arXiv subjects
Publications and source records attributed to Michael Coons.
In a recent paper, Roland Bacher conjectured three identities concerning Stern's sequence and its twist. In this paper we prove Bacher's conjectures. Possibly of independent interest, we also give a way to compute the Stern value (or twisted Stern value) of a number based solely on its binary expansion.
In this paper, we give a new proof and an extension of the following result of Bézivin. Let $f:\B{N}\to K$ be a multiplicative function taking values in a field $K$ of characteristic 0 and write $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ for its generating series. Suppose that $F(z)$ is algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function $χ(n)$ such that $f(n)=n^k χ(n)$ for all $n$, or $f(n)$ is eventually zero. In particular, $F(z)$ is either transcendental or rational. For $K=\B{C}$, we also prove that if $F(z)$ is a $D$-finite generating series of a multiplicative function, then $F(z)$ is either transcendental or rational.
Let $K$ be a field of characteristic 0, $f:\mathbb{N}\to K$ be a multiplicative function, and $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ be algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function $χ(n)$ such that $f(n)=n^k χ(n)$ for all $n$, or $f(n)$ is eventually zero. In particular, the generating function of a multiplicative function $f:\mathbb{N}\to K$ is either transcendental or rational.
The {\em Liouville function} is defined by $\gl(n):=(-1)^{Ω(n)}$ where $Ω(n)$ is the number of prime divisors of $n$ counting multiplicity. Let $\z_m:=e^{2πi/m}$ be a primitive $m$--th root of unity. As a generalization of Liouville's function, we study the functions $\gl_{m,k}(n):=\z_m^{kΩ(n)}$. Using properties of these functions, we give a weak equidistribution result for $Ω(n)$ among residue classes. More formally, we show that for any positive integer $m$, there exists an $A>0$ such that for all $j=0,1,...,m-1,$ we have $$#\{n\leq x:Ω(n)\equiv j (\bmod m)\}=\frac{x}{m}+O(\frac{x}{\log^A x}).$$ Best possible error terms are also discussed. In particular, we show that for $m>2$ the error term is not $o(x^\ga)$ for any $\ga<1$.
A well-known result from Hardy and Ramanujan gives an asymptotic expression for the number of possible ways to express an integer as the sum of smaller integers. In this vein, we consider the general partitioning problem of writing an integer $n$ as a sum of summands from a given sequence $\gL$ of non-decreasing integers. Under suitable assumptions on the sequence $\gL$, we obtain results using associated zeta-functions and saddle-point techniques. We also calculate higher moments of the sequence $\gL$ as well as the expected number of summands. Applications are made to various sequences, including those of Barnes and Epstein types. These results are of potential interest in statistical mechanics in the context of Bose-Einstein condensation.
Denote by $λ(n)$ Liouville's function concerning the parity of the number of prime divisors of $n$. Using a theorem of Allouche, Mendès France, and Peyrière and many classical results from the theory of the distribution of prime numbers, we prove that $λ(n)$ is not $k$--automatic for any $k> 2$. This yields that $\sum_{n=1}^\infty λ(n) X^n\in\mathbb{F}_p[[X]]$ is transcendental over $\mathbb{F}_p(X)$ for any prime $p>2$. Similar results are proven (or reproven) for many common number--theoretic functions, including $ϕ$, $μ$, $Ω$, $ω$, $ρ$, and others.
Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $λ_{A}(n) =(-1)^{Ω_A(n)}$ where $Ω_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville function, $A$ is the set of all primes. Denote $$L_A(n):=\sum_{k\leq n}λ_A(n)\quad{and}\quad R_A:=\lim_{n\to\infty}\frac{L_A(n)}{n}.$$ We show that for every $α\in[0,1]$ there is an $A\subset P$ such that $R_A=α$. Given certain restrictions on $A$, asymptotic estimates for $\sum_{k\leq n}λ_A(k)$ are also given. With further restrictions, more can be said. For {\em character--like functions} $λ_p$ ($λ_p$ agrees with a Dirichlet character $χ$ when $χ(n)\neq 0$) exact values and asymptotics are given; in particular $$\quad\sum_{k\leq n}λ_p(k)\ll \log n.$$ Within the course of discussion, the ratio $ϕ(n)/σ(n)$ is considered.
We give a new proof of Fatou's theorem: {\em if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This result is applied to show that for any non--trivial completely multiplicative function from $\mathbb{N}$ to $\{-1,1\}$, the series $\sum_{n=1}^\infty f(n)z^n$ is transcendental over $\mathbb{Z}[z]$; in particular, $\sum_{n=1}^\infty λ(n)z^n$ is transcendental, where $λ$ is Liouville's function. The transcendence of $\sum_{n=1}^\infty μ(n)z^n$ is also proved.
{\em The Liouville number}, denoted $l$, is defined by $$l:=0.100101011101101111100...,$$ where the $n$th bit is given by ${1/2}(1+\gl(n))$; here $\gl$ is the Liouville function for the parity of prime divisors of $n$. Presumably the Liouville number is transcendental, though at present, a proof is unattainable. Similarly, define {\em the Gaussian Liouville number} by $$γ:=0.110110011100100111011...$$ where the $n$th bit reflects the parity of the number of rational Gaussian primes dividing $n$, 1 for even and 0 for odd. In this paper, we prove that the Gaussian Liouville number and its relatives are transcendental. One such relative is the number $$\sum_{k=0}^\infty\frac{2^{3^k}}{2^{3^k2}+2^{3^k}+1}=0.101100101101100100101...,$$ where the $n$th bit is determined by the parity of the number of prime divisors that are equivalent to 2 modulo 3. We use methods similar to that of Dekking's proof of the transcendence of the Thue--Morse number \cite{Dek1} as well as a theorem of Mahler's \cite{Mahl1}. (For completeness we provide proofs of all needed results.) This method involves proving the transcendence of formal power series arising as generating functions of completely multiplicative functions.