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The Prime Number Theorem
[chapter]

1999
*
Problems and Solutions for Complex Analysis
*

of

doi:10.1007/978-1-4612-1534-9_15
fatcat:avufmzbdufbevod2cqh2g7qx54
*the**Prime**Number**Theorem*" of 1997. ... This concludes*the*proof of*the**Prime**Number**Theorem*. ...*Theorem*. ...##
###
Analytic implication from the prime number theorem
[article]

2021
*
arXiv
*
pre-print

*The*$\psi$-form of

*the*

*prime*

*number*

*theorem*is $\psi(x) =\sum\sb{n \le x}\Lambda(n) =x +O\bigl(x\sp{1-H(x)} \log\sp{2} x\big)$, where $H(x)$ is a certain function of $x$ with $0< H(x) \le \tfrac{1}{2}$ ...

*The*proof involves slightly revising and applying Tur\'an's power sum method and using

*the*Lindel\"of hypothesis in

*the*zero growth rate form, which is proved recently. ... Less known is that

*the*converse is also true. Actually, Turán proved in 1950 that

*the*above ψ-form of

*the*

*prime*

*number*

*theorem*with H(x) =??? ...

##
###
The prime number theorem for generalized primes

1972
*
Journal of Number Theory
*

Suppose

doi:10.1016/0022-314x(72)90066-2
fatcat:tun4gjjzl5fepm2mn7db467cyq
*the*integer-counting function N of a system of generalized*prime**numbers*satisfies N(x) = Ax + 0(x exp{ -c log* x)) for some c and 0~ with c > 0 and 0 < CA < 1. ... This paper improves a result of Malliavin by proving that if b < a/7.91, then*the**prime*-counting function r satisfies n(x) = li x + O(xexp{-logbx}). ... Beurling [l] defined a system, P, of generalized*prime**numbers*as any non-decreasing unbounded sequence of real*numbers*which are greater than one. ...##
###
On the Prime Number Theorems

1942
*
American Journal of Mathematics
*

T and exceeds

doi:10.2307/2371686
fatcat:uh5y5itayvgrtetl52dumkafw4
*the*absolute value of*the*difference (3) whenever 1 < or < 2 (say).Even without this generalization of*the*italicized Tauberian*theorem*,*the**prime**number**theorem*and Landau's result follow ... , since a unified, and much shorter, approach to all these problems (including*the**prime**number**theorem*itself) can be obtained by a straight-forward extension of Ikehara's*theorem*concerning*the*case ...##
###
A Dynamical Proof of the Prime Number Theorem
[article]

2021
*
arXiv
*
pre-print

We present a new, elementary, dynamical proof of

arXiv:2002.04007v4
fatcat:parmb5f76fhvlglhzf4m35lwu4
*the**prime**number**theorem*. ... Finally, we show this implies*the**prime**number**theorem*.*Theorem*2.9.*The**prime**number**theorem*holds, i.e. 1 N n≤N Λ(n) = 1 + o N →∞ (1) Proof. ... Introduction*The**prime**number**theorem*states that # of*primes*≤ N = (1 + o N →∞ (1)) N log N . ...##
###
An alternative proof of the Dirichlet prime number theorem
[article]

2017
*
arXiv
*
pre-print

Dirichlet's

arXiv:1511.03811v6
fatcat:bt6dg22stjfb5drgm2d5s3czbu
*theorem*on arithmetic progressions called as Dirichlet*prime**number**theorem*is a classical result in*number*theory. Atle Selberg\cite{Selberg} gave an elementary proof of this*theorem*. ... Also we get an estimation of*the**prime*counting function in*the*special cases. ... This work is done during*the*stay of*the*author in Strasbourg. ...##
###
A Direct Proof of the Prime Number Theorem using Riemann's Prime-counting Function
[article]

2021
*
arXiv
*
pre-print

In this paper, we develop a novel analytic method to prove

arXiv:2105.05317v5
fatcat:fq4wzyevjvh2hhl5f2q5sth4yy
*the**prime**number**theorem*in de la Vallée Poussin's form: π(x)=li(x)+𝒪(xe^-c√(log x)) Instead of performing asymptotic expansion on Chebyshev ... functions as in conventional analytic methods, this new approach uses contour-integration method to analyze Riemann's*prime*counting function J(x), which only differs from π(x) by 𝒪(√(x)/log x). ... this paper is a new proof of*the**prime**number**theorem*. ...##
###
Sign changes in the prime number theorem
[article]

