Related papers: Average Error of the Prime Number Theorem
We make explicit a theorem of Pintz concerning the error term in the prime number theorem. This gives an improved version of the prime number theorem with error term roughly square-root of that which was previously known. We apply this to a…
Assume the Riemann hypothesis throughout. We obtain some new estimates for the size of the set of large values of the error term in the prime number theorem. Our argument is based on an analysis of the behavior of zeros of the Riemann zeta…
In this paper, we establish some theorems on the distribution of primes in higher-order progressions on average.
We survey the classical results on the prime number theorem
An improved estimate is given for $|\theta(x) -x|$, where $\theta(x) = \sum_{p\leq x} \log p$. Three applications are given: the first to arithmetic progressions that have points in common, the second to primes in short intervals, and the…
In this paper, we establish a theorem on the distribution of primes in quadratic progressions on average.
We present a new, elementary, dynamical proof of the prime number theorem.
We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we…
Definition of the number of prime numbers in the given interval
By combining and improving recent techniques and results, we provide explicit estimates for the error terms $|\pi(x)-\text{li}(x)|$, $|\theta(x)-x|$ and $|\psi(x)-x|$ appearing in the prime number theorem. For example, we show for all…
In this paper we present a short and elementary proof for the error in Simpson's rule.
I give some claims on primorial prime numbers for interested readers in number theory.
We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.
In this note, we approximate the average of prime powers in the decomposition of $n!$ into prime numbers.
We relate the size of the error term in the Hardy-Littlewood conjectured formula for the number of prime pairs to the $L^{1}$ norm of an exponential sum over the primes formed with the von Mangoldt function.
In this article we prove a general theorem which establishes the existence of limiting distributions for a wide class of error terms from prime number theory. As a corollary to our main theorem, we deduce previous results of Wintner (1935),…
We continue our recent work on averages for ternary additive problems with powers of prime numbers.
In this short note, we obtain error estimates for Riemann sums of some singular functions.
This paper is a corrigendum to the article 'On the ideal theorem for number fields`. The main result of this paper proves to be untrue and is replaced by an estimate of a weighted sum with an improved error term.
In this paper, we prove a theorem on the distribution of primes in cubic progressions on average.