Benford's law for the $3x+1$ function

On Benford's Law for multiplicative functions

We provide a criterion to determine whether a real multiplicative function is a strong Benford sequence. The criterion implies that the $k$-divisor functions, where $k \neq 10^j$, and Hecke eigenvalues of newforms, such as Ramanujan tau function, are strong Benford. Moreover, we deduce from the criterion that the collection of multiplicative functions which are not strong Benford forms a group under pointwise multiplication. In contrast to earlier work, our approach is based on Halász's Theorem.

Order Statistics and Benford's Law

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 some set is the same as that of zeta, we say that set exhibits shifted exponential behavior base B (with a shift of log_B C \bmod 1). Let X_1, >..., X_N be independent identically distributed random variables. If the X_i's are drawn from the uniform distribution on [0,L], then as N\to\infty the distribution of the digits of the differences between adjacent o…

Benford's Law for Coefficients of Newforms

Let $f(z)=\sum_{n=1}^\infty λ_f(n)e^{2πi n z}\in S_{k}^{new}(Γ_0(N))$ be a normalized Hecke eigenform of even weight $k\geq2$ on $Γ_0(N)$ without complex multiplication. Let $\mathbb{P}$ denote the set of all primes. We prove that the sequence $\{λ_f(p)\}_{p\in\mathbb{P}}$ does not satisfy Benford's Law in any base $b\geq2$. However, given a base $b\geq2$ and a string of digits $S$ in base $b$, the set \[ A_{λ_f}(b,S):=\{\text{$p$ prime : the first digits of $λ_f(p)$ in base $b$ are given by $S$}\} \] has logarithmic density equal to $\log_b(1+S^{-1})$. Thus $\{λ_f(p)\}_{p\in\mathbb{P}}$ follo…

A characterization of Benford's Law in discrete-time linear systems

A necessary and sufficient condition ("nonresonance") is established for every solution of an autonomous linear difference equation, or more generally for every sequence $(x^\top A^n y)$ with $x,y\in \mathbb{R}^d$ and $A\in \mathbb{R}^{d\times d}$, to be either trivial or else conform to a strong form of Benford's Law (logarithmic distribution of significands). This condition contains all pertinent results in the literature as special cases. Its number-theoretical implications are discussed in the context of specific examples, and so are its possible extensions and modifications.

The first digit frequencies of primes and Riemann zeta zeros tend to uniformity following a size-dependent generalized Benford's law

Prime numbers seem to distribute among the natural numbers with no other law than that of chance, however its global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated sci- entists of all ages to search for local and global patterns in this distribution that eventually could shed light into the ultimate nature of primes. In this work we show that a generalization of the well known first-digit Benford's law, which addresses the rate of appearance of a given leading digit d in data sets, describes with astonishing precision the sta…

环 $\mathbb{Z}(\sqrt{D})$ 中的 Benford 定律

Benford's Law in the ring $\mathbb{Z}(\sqrt{D})$

本文研究了二次整数环 $\mathbb{Z}(\sqrt{D})$ 中满足特定方程的子集 $\mathbb{Z}_k(\sqrt{D})$，证明其自然排列序列中的整数 $x$ 和 $y$ 均构成强 Benford 序列。

Benford Behavior of Generalized Zeckendorf Decompositions

We prove connections between Zeckendorf decompositions and Benford's law. Recall that if we define the Fibonacci numbers by $F_1 = 1, F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$, every positive integer can be written uniquely as a sum of non-adjacent elements of this sequence; this is called the Zeckendorf decomposition, and similar unique decompositions exist for sequences arising from recurrence relations of the form $G_{n+1}=c_1G_n+\cdots+c_LG_{n+1-L}$ with $c_i$ positive and some other restrictions. Additionally, a set $S \subset \mathbb{Z}$ is said to satisfy Benford's law base 10 if the densi…

Significant Digits of Primes in Subsets

Benford's Law describes the prevalence of small numbers as the leading digits of numbers in many sets of integers. We prove a variant of Benford's law for many positive-density subsets of the primes. This follows from a more general result over number fields.

论对数 Benford 序列

On Logarithmically Benford Sequences

本文研究受多项式界控制且归一化系数关于对称测度等分布的序列，刻画其在算术密度下不服从而在对数密度下可服从 Benford 定律的条件，并应用于偶权 newform 的 Frobenius 迹。

Two problems on the distribution of Carmichael's lambda function

Let $λ(n)$ denote the exponent of the multiplicative group modulo $n$. We show that when $q$ is odd, each coprime residue class modulo $q$ is hit equally often by $λ(n)$ as $n$ varies. Under the stronger assumption that $\gcd(q,6)=1$, we prove that equidistribution persists throughout a Siegel--Walfisz-type range of uniformity. By similar methods we show that $λ(n)$ obeys Benford's leading digit law with respect to natural density. Moreover, if we assume GRH, then Benford's law holds for the order of $a$ mod $n$, for any fixed integer $a\notin \{0,\pm 1\}$.