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Christian Axler

Publications and source records attributed to Christian Axler.

18 recordsLinked to original sources
arXiv2024

Inequalities involving the primorial counting function

Let $\varphi(n)$ denote the Euler totient function. In this paper, we first establish a new upper bound for $n/\varphi(n)$ involving $K(n)$, the function that counts the number of primorials not exceeding $n$. In particular, this leads to an answer to a question raised by Aoudjit, Berkane, and Dusart concerning an upper bound for the sum-of-divisors function $\sigma(n)$. Furthermore, we give some lower bounds for $N_k/\varphi(N_k)$ as well as for $\sigma(N_k)/N_k$, where $N_k$ denotes the $k$th primorial.

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arXiv2022

On Ramanujan's prime counting inequality

In this paper, we give a new upper bound for the number $N_{\mathcal{R}}$ which is defined to be the smallest positive integer such that a certain inequality due to Ramanujan involving the prime counting function $\pi(x)$ holds for every $x \geq N_{\mathcal{R}}$.

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arXiv2022

New estimates for some integrals of functions defined over primes

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 $\pi(x)$ and the Chebyshev $\vartheta$-function. Some of these estimates depend on the correctness of the Riemann hypothesis on the nontrivial zeros of the Riemann zeta function $\zeta(s)$.

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arXiv2022

Effective estimates for some functions defined over primes

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 $\vartheta$-function holds for every $x \geq x_0$. Then we give some new results concerning the existence of prime numbers in short intervals. Also we derive new upper and lower bounds for some functions defined over prime numbers, for instance the prime counting function $\pi(x)$, which improve current best estimates of similar shape.

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arXiv2021

On Robin's inequality

Let $\sigma(n)$ denotes the sum of divisors function of a positive integer $n$. Robin proved that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma}n \log \log n$ holds for every positive integer $n \geq 5041$, where $\gamma$ is the Euler-Mascheroni constant. In this paper we establish a new family of integers for which Robin's inequality $\sigma(n) < e^{\gamma}n \log \log n$ hold. Further, we establish a new unconditional upper bound for the sum of divisors function. For this purpose, we use an approximation for Chebyshev's $\vartheta$-function and for some product defined over prime numbers.

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arXiv2019

Some Results on a Conjecture of Hardy and Littlewood

Let $m$ and $n$ be positive integers with $m,n \geq 2$. The second Hardy-Littlewood conjecture states that the number of primes in the interval $(m,m+n]$ is always less than or equal to the number of primes in the interval $[2,n]$. Based on new explicit estimates for the prime counting function $\pi(x)$, we give some new ranges in which this conjecture holds.

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arXiv2017

On the number of primes up to the $n$th Ramanujan prime

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x \geq R_n$ the interval $(x/2, x]$ contains at least $n$ primes. In this paper we undertake a study of the sequence $(\pi(R_n))_{n \in \mathbb{N}}$, which tells us where the $n$th Ramanujan prime appears in the sequence of all primes. In the first part we establish new explicit upper and lower bounds for the number of primes up to the $n$th Ramanujan prime, which imply an asymptotic formula for $\pi(R_n)$ conjectured by Yang and Togb\'e. In the second part of this paper, we use these explicit estimates to derive a result concerning an inequality involving $\pi(R_n)$ conjectured by of Sondow, Nicholson and Noe.

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arXiv2017

New estimates for the $n$th prime number

In this paper we establish a new explicit upper and lower bound for the $n$-th prime number, which improve the currently best estimates given by Dusart in 2010. As the main tool we use some recently obtained explicit estimates for the prime counting function. A further main tool is the usage of estimates concerning the reciprocal of $\log p_n$. As an application we derive refined estimates for $\vartheta(p_n)$ in terms of $n$, where $\vartheta(x)$ is Chebyshev's $\vartheta$-function.

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arXiv2017

New estimates for some functions defined over primes

In this paper we first establish new explicit estimates for Chebyshev's $\vartheta$-function. Applying these new estimates, we derive new upper and lower bounds for some functions defined over the prime numbers, for instance the prime counting function $\pi(x)$, which improve the currently best ones. Furthermore, we use the obtained estimates for the prime counting function to give two new results concerning the existence of prime numbers in short intervals.

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arXiv2017

Estimates for $\pi(x)$ for large values of $x$ and Ramanujan's prime counting inequality

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 for large values of $x$. As an application we find an upper bound for the number $H_0$ which is defined to be the smallest positive integer so that Ramanujan's prime counting inequality holds for every $x \geq H_0$.

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arXiv2016

On the arithmetic and geometric means of the prime numbers

In this paper we establish explicit upper and lower bounds for the ratio of the arithmetic and geometric means of the prime numbers, which improve the current best estimates. Further, we prove several conjectures related to this ration stated by Hassani. In order to do this, we use explicit estimates for the prime counting function, Chebyshev's $\vartheta$-function and the sum of the first $n$ prime numbers.

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arXiv2016

New bounds for the sum of the first $n$ prime numbers

In this paper we establish a general asymptotic formula for the sum of the first $n$ prime numbers, which leads to a generalization of the most accurate asymptotic formula given by Massias and Robin. Further we prove a series of results concerning Mandl's inequality on the sum of the first $n$ prime numbers. We use these results to find new explicit estimates for the sum of the first $n$ prime numbers, which improve the currently best known estimates.

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arXiv2016

New bounds for the prime counting function π(x)

In this paper we establish a number of new estimates concerning the prime counting function π(x), which improve the estimates proved in the literature. As an application, we deduce a new result concerning the existence of prime numbers in small intervals.

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arXiv2016

On generalized Ramanujan primes

In this paper we establish several results concerning the generalized Ramanujan primes. For $n\in\mathbb{N}$ and $k \in \mathbb{R}_{> 1}$ we give estimates for the $n$th $k$-Ramanujan prime which lead both to generalizations and to improvements of the results presently in the literature. Moreover, we obtain results about the distribution of $k$-Ramanujan primes. In addition, we find explicit formulae for certain $n$th $k$-Ramanujan primes. As an application, we prove that a conjecture of Mitra, Paul and Sarkar concerning the number of primes in certain intervals holds for every sufficiently large positive integer.

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arXiv2015

On a sequence involving the prime numbers

In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.

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arXiv2014

On the sum of the first n prime numbers

In this paper we establish a general asymptotic formula for the sum of the first n prime numbers, which leads to a generalization of the most accurate asymptotic formula given by Massias and Robin in 1996.

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