# Updating the error term in the prime number theorem

@article{Trudgian2014UpdatingTE,
title={Updating the error term in the prime number theorem},
author={Tim Trudgian},
journal={The Ramanujan Journal},
year={2014},
volume={39},
pages={225-234}
}
• T. Trudgian
• Published 2014
• Mathematics
• The Ramanujan Journal
An improved estimate is given for $$|\theta (x) -x|$$|θ(x)-x|, where $$\theta (x) = \sum _{p\le x} \log p$$θ(x)=∑p≤xlogp. Four applications are given: the first to arithmetic progressions that have points in common, the second to primes in short intervals, the third to a conjecture by Pomerance and the fourth to an inequality studied by Ramanujan.
41 Citations

#### Tables from this paper

Estimates for π(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 estimatesExpand
Explicit estimates of some functions over primes
New results have been found about the Riemann hypothesis. In particular, we noticed an extension of zero-free region and a more accurate location of zeros in the critical strip. The RiemannExpand
On the first sign change of $\theta(x) - x$
• Mathematics
• 2014
Let θ(x) = ∑ p≤x log p. We show that θ(x) < x for 2 < x < 1.39 · 10. We also show that there is an x < exp(727.951332668) for which θ(x) > x. AMS Codes: 11M26, 11Y35
Explicit bounds for primes in arithmetic progressions
• Mathematics
• 2018
We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, andExpand
Explicit Estimates in the Theory of Prime Numbers
It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove thatExpand
An Explicit Result for Primes Between Cubes
We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.217))$. Our new tool which we derive is a version of Landau's explicit formula for the Riemann zeta-functionExpand
New estimates for some prime functions
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 theExpand
New Estimates for Some Functions Defined Over Primes
New explicit estimates for Chebyshev's $\vartheta$-function are established and new upper and lower bounds for some functions defined over the prime numbers are derived, for instance the prime counting function $\pi(x)$, which improve the currently best ones. Expand
The error term in the prime number theorem
• Computer Science, Mathematics
• Math. Comput.
• 2021
An improved version of the prime number theorem with error term roughly square-root of that which was previously known is given, applied to a long-standing problem concerning an inequality studied by Ramanujan. Expand
Primes between consecutive powers
This paper updates two explicit estimates for primes between consecutive powers. We find at least one prime between n3 and (n+1)3 for all n ≥ exp(exp(32.9)), and at least one prime in (n296, (n +Expand

#### References

SHOWING 1-10 OF 36 REFERENCES
Estimating π(x) and related functions under partial RH assumptions
• Jan Büthe
• Computer Science, Mathematics
• Math. Comput.
• 2016
The aim of this paper is to give a direct interpretation of the validity of the Riemann hypothesis up to a certain height $T$ in terms of the prime-counting function $\pi(x)$. This is done by provingExpand
An Explicit Result for Primes Between Cubes
We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.217))$. Our new tool which we derive is a version of Landau's explicit formula for the Riemann zeta-functionExpand
Major arcs for Goldbach's problem
The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer n greater than 5 is the sum of three primes. The present paper proves this conjecture. Both the ternaryExpand
Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$
• Mathematics
• 1975
Abstract : The authors demonstrate a wider zero-free region for the Riemann zeta function than has been given before. They give improved methods for using this and a recent determination thatExpand
A note on the least prime in an arithmetic progression
Abstract Let k , l denote positive integers with ( k , l ) = 1. Denote by p ( k , l ) the least prime p ≡ l (mod k ). Let P ( k ) be the maximum value of p ( k , l ) for all l . We show lim inf P(k)Expand
New bounds for π(x)
• Computer Science, Mathematics
• Math. Comput.
• 2015
The proof relies on two new arguments: smoothing the prime counting function which allows to generalize the previous approaches, and a new explicit zero density estimate for the zeros of the Riemann zeta function. Expand
Vinogradov's Integral and Bounds for the Riemann Zeta Function
The main result is an upper bound for the Riemann zeta function in the critical strip: $\zeta(\sigma + it) \le A|t|^{B(1 - \sigma)^{3/2}} \log^{2/3} |t|$ with $A = 76.2$ and $B = 4.45$, valid forExpand
Estimates of Some Functions Over Primes without R.H.
Some computations made about the Riemann Hypothesis and in particular, the verification that zeroes of zeta belong on the critical line and the extension of zero-free region are useful to get betterExpand
On a conjecture of Pomerance
• Mathematics
• 2011
We say that k is a P -integer if the first φ(k) primes coprime to k form a reduced residue system modulo k. In 1980 Pomerance proved the finiteness of the set of P -integers and conjectured that 30Expand
A Strong Form of a Problem of R. L. Graham
• K. Ford
• Mathematics
Abstract If $A$ is a set of $M$ positive integers, let $G(A)$ be the maximum of ${{a}_{i}}/\,\gcd ({{a}_{i}},\,\,{{a}_{j}}\,)$ over ${{a}_{i}},\,{{a}_{j}}\,\,\in \,\,A.$ We show that if $G(A)$ is notExpand