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Generalizations of the Erdős–Kac Theorem and the Prime Number Theorem
In this paper, we study the linear independence between the distribution of the number of prime factors of integers and that of the largest prime...
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Refinements to the prime number theorem for arithmetic progressions
We prove a version of the prime number theorem for arithmetic progressions that is uniform enough to deduce the Siegel–Walfisz theorem, Hoheisel’s...
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Multiplicative Functions with Sum Zero Over Beurling Generalized Prime Number Systems
Completely multiplicative arithmetic functions with sum zero (in short CMO functions) were initially introduced by Kahane and Saïas (Expo Math...
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A conditional explicit result for the prime number theorem in short intervals
This paper gives an explicit bound for the prime number theorem in short intervals under the assumption of the Riemann hypothesis.
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Explicit Kronecker–Weyl theorems and applications to prime number races
We prove explicit versions of the Kronecker–Weyl theorems, both in a discrete and a continuous settings, without any linear independence hypothesis....
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Wirsing’s Elementary Proofs of the Prime Number Theorem with Remainder Terms
We survey Wirsing’s two papers giving elementary proofs of the Prime Number Theorem with remainder terms.
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An asymptotic formula for the number of prime solutions for multivariate linear equations
In this paper, we study the multivariate linear equations with arbitrary positive integral coefficients. Under the Generalized Riemann Hypothesis, we...
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The prime number theorem and pair correlation of zeros of the Riemann zeta-function
We prove that the error in the prime number theorem can be quantitatively improved beyond the Riemann Hypothesis bound by using versions of...
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Properties of Boolean Functions with Extremal Number of Prime Implicants
AbstractThe known lower bound for the maximum number of prime implicants (maximal faces) of a Boolean function differs from the upper bound by a...
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Euclidean ideal classes in Galois number fields of odd prime degree
Weinberger [
16 ] in 1972, proved that the ring of integers of a number field with unit rank at least 1 is a principal ideal domain if and only if it... -
A pair of equations in two prime squares, four prime cubes and powers of two
In this paper, it is proved that every sufficiently large even number can be expressed as a pair of equations, each containing two prime squares,...
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A Cesàro average for an additive problem with an arbitrary number of prime powers and squares
In this paper we extend and improve all the previous results known in literature about weighted average, with Cesàro weight, of representations of an...
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Quantum and classical study of prime numbers, prime gaps and their dynamics
A wave function constructed from prime counting function is employed to study the properties of primes using quantum dynamics. The prime gaps are...
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Prime Number Counting Function
On the way to showing the significance of the zeta zeros for counting prime numbers up to a given magnitude, Riemann introduces an important weighted...
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Proving the Prime Number Theorem
This is slightly off our main road, but we have in fact all the ingredients at our disposal to prove it! This will give us the opportunity to show a...
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Additive problems with almost prime squares
We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost...
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Right-angled triangles with almost prime hypotenuse
The sequence OEIS A281505 consists of distinct odd legs in right triangles with integer sides and prime hypotenuse. In this paper, we count the...