Related papers: A central limit theorem for partitions involving g…

Let $\mathcal{A}=\left(a_i\right)_{i=1}^\infty$ be a weakly increasing sequence of positive integers and let $k$ be a fixed positive integer. For an arbitrary integer $n$, the restricted partition $p_\mathcal{A}(n,k)$ enumerates all the…

We present a class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ with bounded partial sums. The novelty here is that our functions do not need to have modulus bounded by $1$. The key feature is that they pretend to be the constant…

Inspired by a question asked on the list {\tt mathfun}, we revisit {\em Kempner-like series}, i.e., harmonic sums $\sum' 1/n$ where the integers $n$ in the summation have ``restricted'' digits. First we give a short proof that $\lim_{k \to…

We investigate when the exponential sum $S_f(x,\alpha) := \sum_{n\le x}f(n)\mathrm{e}(n\alpha)$ is bounded, for a multiplicative function $f$ and $\alpha\in\mathbb{R}$. We show that under natural assumptions, $S_f(x,\alpha)$ is bounded only…

A generalised summation method is considered based on the Fourier series of periodic distributions. It is shown that $$ e^{it}-2e^{2it}+3e^{3it}-4e^{4it}+-\cdots = {\mathrm P\mathrm f} {\displaystyle \frac{e^{it}}{(1+e^{it})^2}} +i\pi…

We extend the Matom\"{a}ki-Radziwi\l\l{} theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a…

We find general solutions to the generating-function equation sum c_q^{(X)} z^q = F(z)^X, where X is a complex number and F(z) is a convergent power series with |F(0)| >0. We then use these results to derive finite expressions containing…

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$. Let $F(s)= \sum_{n=1}^\infty f(n)n^{-s}$ be the associated Dirichlet series and $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ be the truncated Dirichlet…

The Central Limit Theorem (CLT) establishes that sufficiently large sequences of independent and identically distributed random variables converge in probability to a normal distribution. This makes the CLT a fundamental building block of…

We establish several results concerning the expected general phenomenon that, given a multiplicative function $f:\mathbb{N}\to\mathbb{C}$, the values of $f(n)$ and $f(n+a)$ are "generally" independent unless $f$ is of a "special" form.…

We define new generalized factorials in several variables over an arbitrary subset $\underline{S} \subseteq R^n,$ where $R$ is a Dedekind domain and $n$ is a positive integer. We then study the properties of the fixed divisor…

We obtain the empirical strong law of large numbers, empirical Glivenko-Cantelli theorem, central limit theorem, functional central limit theorem for various nonparametric Bayesian priors which include the Dirichlet process with general…

In this paper, we consider the general divisor functions over Piatetski-Shapiro sequences. We can give some general results which contain some special divisor functions. Precisely, we extend the divisor problem over Piatetski-Shapiro…

Divisor functions have attracted the attention of number theorists from Dirichlet to the present day. Here we consider associated divisor functions $c_j^{(r)}(n)$ which for non-negative integers $j, r$ count the number of ways of…

We investigate fractional sums of arithmetic functions over products of two or three integers, with emphasis on fixed greatest common divisors and multiplicative weights. Let $f$ be an arithmetic function satisfying $f(n) \ll n^\alpha$ for…

We prove some distribution results for the $k$-fold divisor function in arithmetic progressions to moduli that exceed the square-root of length $X$ of the sum, with appropriate constrains and averaging on the moduli, saving a power of $X$…

In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function $\sigma_{\alpha}(n):=\sum_{d|n}d^{\alpha}$. We obtain an exact identity relating the Dirichlet series…

DP-means clustering was obtained as an extension of $K$-means clustering. While it is implemented with a simple and efficient algorithm, it can estimate the number of clusters simultaneously. However, DP-means is specifically designed for…

The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. We find the behavior of coefficients in the partial fraction decomposition of this product as $N \to…

Let $f: \mathbb{N}^2 \mapsto \mathbb{C}$ be an arithmetic function of two variables. We study the existence of the limit: \[\displaystyle \lim_{x \to \infty} \frac{1}{x^2 (\log x)^{k-1}} \sum_{n_1 , n_2 \le x} f (n_1, n_2) \] where $k$ is a…