Related papers: The normality of digits in almost constant additiv…

Let $\mathcal{F}\subset\mathcal{M}(D)$ and let $a, b$ and $c$ be three distinct complex numbers. If, there exist a holomorphic function $h$ on $D$ and a positive constant $\rho$ such that for each $f\in\mathcal{F},$ $f$ and $f^{'}$…

Generalized L\"uroth series generalize $b$-adic representations as well as L\"uroth series. Almost all real numbers are normal, but it is not easy to construct one. In this paper, a new construction of normal numbers with respect to…

We prove that a point $x$ is normal with respect to an ergodic, number-theoretic transformation $T$ if and only if $x$ is normal with respect to $T^n$ for any $n\ge 1$. This corrects an erroneous proof of Schweiger. Then, using some…

The Landau-Selberg-Delange method provides an asymptotic formula for the partial sums of a multiplicative function whose average value on primes is a fixed complex number $v$. The shape of this asymptotic implies that $f$ can get very small…

Fix a base B and let zeta have the standard exponential distribution; the distribution of digits of zeta base B is known to be very close to Benford's Law. If there exists a C such that the distribution of digits of C times the elements of…

Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0, 1, ..., s-1 occurs with the same frequency 1/s. Let X be the set of positive integers that are not perfect…

Champernowne famously proved that the number $0.(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)...$ formed by concatenating all the integers one after another is normal base 10. We give a generalization of Champernowne's construction to various…

The decimal digits of $\pi$ are widely believed to behave like as statistically independent random variables taking the values $0, 1, 2, 3, 4, 5$, $6, 7, 8, 9$ with equal probabilities $1/10$. In this article, first, another similar…

We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…

The average value of log s(n)/n taken over the first N even integers is shown to converge to a constant lambda when N tends to infinity; moreover, the value of this constant is approximated and proven to be less than 0. Here s(n) sums the…

When numbers are added in base $b$ in the usual way, carries occur. If two random, independent 1-digit numbers are added, then the probability of a carry is $\frac{b-1}{2b}$. Other choices of digits lead to less carries. In particular, if…

In this paper, we show that the concatenation of the Fibonacci sequence is \textit{normal} in base $10$, meaning every string of a given length, $k$, occurs as frequently as every other string of length $k$ (there are as many $1$'s as $2$'s…

Adler, Keane, and Smorodinsky showed that if one concatenates the finite continued fraction expansions of the sequence of rationals \[ \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{1}{5}, \cdots \] into…

In the present paper the authors construct normal numbers in base $q$ by concatenating $q$-adic expansions of prime powers $\lfloor\alpha p^\theta\rfloor$ with $\alpha>0$ and $\theta>1$.

The \textit{ternary digits of $2^n$} are a finite sequence of 0s, 1s, and 2s. It is a natural question to ask whether the frequency of any string of 0s, 1s, and 2s in this sequence approaches the same limit for all strings of the same…

Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation…

A real number $x$ is considered normal in an integer base $b \geq 2$ if its digit expansion in this base is ``equitable'', ensuring that for each $k \geq 1$, every ordered sequence of $k$ digits from $\{0, 1, \ldots, b-1\}$ occurs in the…

In 2008 or earlier, Michel Mend\`es France asked for an instance of a real number $x$ such that both $x$ and $1/x$ are simply normal to a given integer base $b$. We give a positive answer to this question by constructing a number $x$ such…

For an integer b>=2, let s_b(n) be the sum of the digits of the integer n when written in base b, and let S_b(N) be the sum of s_b(n) over n=0,...,N-1, so that S_b(N) is the sum of all b-ary digits needed to write the numbers 0,1,...,N-1.…

Let $s_2$ be the sum-of-digits function in base $2$, which returns the number of non-zero binary digits of a nonnegative integer $n$. We study $s_2$ alon g arithmetic subsequences and show that --- up to a shift --- the set of $m$-tuples of…