scholarly journals Asymptotic expansion for the sum of inverses of arithmetical functions involving iterated logarithms

2011 ◽  
Vol 5 (1) ◽  
pp. 80-86 ◽  
Author(s):  
Hacène Belbachir ◽  
Djamel Berkane
Keyword(s):  
Asymptotic Expansion ◽  
Large Class ◽  
Counting Function ◽  
Arithmetical Functions ◽  
Iterated Logarithms ◽  
Prime Counting Function

A generalized formula is obtained for the sum of inverses of the prime counting function for a large class of arithmetical functions related to the iterated logarithms.

scholarly journals Prime numbers with a certain extremal type property

2018 ◽  
Vol 17 (1) ◽  
pp. 127-151
Author(s):  
Edward Tutaj
Keyword(s):  
Riemann Hypothesis ◽  
Convex Set ◽  
Counting Function ◽  
Numerical Data ◽  
Prime Numbers ◽  
Affine Function ◽  
Piecewise Affine ◽  
Order Of Magnitude ◽  
Type Property ◽  
Prime Counting Function

Abstract The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points $({e_k},\pi ({e_k}))_1^\infty $ . The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then ${{{e_k} + 1} \over {{e_k}}} = 1$ . The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.

On a Constant Related to the Prime Counting Function

2015 ◽  
Vol 13 (3) ◽  
pp. 929-938 ◽  
Author(s):  
Djamel Berkane ◽  
Pierre Dusart
Keyword(s):  
Counting Function ◽  
Prime Counting Function

scholarly journals Proof the Skewes’ number is not an integer using lattice points and tangent line

2021 ◽  
Vol 17 (2) ◽  
pp. 5-18
Author(s):  
V. Ďuriš ◽  
T. Šumný ◽  
T. Lengyelfalusy
Keyword(s):  
Counting Function ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Integral Function ◽  
Lattice Points ◽  
Tangent Line ◽  
Miller Test ◽  
Logarithmic Integral ◽  
The Difference ◽  
Prime Counting Function

Abstract Skewes’ number was discovered in 1933 by South African mathematician Stanley Skewes as upper bound for the first sign change of the difference π (x) − li(x). Whether a Skewes’ number is an integer is an open problem of Number Theory. Assuming Schanuel’s conjecture, it can be shown that Skewes’ number is transcendental. In our paper we have chosen a different approach to prove Skewes’ number is an integer, using lattice points and tangent line. In the paper we acquaint the reader also with prime numbers and their use in RSA coding, we present the primary algorithms Lehmann test and Rabin-Miller test for determining the prime numbers, we introduce the Prime Number Theorem and define the prime-counting function and logarithmic integral function and show their relation.

scholarly journals On a Family of Sequences Related to Prime Counting Function

2018 ◽  
Vol 7 (6) ◽  
pp. 149
Author(s):  
Altug Alkan ◽  
Orhan Ozgur Aybar
Keyword(s):  
Initial Conditions ◽  
Counting Function ◽  
Integer Sequences ◽  
New Family ◽  
Prime Counting Function

In this study, we introduce a new family of integer sequences which are related to prime-counting function and we focus on some properties of these sequences. Sequence A316434 in OEIS is the fundamental member of solution family that we study. More precisely, we investigate the solutions of recurrence a(n) = a(π(n)) + a(n-π(n)) with some natural initial conditions where π(n) is defined by A000720 in OEIS.

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