Related papers: A relation to a remainder terms in an asymptotic f…

We present a proof given by Euler in his paper {\it ``De serierum determinatione seu nova methodus inveniendi terminos generales serierum"} \cite{E189} (E189:``On the determination of series or a new method of finding the general terms of…

We prove certain conjecture holds true for a finite category which has M\"obius inversion. The conjecture states a relationship between the zeta function of a finite category and the Euler characteristic of a finite category.

It is shown that a known asymptotic expansion of the Kummer function $U(a,b,z)$ as $a$ tends to infinity is valid for $z$ on the full Riemann surface of the logarithm. A corresponding result is also proved in a more general setting…

Euler Maclaurin formulas for a polytope express the sum of the values of a function over the lattice points in the polytope in terms of integrals of the function and its derivatives over faces of the polytope or its expansions. Exact Euler…

In the work we shall present formulas to sum Lambert series using Euler's q-exponential functions, and several Lambert series associated with well-known arithmetic functions are given as examples. These functions are: the M\"{o}bius…

Determination of linear combination of exponential functions with unknown rate constants from its sampled values is a problem of considerable interest. Here we present a constructive and explicit solution to this problem. Moments of such…

We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients are not too large and do not differ from each other by too much. Additionally, we prove a number of…

We give a simple proof of a classical theorem by A.M\'at\'e, P.Nevai, and V.Totik on asymptotic behavior of orthogonal polynomials on the unit circle. It is based on a new real-variable approach involving an entropy estimate for the…

In 1980 Montgomery made a conjecture about the true order of the error term in the prime number theorem. In 2012 the author made an analogous conjecture for the true order of the sum of the M\"{o}bius function, $M(x)$. This refined an…

Given a function $f: (a,b) \rightarrow \mathbb{R},$ L\"owner's theorem states $f$ is monotone when extended to self-adjoint matrices via the functional calculus, if and only if $f$ extends to a self-map of the complex upper half plane. In…

We prove that entire transcendental holomorphic functions with an omitted value have infinite entropy. A proof for general transcendental entire functions will be given in an upcoming paper.

Let $H_k = 1 + 1/2 + 1/3 + \cdots + 1/k$ denote the $k$th harmonic number. We present an easy-to-implement algorithm for the computation of explicit closed-form evaluations, in terms of the digamma and polygamma functions, for Euler sums of…

In the framework of the planar Euler problem in the quasi--periodic regime, the formulae of the periods available in the literature are simple only on one side of their singularity. In this paper, we complement such formulae with others,…

This paper deals with the evaluation of some definite Euler-type integrals in terms of the Wright hypergeometric function. We obtain a theorem on the Wright hypergeometric function and then use this theorem to evaluate some definite…

In this paper, we obtain some formulas for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. By using these formulas, we give new closed form sums of several quadratic Euler series through Riemann zeta…

In this paper certain Tur\'an type inequalities for some Lommel functions of the first kind are deduced. The key tools in our proofs are the infinite product representation for these Lommel functions of the first kind, a classical result of…

We obtain an upper bound for the sum $\sum_{n\leq N} (a_{n}/\varphi (a_{n}))^{s}$, where $\varphi$ is Euler's totient function, $s\in \mathbb{N}$, and $a_{1},\ldots, a_{N}$ are positive integers (not necessarily distinct) with some…

Motivated by the study of the summatory $k$-free indicator and totient functions in the classical setting, we investigate their function field analogues. First, we derive an expression for the error terms of the summatory functions in terms…

Using functional equations, we define functors that generalize standard examples from calculus of one variable. Examples of such functors are discussed and their Taylor towers are computed. We also show that these functors factor through…

We provide an explicit $O(x\log x/T)$ error term for the Riemann--von Mangoldt formula by making results of Wolke (1983) and Ramar\'e (2016) explicit. We also include applications to primes between consecutive powers, the error term in the…