11 citations · 23 across the 17 of their papers we have counts for
18 papers
Inequalities involving the primorial counting function
Christian Axler
Let denote the Euler totient function. In this paper, we first establish a new upper bound for involving , the function that counts the number of primorials n…
On Ramanujan's prime counting inequality
Christian Axler
In this paper, we give a new upper bound for the number which is defined to be the smallest positive integer such that a certain inequality due to Ramanujan invol…
New estimates for some integrals of functions defined over primes
Christian Axler
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 and t…
Effective estimates for some functions defined over primes
Christian Axler
In this paper we give effective estimates for some classical arithmetic functions defined over prime numbers. First we find the smallest real number so that some inequality i…
On Robin's inequality
Christian Axler
Let denotes the sum of divisors function of a positive integer . Robin proved that the Riemann hypothesis is true if and only if the inequality …
Some Results on a Conjecture of Hardy and Littlewood
Christian Axler
Let and be positive integers with . The second Hardy-Littlewood conjecture states that the number of primes in the interval is always less than or equ…
On the number of primes up to the th Ramanujan prime
Christian Axler
The th Ramanujan prime is the smallest positive integer such that for all the interval contains at least primes. In this paper we undertake a s…
Improving the Estimates for a Sequence Involving Prime Numbers
Christian Axler
Based on new explicit estimates for the prime counting function, we improve the currently known estimates for the particular sequence , ,…
New estimates for the th prime number
Christian Axler
In this paper we establish a new explicit upper and lower bound for the -th prime number, which improve the currently best estimates given by Dusart in 2010. As the main tool we…
Estimates for for large values of and Ramanujan's prime counting inequality
Christian Axler
In this paper we use refined approximations for Chebyshev's -function to establish new explicit estimates for the prime counting function , which improve the curre…
New estimates for some functions defined over primes
Christian Axler
In this paper we first establish new explicit estimates for Chebyshev's -function. Applying these new estimates, we derive new upper and lower bounds for some functions…
On the arithmetic and geometric means of the prime numbers
Christian Axler
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. Furth…
New bounds for the sum of the first prime numbers
Christian Axler
In this paper we establish a general asymptotic formula for the sum of the first prime numbers, which leads to a generalization of the most accurate asymptotic formula given by…
An explicit upper bound for the first -Ramanujan prime
Christian Axler, Thomas Leßmann
In this paper we establish an explicit upper bound for the first -Ramanujan prime by using a recent result concerning the existence of prime numbers in small interva…
On a sequence involving the prime numbers
Christian Axler
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 estimate…
On the sum of the first n prime numbers
Christian Axler
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 M…
New bounds for the prime counting function π(x)
Christian Axler
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…
On generalized Ramanujan primes
Christian Axler
In this paper we establish several results concerning the generalized Ramanujan primes. For and we give estimates for the th -Ramanu…