## 316 Citations

New estimates for some integrals of functions defined over primes

- Mathematics
- 2022

In this paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the prime counting function π(x) and the Chebyshev…

A Simple and Accurate Relation Between the Logarithm Integral Li(x) and the Primes Counting Function π(x) is Derived Making use of the O.E.I.S. Prime Numbers “Sequences”

- MathematicsJournal of Mathematics and Statistics
- 2021

Email: ascarelli.p@gmail.com Abstract: Today the prime numbers π(x) contained under the number x appears to be somewhat overestimated by the logarithm integral function Li(x) and underestimated by…

The Elementary Proof of the Riemann's Hypothesis

- Mathematics
- 2020

This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to…

On the Asymptotic Density of Prime k-tuples and a Conjecture of Hardy and Littlewood

- Mathematics, Computer ScienceComputational Methods in Science and Technology
- 2019

"Skewes numbers" are found for 8 more prime k-tuples and provided numerical data in support of the Hardy-Littlewood conjecture and several algorithms to compute such numbers are presented.

Irrégularités dans la distribution des nombres premiers et des suites plus générales dans les progressions arithmétiques

- Mathematics
- 2011

Effective estimates for some functions defined over primes

- Mathematics
- 2022

. In this paper we give eﬀective estimates for some classical arithmetic functions deﬁned over prime numbers. First we ﬁnd the smallest real number x 0 so that some inequality involving Chebyshev’s ϑ…

An analytic approach to the Riemann hypothesis

- Mathematics
- 2019

In this work we consider a new functional equation for the Riemann zeta-function in the critical half-strip. With the help of this equation we prove that finding non-trivial zeros of the Riemann…

New Estimates for Some Functions Defined Over Primes

- MathematicsIntegers
- 2018

New explicit estimates for Chebyshev's $\vartheta$-function are established and new upper and lower bounds for some functions defined over the prime numbers are derived, for instance the prime counting function $\pi(x)$, which improve the currently best ones.

Estimates for π(x) for Large Values of x and Ramanujan's Prime Counting Inequality

- MathematicsIntegers
- 2018

In this paper we use refined approximations for Chebyshev's $\vartheta$-function to establish new explicit estimates for the prime counting function $\pi(x)$, which improve the current best estimates…

Explicit results on primes

- Mathematics
- 2014

Let p be a prime number. In this thesis we prove the following theorems. Theorem -1.0.1. Let x0 ≥ 4·1018 be a fixed constant and let x > x0. Then there exists at least one prime p such that (1 −∆−1)x…