Related papers: Summatory Multiplicative Arithmetic Functions: Sca…

Approximations of non-smooth multivariate functions return low-order approximations in the vicinities of the singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating…

From the optimization point of view, a difficulty with parallel MRI with simultaneous coil sensitivity estimation is the multiplicative nature of the non-linear forward operator: the image being reconstructed and the coil sensitivities…

In this paper, we investigate the asymptotics of a class of weighted sums over multiplicative functions and apply our results to deduce a stronger asymptotic form of Yitang Zhang's smoothened GPY sieve with coefficient expressions that are…

We consider the sum of the reciprocals of the middle prime factor of an integer, defined according to multiplicity or not. We obtain an asymptotic expansion in the first case and an asymptotic formula involving an implicit parameter in the…

Special arithmetic series $f(x)=\sum_{n=0}^{\infty}c_nx^n$, whose coefficients $c_n$ are normally given as certain binomial sums, satisfy "self-replicating" functional identities. For example, the equation…

We study the optimization of functions with $n>2$ arguments that have a representation as a sum of several functions that have only $2$ of the $n$ arguments each, termed sums of bivariates, on finite domains. The complexity of optimizing…

Let $p \in (0, \infty)$ be a constant and let $\{\xi_n\} \subset L^p(\Omega, {\mathcal F}, \P)$ be a sequence of random variables. For any integers $m, n \ge 0$, denote $S_{m, n} = \sum_{k=m}^{m + n} \xi_k$. It is proved that, if there…

Let $f$ be a multiplicative function which satisfies \[ f(a^2+b^2+c^2+d^2) = f(a^2+b^2)+f(c^2+d^2) \] for positive integers $a$, $b$, $c$, and $d$. We show that $f$ is the identity function provided that $f(3)\,f(11) \ne 0$. Otherwise,…

Let $f$ be a Rademacher or Steinhaus random multiplicative function. For various arithmetically interesting subsets $\mathcal A\subseteq [1, N]\cap\mathbb N$ such that the distribution of $\sum_{n\in \mathcal A} f(n)$ is approximately…

A brief survey of the zeta function regularization and multiplicative anomaly issues when the associated zeta function of fluctuation operator is the regular at the origin (regular case) as well as when it is singular at the origin…

Let $G(g;x):=\sum_{n\leq x}g(n)$ be the summatory function of an arithmetical function $g(n)$. In this paper, we prove that we can write weighted averages of an arbitrary fixed number $N$ of arithmetical functions $g_{j}(n),\,j\in\left\{…

In this paper, we consider the mean value of the product of two real valued multiplicative functions with shifted arguments. The functions $F$ and $G$ under consideration are close to two nicely behaved functions $A$ and $B$, such that the…

Let $n=\prod_p p^{\nu_p(n)}$ denote the canonical factorization of $n\in \N$. The binomial convolution of arithmetical functions $f$ and $g$ is defined as $(f\circ g)(n)=\sum_{d\mid n} (\prod_p \binom{\nu_p(n)}{\nu_p(d)}) f(d)g(n/d),$ where…

We introduce a general class $F_0$ of additive functions $f$ such that $f(p) = 1$ and prove a tight bound for exponential sums of the form $\sum_{n \le x} f(n) e(\alpha n)$ where $f \in F_0$ and $e(\theta) = \exp(2\pi i \theta)$. Both…

Let $f(n)$ be an arithmetic function with $f(1)\neq0$ and let $f^{-1}(n)$ be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behaviour of $|f^{-1}(n)|$ with regard to the asymptotic behaviour of $|f(n)|$…

We investigate multicritical phenomena in O(N)+O(M)-models by means of nonperturbative renormalization group equations. This constitutes an elementary building block for the study of competing orders in a variety of physical systems. To…

Using a renormalization approach, we study the asymptotic limit distribution of the maximum value in a set of independent and identically distributed random variables raised to a power q(n) that varies monotonically with the sample size n.…

We study the growth rate of the summatory function of the M\"obius function in the context of an algebraic curve over a finite field. Our work shows a strong resemblance to its number field counterpart, which was proved by Ng in 2004. We…

We consider the method of self-similar renormalization for calculating critical temperatures and critical indices. A new optimized variant of the method for an effective summation of asymptotic series is suggested and illustrated by several…

We consider the recovery of real-valued bandlimited functions from the absolute values of their samples, possibly spaced nonuniformly. We show that such a reconstruction is always possible if the function is sampled at more than twice its…