Related papers: Approximating the Prime Counting Function via an O…
We construct a smooth real-valued function P(n) in [0,1], defined via a triple integral with a periodic kernel, that approximates the characteristic function of prime numbers. The function is built to suppress when n is divisible by some m…
The numbers of even and odd permutations with a given ascent number are investigated using an operator that was previously introduced by the author. Their difference is called a signed Eulerian number. By means of the operator the…
Using a sieve procedure akin to the sieve of Eratosthenes we show how for each prime $p$ to build the corresponding M\"obius prime-function, which in the limit of infinitely large primes becomes identical to the original M\"obius function.…
We shed some new light to the problem of characterizing those functions of several arguments that have a unique identification minor. The 2-set-transitive functions are known to have this property. We describe another class of functions…
In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vall\'ee Poussin's form: $$ \pi(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) $$ Instead of performing asymptotic expansion on Chebyshev…
We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set…
Based on Euclid's algorithm, we find a kind of special sequences which play an interesting role in the study of primes. We call them W Sequences. They not only ties up the distribution of primes in short interval but also enables us to give…
We report on prime number decomposition by use of the Talbot effect, a well-known phenomenon in classical near field optics whose description is closely related to Gauss sums. The latter are a mathematical tool from number theory used to…
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.
This document seeks to prove there are infinitely many primes whose difference is 2, referred to as twin prime pairs. This proof's methodology involves constructing a function that approximates the number of positive integers, less than a…
A parking function of length $n$ is prime if we obtain a parking function of length $n-1$ by deleting one 1 from it. In this note we give a new direct proof that the number of prime parking functions of length $n$ is $(n-1)^{n-1}$. This…
We give an informal survey of the historical development of computations related to prime number distribution and zeros of the Riemann zeta function.
A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of…
In this note, we approximate the average of prime powers in the decomposition of $n!$ into prime numbers.
We showed that the prime gap for a prime number p is less than or equal to the prime count of the prime number.
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…
An unconditional inequality of the totient function is contributed to the literature. This result is associated with various problems about the distribution of prime numbers.
In the work we discuss the benefit of using bitwise operations in programming. Some interesting examples in this respect have been shown. What is described in detail is an algorithm for sorting an integer array with the substantial use of…
This short paper presents an exact formula for counting twin prime pairs less than or equal to x in terms of the classical Smarandache Function. An extension of the formula to count prime pairs (p, p+2n), n > 1, is also given.
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…