$, remains preserved if we require moreover $\mathbb Q(\sqrt D)$ to contain an irrational with partial quotients bounded by $A$.

We show that, for any fixed $\varepsilon > 0$ and almost all primes $p$, the $g$-ary expansion of any fraction $m/p$ with $\gcd(m,p) = 1$ contains almost all $g$-ary strings of length $k < (5/24 - \varepsilon) \log_g p$. This complements a result of J. Bourgain, S. V. Konyagin, and I. E. Shparlinski that asserts that, for almost all primes, all $g$-ary strings of length $k < (41/504 -\varepsilon) \log_g p$ occur in the $g$-ary expansion of $m/p$.

Let $u\neq \pm 1,v^2$ be a fixed integer, let $p\geq 2$ be a prime, and let $\text{ord}_p(u)=d \:|\: p-1$ be the order of $u \text{ mod } p$. This note provides an effective lower bound $\pi_u(x)=\# \{ p\leq x:\text{ord}_p(u)=p-1 \}\gg x (\log x)^{-1}$ for the number of primes $p\leq x$ with a fixed primitive root $u \text{ mod } p$ for all large numbers $x\geq 1$. The current results in the literature have the lower bound $\pi_u(x)=\# \{ p\leq x:\text{ord}_p(u)=p-1 \}\gg x (...

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for $\alpha\in\mathbb{R}\backslash\mathbb{Q},\,\beta\in\mathbb{R}$ and $0<\theta<10/1561$, there exist infinitely many primes $p$, such that $\|\alpha p^2+\beta\|<p^{-\theta}$ and $p+2=\mathcal{P}_4$, which constitutes an improvement upon the previous result.

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $\varphi(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12, 14$, the generalized Riemann Hypothesis for the Dedekind zeta function of the cyclotomic field $\mathbb{Q}(e^{2\pi i/q})$ is true if and only if for all integers $k\geq 1$ we have \[\frac{\bar{N}_k}{\varphi(\bar{N}_k)(\log(\varphi(q)\log{\bar...

In this short note we present some remarks and conjectures on two of Erd\"os's open problems in number theory.

Let $q\ne \pm1,v^2$ be a fixed integer, and let $x\geq 1$ be a large number. The least prime number $p \geq3 $ such that $q$ is a primitive root modulo $p$ is conjectured to be $p\ll (\log q)(\log \log q)^3),$ where $\gcd(p,q)=1$. This note proves the existence of small primes $p\ll(\log x)^c$, where $c>0$ is a constant, a close approximation to the conjectured upper bound.

Let $\|\cdot\|$ denote the minimum distance to an integer. For $0<\gamma< 1$, $\theta>0$ and $(\alpha, \beta) \in \mathbb{R} \setminus \{0\} \times \mathbb{R}$ we study when \begin{equation*} \|\alpha p^{\gamma}+\beta \|<p^{-\theta}, \end{equation*} holds for infinitely many primes $p$ of a special type. In particular, we consider when this inequality holds for primes $p$ such that $p+2$ has few prime factors counted with multiplicity. This is done using an exponential sum es...

It is shown that there is an absolute constant $C$ such that any rational $\frac bq\in]0, 1[, (b, q)=1$, admits a representation as a finite sum $\frac bq=\sum_\alpha\frac {b_\alpha}{q_\alpha}$ where $\sum_\alpha\sum_ia_i(\frac {b_\alpha}{q_\alpha})<C\log q$ and $\{a_i(x)\}$ denotes the sequence of partial quotients of $x$.