Related papers: Some remarks on Euler's totient function
The theory of differential equations has an arithmetic analogue in which derivatives of functions are replaced by Fermat quotients of numbers. Many classical differential equations (Riccati, Weierstrass, Painlev\'{e}, etc.) were previously…
A result of Legendre asserts that the difference between the numbers of (length) even and odd partitions of $n$ into distinct parts is $0$, $1$, or $-1$; this also follows from Euler's pentagonal number theorem. We establish an analogous…
Let $\phi(n)$ be the Euler totient function and $\sigma(n)$ denote the sum of divisors of $n$. In this note, we obtain explicit upper bounds on the number of positive integers $n\leq x$ such that $\phi(\sigma(n)) > cn$ for any $c>0$. This…
We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural…
This study proposes a novel approach to extract topological properties, specifically the Euler characteristic, from input images using neural networks without relying on large pre-existing datasets but with a single geometric image.…
We begin by introducing an interesting class of functions, known as the Schemmel totient functions, that generalizes the Euler totient function. For each Schemmel totient function $L_m$, we define two new functions, denoted $R_m$ and $H_m$,…
A recurrent formula is presented, for the enumeration of the compositions of positive integers as sums over multisets of positive integers, that closely resembles Euler's recurrence based on the pentagonal numbers, but where the…
Let $\sigma(n)$ be the sum of the positive divisors of $n$, and let $A(t)$ be the natural density of the set of positive integers $n$ satisfying $\sigma(n)/n \ge t$. We give an improved asymptotic result for $\log A(t)$ as $t$ grows…
We obtain recurrences for smallest parts functions which resemble Euler's recurrence for the ordinary partition function. The proofs involve the holomorphic projection of non-holomorphic modular forms of weight 2.
Finding the mean square averages of the Dirichlet $L$-functions over Dirichlet characters $\chi$ of same parity is an active problem in number theory. Here we explicitly evaluate such averages of $L(3,\chi)$ and $L(4,\chi)$ using certain…
For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^{\alpha})=\begin{cases} 0, & \mbox{if } p\leq r; \\ p^{\alpha-1}(p-r), & \mbox{if } p>r…
Let phi(n) be Euler's totient function and let sigma(n) be the sum of the positive divisors of n. We show that most phi-values (integers in the range of phi) are not sigma-values and vice versa.
A pair of numbers is amicable if each number equals the sum of the proper divisors of the other. This paper after exploring the history and evolution of amicable numbers, introduces a novel characterization of amicable pairs whose greatest…
Lehmer's totient problem asks if there exists a composite number $d$ such that its totient divide $d-1$. In this article we generalize the Lehmer's totient problem in algebraic number fields. We introduce the notion of a Lehmer number.…
The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by…
An elementary particle such as a photon cannot be cut in two pieces. Still it must be possible to truncate a photon with an optical shutter. The result is neither another photon nor a mix of a photon and a vacuum. Instead it is a…
The Eulerian numbers form a triangular array with many interesting properties. The numbers arise from various combinatorial and probabilistic interpretations, and have been studied in a variety of mathematical contexts. In this article we…
The purpose of this article is to introduce the concept of invariance and its properties. These properties can be used to check the primality of a number. Combining these properties with the Euler theorem, it is possible to generalize this…
Recently, Bordell\'{e}s, Dai, Heyman, Pan and Shparlinski in \cite{Igor} considered a partial sum involving the Euler totient function and the integer parts $\lfloor x/n\rfloor$ function. Among other things, they obtained reasonably tight…
In this paper, we use the Lambert series generating function for Euler's totient function to introduce a new identity for the number of $1$'s in the partitions of $n$. A new expansion for Euler's partition function $p(n)$ is derived in this…