18 problems

Let $L(s,f)$ be a suitably normalized nice $L$-function, and call a sequence of positive real values Benford when the frequencies of its leading digits follow Benford's probabiliti…

Naturalness conjecture for Benford's law. Benford's law is an act of nature and can be used to indicate the randomness of our world.

Newcomb's first-digit conjecture. The first digit is $1$ about $30.1\%$ of the time, whereas it is $9$ only about $4.6\%$ of the time. This is the numerical form of the proposed lo…

Fix an integer $k\geq2$. In the discrete stick-fragmentation process, break each stick into $k$ pieces by choosing $k-1$ cut points recursively according to the uniform distributio…

Fix an even modulus $n\geq2$, a subset $S\subset\{0,\dots,n-1\}$ of residue classes, and the associated discrete stick-fragmentation process with stopping set determined by $S$. Le…

Let $\mathfrak{S}\subseteq\mathbb{Z}_+$ be a stopping set such that the natural density exists, and write … Assume that $r>0$. Density-half conjecture. The set of dead stick length…

Consider chained probability distributions and the leading digits of the resulting random quantities. Kossovsky's conjecture. As the length of the chain increases, the leading-digi…

Lower-dimensional volume conjecture. A $d$-dimensional volume for an $m$-dimensional object, where $d<m$, undergoing a fragmentation process follows the Benford distribution for th…

Rectangle-perimeter conjecture. The perimeters of the sub-rectangles from a rectangle undergoing an unrestricted fragmentation process converge to Benford behavior.

One-piece fragmentation conjecture. If only one piece is fragmented at a time, the resulting pieces still converge to Benford behavior.

Lower-dimensional volume conjecture. Even lower-dimensional volumes under the unrestricted fragmentation process follow Benford's Law.

A random walk on the real line is a sequence of partial sums of independent identically distributed real-valued increments. A path is Benford when its significand behavior follows…

Let $a\geq 2$ be an integer that is not a power of $10$, let $d\in\{1,2,\dots,9\}$, and let $E_d(N,\{a^n\})$ denote the Benford error. Nonnormal distribution conjecture for Benford…

Let $a>0$ be a real number satisfying the relevant condition, and write $\alpha=\log_{10}a$. Suppose that the continued fraction expansion of $\alpha$ satisfies … For a digit…

Fix a monotonically increasing sequence $\{a_n\}$. In the additive decomposition process, a stick whose length belongs to $\{a_n\}$ does not decompose further; sticks of length $1$…

Consider an additive decomposition process starting from a stick of integer length $L$: choose uniformly an integer cut point $c\in[1,L-1]$, replace the stick by pieces of lengths…

3x+1 Benford obstruction. The main $3x+1$ conjecture predicts that this sequence is eventually periodic; consequently, the full infinite sequence cannot satisfy the asserted Benfor…

Kossovsky's conjecture. As the length of a chain of probability distributions increases, the distribution of leading digits converges to Benford's law. The conjecture proposes a ge…