2019
*
arXiv
*
pre-print

Let $V(T)$ denote

arXiv:1910.14203v3
fatcat:bfhcocxvx5gdfnfrv5swogqxze
*the**number*of sign changes in $\psi(x) - x$ for $x\in[1, T]$. ... We show that $\liminf_{\;T\rightarrow\infty} V(T)/\log T \geq \gamma_{1}/\pi + 1.867\cdot 10^{-30}$, where $\gamma_{1} = 14.13\ldots$ denotes*the*ordinate of*the*lowest-lying non-trivial zero of*the*Riemann ... Acknowledgements We should like to acknowledge Jerzy Kaczorowski for his valuable comments, and Jonathan Bober, Ben Green, David Harvey, and Jesse Thorner for spotting a howler in*the*abstract. ...##
###
Scale Free Analysis and the Prime Number Theorem
[article]

2010
*
arXiv
*
pre-print

We present an elementary proof of

arXiv:1001.1490v3
fatcat:4sfl2hzhdrekjjgaijpv4q65fa
*the**prime**number**theorem*.*The*relative error follows a golden ratio scaling law and respects*the*bound obtained from*the*Riemann's hypothesis. ...*The*extended real*number*system is realized as a completion of*the*field of rational*numbers*$Q$ under a {\em new} nonarchimedean absolute value, which treats arbitrarily small and large*numbers*separately ... Introduction We present a new proof of*the**Prime**Number**Theorem*[1] . ...##
###
A new elementary proof of the Prime Number Theorem
[article]

2020
*
arXiv
*
pre-print

We give a new elementary proof of

arXiv:2002.03255v1
fatcat:soslnqswtnd5lgktz4qr2wta34
*the**Prime**Number**Theorem*by comparing averages of*the*M\"obius function dilated by*primes*to those dilated by almost*primes*. ... Acknowledgments:*The*author thanks Vitaly Bergelson and Redmond McNamara for helpful comments. This work is supported by*the*National Science Foundation under grant*number*DMS 1901453. ... Introduction One of*the*most fundamental results in mathematics is*the**Prime**Number**Theorem*, which describes*the*asymptotic law of*the*distribution of*prime**numbers*in*the*integers. (1.1)*The**Prime**Number*...##
###
Chapter 4 The Prime Number Theorem
[chapter]

1974
*
Pure and Applied Mathematics
*

*prime*

*number*

*theorem*(PNT). ... Therefore f (x)/x → c as x → ∞, completing

*the*proof of both

*the*corollary and

*the*

*prime*

*number*

*theorem*. ♣

*The*

*prime*

*number*

*theorem*has a long and interesting history. ...

##
###
The Prime Number Theorem from logn!

1964
*
Proceedings of the American Mathematical Society
*

During

doi:10.2307/2034529
fatcat:5dgyfnx3dfdovf7mvkrblruwtq
*the*nineteenth century attempts were made to prove*the**prime**number**theorem*from*the*formula [*THEOREM*. ... Pitt that a differenit proof of*the**prime**number**theorem*based on log n! and using Wiener's*theorem*was given by A. E. Ingham, Some Tauberian*theorems*connected with*the**prime**number**theorem*, J. ...##
###
Note on the Prime Number Theorem
[article]

2005
*
arXiv
*
pre-print

We survey

arXiv:math/0502062v1
fatcat:fodmn55p6zg2tlftdyoxhxjt4y
*the*classical results on*the**prime**number**theorem*...*The*truth of this assertion is*the*core of*the**prime**number**theorem*. ... Hence we complete*the*proof.*Theorem*5.10[*Prime**Number**Theorem*]. If π(x) denotes*the**number*of*prime**numbers*p ≤ x, then we have that π(x) ∼ x ln x . Proof. ...##
###
The error term in the prime number theorem
[article]

2020
*
arXiv
*
pre-print

We make explicit a

arXiv:1809.03134v2
fatcat:gqgetxd47zbrtlx4mui2embu44
*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. ... Acknowledgements We wish to thank Habiba Kadiri, Nathan Ng, and Allysa Lumley, and*the*referees for very useful feedback. ...##
###
A Simple Proof Of The Prime Number Theorem
[article]

2018
*
arXiv
*
pre-print

It is shown that

arXiv:1510.03465v2
fatcat:o2zrulourfe3jnpdcacfkn5lze
*the*Mean Value*Theorem*for arithmetic functions, and simple properties of*the*zeta function are sufficient to assemble proofs of*the**Prime**Number**Theorem*, and Dirichlet*Theorem*. ... These are among*the*simplest proofs of*the*asymptotic formulas of*the*corresponding*prime*counting functions. ... Introduction*The**Prime**Number**Theorem*is an asymptotic formula π(x) = x/ log x + o(x/ log x) for*the**number*of*primes*p ≥ 2 up to a*number*x ≥ 1. ...
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