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For an n$n$‐bit positive integer a$a$ written in binary as a=∑j=0n−1εj(a)2j,$ {a} = \sum _{j=0}^{n-1} \varepsilon _{j}(a) \,2^j,$ where εj(a)∈{0,1}$\varepsilon _j(a) \in \lbrace 0,1\rbrace$, j∈{0,...,n−1}$j\in \lbrace 0, \ldots , n-1\rbrace$, εn−1(a)=1$\varepsilon _{n-1}(a)=1$, let us define a←=∑j=0n−1εj(a)2n−1−j,$ \overleftarrow{a} = \sum _{j=0}^{n-1} \varepsilon _j(a)\,2^{n-1-j},$ the digital reversal of a$a$. Also let Bn={2n−1⩽a<2n:aodd}${\mathcal {B}}_n= \lbrace 2^{n-1}\leqslant a<2^n:\nobreakspace a \text{ odd}\rbrace$. With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of p∈Bn$p \in {\mathcal {B}}_n$ such that p$p$ and p←$\overleftarrow{p}$ are prime. We also prove that for sufficiently large n$n$, {a∈Bn:max{Ω(a),Ω(a←)}⩽8}⩾c2nn2,$$\begin{equation*} \hspace*{30pt}{\left| \lbrace a \in {\mathcal {B}}_n:\nobreakspace \max \lbrace \Omega (a), \Omega (\overleftarrow{a})\rbrace \leqslant 8 \rbrace \right|} \geqslant c\, \frac{2^n}{n^2}, \end{equation*}$$where Ω(n)$\Omega (n)$ denotes the number of prime factors counted with multiplicity of n$n$ and c>0$c > 0$ is an absolute constant. Finally, we provide an asymptotic formula for the number of n$n$‐bit integers a$a$ such that a$a$ and a←$\overleftarrow{a}$ are both squarefree. Our method leads us to provide various estimates for the exponential sum ∑a∈Bnexp2πi(αa+ϑa←)(α,ϑ∈R).$$\begin{equation*} \hspace*{30pt}\sum _{a \in {\mathcal {B}}_n} \exp {\left(2\pi i (\alpha a + \vartheta \overleftarrow{a})\right)} \quad (\alpha,\vartheta \in \mathbb {R}). \end{equation*}$$ [ABSTRACT FROM AUTHOR]

Bourgain [Israel J. Math. 206 (2015) 165-182] estimated the number of prime numbers with a proportion c > 0 of preassigned digits in base 2 (c is an absolute constant not specified). We establish a generalization of this result in any base g ≥ 2 and we provide explicit admissible values for the proportion c depending on g. Our proof, which develops and enhances Bourgain's arguments, is based on the circle method and combines techniques from harmonic analysis together with results on zeros of Dirichlet L-functions, notably a very strong zero-free region due to Iwaniec. [ABSTRACT FROM AUTHOR]