Related papers: Multiplicative functions in large arithmetic progr…
We establish several new $\Omega$-theorems for logarithmic derivatives of the Riemann zeta function and Dirichlet $L$-functions. In particular, this improves on earlier work of Landau (1911), Bohr-Landau (1913), and recent work of Lamzouri.
For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions…
We introduce and study properties of certain new harmonic function spaces on products of upper half-spaces.Norm estimates for the so-called expanded Bergman projections are obtained.Sharp theorems on multipliers acting on certain Sobolev…
We prove that the divisor function $d(n)$ counting the number of divisors of the integer $n$, is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system $(X, {\mathcal A},\nu,\tau)$ and any $f\in…
In the present paper, we were mainly concerned with obtaining estimates for the general Taylor-Maclaurin coefficients for functions in a certain general subclass of analytic bi-univalent functions. For this purpose, we used the Faber…
We establish a higher dimensional counterpart of Bourgain's pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates $V_r$ on $L^p$ spaces for all $1<p<\infty$ and…
We answer a number of questions of Erd\H{o}s on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a…
We establish results of Bombieri-Vinogradov type for the von Mangoldt function $\Lambda(n)$ twisted by a nilsequence. In particular, we obtain Bombieri-Vinogradov type results for the von Mangoldt function twisted by any polynomial phase…
Several new estimates for the 2-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as…
An extension of the empirical copula is considered by combining an estimator of a multivariate cumulative distribution function with estimators of the marginal cumulative distribution functions for marginal estimators that are not…
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…
We give two improved explicit versions of the prime number theorem for primes in arithmetic progression: the first isolating the contribution of the Siegel zero and the second completely explicit, where the improvement is for medium-sized…
Extensions of the $Stirling$ numbers of the second kind and $Dobinski$ -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper $ [1]$.…
We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that…
We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is 'approximately multiplicative' and uniformly distributed on short…
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…
We use the properties of Hermite and Kamp\'e de F\'eriet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher-order polynomial. The results we obtain are extended to product of…
For a function of a type $ \left| \mathbf{r}_1{+}\ldots {+}\mathbf{r}_{_N} \right|^{-\nu} \in \mathbb{R} $ from the many-dimensional vectors $ \mathbf{r}_s $ in Euclidean space, the successive algebraic approach is the derivation of the…
We present some properties of the gradient of a mu-differentiable function. The Method of Lagrange Multipliers for mu-differentiable functions is then exemplified.
We revisit several hybrid multiplicative-to-additive type functions from a recent preprint article. These functions, $g(n)$ with Dirichlet generating function (DGF) $\zeta(s)^{-1} (1+P(s))^{-1}$ for $\Re(s) > 1$ where $P(s) = \sum_p p^{-s}$…