A superadditive property of Hadamard’s gamma function

Hadamard’s gamma function is defined by

Absolutely monotonic functions related to Euler’s gamma function and Barnes’ double and triple gamma function

It is shown that the remainders in asymptotic expansions of the logarithm of Barnes double and triple gamma function and Euler’s gamma function are Laplace transformations of positive multiples of absolutely monotonic functions. Applications concerning positivity of sums involving Bernoulli numbers are given.     

Absolutely monotonic functions related to Euler’s gamma function and Barnes’ double and triple gamma function

It is shown that the remainders in asymptotic expansions of the logarithm of Barnes double and triple gamma function and Euler’s gamma function are Laplace transformations of positive multiples of absolutely monotonic functions. Applications concerning positivity of sums involving Bernoulli numbers are given.     

A monotonicity property of Riemann’s xi function and a reformulation of the Riemann hypothesis

We prove that Riemann’s xi function is strictly increasing (respectively, strictly decreasing) in modulus along every horizontal half-line in any zero-free, open right (respectively, left) half-plane. A corollary is a reformulation of the Riemann Hypothesis.     

Complex Euler’s groups and values of Euler’s function at complex integer Gauss points

The complex Euler group is defined associating to an integer complex number z the multiplicative group of the complex integers residues modulo z, relatively prime to z. This group is calculated for z=(3+0i) ⁿ : it is isomorphic to the product of three cyclic group or orders (8, 3 ⁿ⁻¹ and 3 ⁿ⁻¹).     

Sets of monotonicity for Euler’s totient function

We study subsets of [1,x] on which the Euler φ-function is monotone (nondecreasing or nonincreasing). For example, we show that for any ?>0, every such subset has size smaller than ?x, once x>x ₀(?). This confirms a conjecture of the second author.     

A generalization of Euler’s constant

The purpose of this paper is to evaluate the limit γ(a) of the sequence , where a ∈ (0, + ∞ ).      

On the oscillation property of Green’s function of a fourth-order discontinuous boundary-value problem

The paper deals with conditions under which the Green function of a multipoint boundary-value problem for fourth-order equations describing small strains of a rod fastened to a solid elastic basement and additionally fixed by “concentrated” elastic supports at separate points has the oscillation property. It is shown that the condition that the Green function is positive is necessary and sufficient for the Green function to have the oscillation property.     

On a new constant related to Euler’s constant

This paper introduces a new constant \(\kappa \), with a definition closely related to that of the Euler–Mascheroni’s constant \(\gamma \). Some integrals and infinite sums are evaluated in terms of \(\kappa \).     

Two remarks on iterates of Euler’s totient function

Let φ _k denote the kth iterate of Euler’s φ-function. We study two questions connected with these iterates. First, we determine the average order of φ _k and 1/φ _k ; e.g., we show that for each k ≥ 0,

$\sum_{n \leq x} \varphi_{k+1}(n) \sim \frac{3}{k! {\rm e}^{k\gamma}\pi^2}\frac{x^2}{(\log_3{x})^k}\qquad (x\to\infty),$

where γ is the Euler–Mascheroni constant. Second, for prime values of p, we study the number of distinct primes dividing \({\prod_{k=1}^{\infty}\varphi_k(p)}\). These prime divisors are precisely the primes appearing in the Pratt tree for p, which has been the subject of recent work by Ford, Konyagin, and Luca. We show that for each \({\epsilon > 0}\), the number of distinct primes appearing in the Pratt tree for p is \({ > ({\rm log}{p})^{1/2-\epsilon}}\) for all but x ^o(1) primes p ≤ x.

     

A new proof of a generalization of Gerzon’s bound

Periodica Mathematica Hungarica - In this paper we give a short, new proof of a natural generalization of Gerzon’s bound. This bound improves the Delsarte, Goethals and Seidel’s upper...     

A tiling proof of Euler’s Pentagonal Number Theorem and generalizations

The Ramanujan Journal - In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider...     

A variety of Euler’s sum of powers conjecture

Czechoslovak Mathematical Journal - We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $$left{{matrix{{n = {a_1} + {a_2} + ldots + {a_{s -...     

A generalization of Eagon–Reiner’s theorem and a characterization of bi-CMt bipartite and chordal graphs

We give a generalization of Eagon-Reiner’s theorem relating Betti numbers of the Stanley-Reisner ideal of a simplicial complex and the CM_t property of its Alexander dual. Then we characterize bi-CM_t bipartite graphs and bi-CM_t chordal graphs. These are generalizations of recent results due to Herzog and Rahimi.     

A quicker continued fraction approximation of the gamma function related to the Windschitl’s formula

In this paper, based on the Windschitl’s formula, a new continued fraction approximation and some inequalities for the gamma function are established. Finally, for demonstrating the superiority of our new series over the classical ones, some numerical computations are given.     

A new proof of Stetsenko’s theorem

A short proof of V.A. Stetsenko’s theorem on weakly read-many Boolean functions is given, which is based on the technique of representing read-once functions in the form of trees.     

A unification of two refinements of Euler’s partition theorem

We obtain a unification of two refinements of Euler’s partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt’s insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.     

Some properties of extended remainder of binet’s first formula for logarithm of gamma function

In the paper, we extend Binet’s first formula for the logarithm of the gamma function and investigate some properties, including inequalities, star-shaped and sub-additive properties and the complete monotonicity, of the extended remainder of Binet’s first formula for the logarithm of the gamma function and related functions.