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Inequalities concerning the function π(x): Applications

Laurenţiu Panaitopol (2000)

Acta Arithmetica

Introduction. In this note we use the following standard notations: π(x) is the number of primes not exceeding x, while θ ( x ) = p x l o g p . The best known inequalities involving the function π(x) are the ones obtained in [6] by B. Rosser and L. Schoenfeld: (1) x/(log x - 1/2) < π(x) for x ≥ 67 (2) x/(log x - 3/2) > π(x) for x > e 3 / 2 . The proof of the above inequalities is not elementary and is based on the first 25 000 zeros of the Riemann function ξ(s) obtained by D. H. Lehmer [4]. Then Rosser, Yohe and Schoenfeld...

Inequalities for Taylor series involving the divisor function

Horst Alzer, Man Kam Kwong (2022)

Czechoslovak Mathematical Journal

Let T ( q ) = k = 1 d ( k ) q k , | q | < 1 , where d ( k ) denotes the number of positive divisors of the natural number k . We present monotonicity properties of functions defined in terms of T . More specifically, we prove that H ( q ) = T ( q ) - log ( 1 - q ) log ( q ) is strictly increasing on ( 0 , 1 ) , while F ( q ) = 1 - q q H ( q ) is strictly decreasing on ( 0 , 1 ) . These results are then applied to obtain various inequalities, one of which states that the double inequality α q 1 - q + log ( 1 - q ) log ( q ) < T ( q ) < β q 1 - q + log ( 1 - q ) log ( q ) , 0 < q < 1 , holds with the best possible constant factors α = γ and β = 1 . Here, γ denotes Euler’s constant. This refines a result of Salem, who proved the inequalities...

Inequalities for the arithmetical functions of Euler and Dedekind

Horst Alzer, Man Kam Kwong (2020)

Czechoslovak Mathematical Journal

For positive integers n , Euler’s phi function and Dedekind’s psi function are given by φ ( n ) = n p n p prime 1 - 1 p and ψ ( n ) = n p n p prime 1 + 1 p , respectively. We prove that for all n 2 we have 1 - 1 n n - 1 1 + 1 n n + 1 φ ( n ) n φ ( n ) ψ ( n ) n ψ ( n ) and φ ( n ) n ψ ( n ) ψ ( n ) n φ ( n ) 1 - 1 n n + 1 1 + 1 n n - 1 . The sign of equality holds if and only if n is a prime. The first inequality refines results due to Atanassov (2011) and Kannan & Srikanth (2013).

Infinite families of noncototients

A. Flammenkamp, F. Luca (2000)

Colloquium Mathematicae

For any positive integer n let ϕ(n) be the Euler function of n. A positive integer n is called a noncototient if the equation x-ϕ(x)=n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression ( 2 m k ) m 1 consists entirely of noncototients. We then use computations to detect seven such positive integers k.

Integer matrices related to Liouville's function

Shea-Ming Oon (2013)

Czechoslovak Mathematical Journal

In this note, we construct some integer matrices with determinant equal to certain summation form of Liouville's function. Hence, it offers a possible alternative way to explore the Prime Number Theorem by means of inequalities related to matrices, provided a better estimate on the relation between the determinant of a matrix and other information such as its eigenvalues is known. Besides, we also provide some comparisons on the estimate of the lower bound of the smallest singular value. Such discussion...

Integers with a maximal number of Fibonacci representations

Petra Kocábová, Zuzana Masáková, Edita Pelantová (2005)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We study the properties of the function R ( n ) which determines the number of representations of an integer n as a sum of distinct Fibonacci numbers F k . We determine the maximum and mean values of R ( n ) for F k n &lt; F k + 1 .

Integers with a maximal number of Fibonacci representations

Petra Kocábová, Zuzana Masáková, Edita Pelantová (2010)

RAIRO - Theoretical Informatics and Applications

We study the properties of the function R(n) which determines the number of representations of an integer n as a sum of distinct Fibonacci numbers Fk. We determine the maximum and mean values of R(n) for Fk ≤ n < Fk+1.

Introduction to Diophantine Approximation

Yasushige Watase (2015)

Formalized Mathematics

In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].

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