Related papers: Spectrum of multiplicative functions over powerful…
We achieve the multifractal analysis of a class of complex valued statistically self-similar continuous functions. For we use multifractal formalisms associated with pointwise oscillation exponents of all orders. Our study exhibits new…
This paper considers MEP - Mixed Exponential Polynomials as one class of real exponential polynomials. We introduce a method for proving the positivity of MEP inequalities over positive intervals using the Maclaurin series to approximate…
We develop a variant of the generalized slow roll approach for calculating the curvature power spectrum that is well-suited for order unity deviations in power caused by sharp features in the inflaton potential. As an example, we show that…
We construct a smooth real-valued function P(n) in [0,1], defined via a triple integral with a periodic kernel, that approximates the characteristic function of prime numbers. The function is built to suppress when n is divisible by some m…
We initiate a program of average smoothness analysis for efficiently learning real-valued functions on metric spaces. Rather than using the Lipschitz constant as the regularizer, we define a local slope at each point and gauge the function…
We show that if $f$ is the random completely multiplicative function, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is positive for every $x$ is at least $1-10^{-45}$, while also strictly smaller than $1$. For large $x$, we prove an…
We determine the special values at positive integers of the spectral zeta function associated with the combinatorial Laplacian on the regular tree. These values admit explicit formulas in terms of certain polynomials, which we show to be…
In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank four Lie groups with biinvariant Riemannian metric, corresponding to root systems…
Let $(\mathcal{A},\mathrm{tr})$ be a von Neumann algebra with a faithful, normal trace $\mathrm{tr}:\mathcal{A}\rightarrow\mathbb{C}.$ For each $a\in\mathcal{A},$ define \[…
Let $f\colon\mathbb{N}\rightarrow\mathbb{N}_0$ be a multiplicative arithmetic function such that for all primes $p$ and positive integers $\alpha$, $f(p^{\alpha})<p^{\alpha}$ and $f(p)\vert f(p^{\alpha})$. Suppose also that any prime that…
We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs.…
In this paper we study some fundamental algebraic properties of slice functions and slice regular functions over an alternative $^*$-algebra $A$ over $\mathbb{R}$. These recently introduced function theories generalize to higher dimensions…
We prove a generalized version of Rogers' mean value formula in the space $X_n$ of unimodular lattices in $R^n$, which gives the mean value of a multiple sum over a lattice $L$ and its dual $L^*$. As an application, we prove that for $L$…
We provide uniform bounds on mean values of multiplicative functions under very general hypotheses, detecting certain power savings missed in known results in the literature. As an application, we study the distribution of the…
We characterize maps $\phi_i: \mathcal{S} \to \mathcal{S}$, $i=1, \ldots, m$ and $m\ge 1$, that have the multiplicative spectrum or trace preserving property: \begin{eqnarray*} \textrm{spec} (\phi_1(A_1)\cdots \phi_m(A_m)) &=& \textrm{spec}…
We study the complex spectrum of the partial theta function \[ \Theta(q,x)=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j, \qquad |q|<1, \] where a spectral value is a parameter for which \(\Theta(q,\cdot)\) has a multiple zero. Since the function is…
The (classical) Lagrange spectrum is a closed subset of the positive real numbers defined in terms of diophantine approximation. Its structure is quite involved. This article describes a polynomial time algorithm to approximate it in…
A bounded measurable set $\Omega\subset{\mathbb R}^d$ is called a spectral set if it admits some exponential orthonormal basis $\{e^{2\pi i \langle\lambda,x\rangle}: \lambda\in\Lambda\}$ for $L^2(\Omega)$. In this paper, we show that in…
The distance of an operation from being associative can be "measured" by its associative spectrum, an appropriate sequence of positive integers. Associative spectra were introduced in a publication by B. Cs\'ak\'any and T. Waldhauser in…
Recently there have been differing viewpoints on how to evaluate the curvature power spectrum generated during inflation. Since the primordial curvature power spectrum is the seed for structure formation and provides a link between…