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The Prime Number Theorem [chapter]

Rami Shakarchi
1999 Problems and Solutions for Complex Analysis
of the Prime Number Theorem" of 1997.  ...  This concludes the proof of the Prime Number Theorem.  ...  Theorem.  ...

Analytic implication from the prime number theorem [article]

Yuanyou Cheng, Glenn Fox, Mehdi Hassani
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

R.S Hall
1972 Journal of Number Theory
Suppose 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

Aurel Wintner
1942 American Journal of Mathematics
T and exceeds 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]

Redmond McNamara
2021 arXiv   pre-print
We present a new, elementary, dynamical proof of 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]

Haifeng Xu
2017 arXiv   pre-print
Dirichlet's 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]

Zihao Liu
2021 arXiv   pre-print
In this paper, we develop a novel analytic method to prove 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]

Thomas Morrill, Dave Platt, Tim Trudgian
2019 arXiv   pre-print
Let $V(T)$ denote 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]

Dhurjati Prasad Datta, Anuja Roy Choudhuri
2010 arXiv   pre-print
We present an elementary proof of 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  .  ...

A new elementary proof of the Prime Number Theorem [article]

Florian K. Richter
2020 arXiv   pre-print
We give a new elementary proof of 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!

N. Levinson
1964 Proceedings of the American Mathematical Society
During 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]

Yong-Cheol Kim
2005 arXiv   pre-print
We survey 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]

Dave Platt, Tim Trudgian
2020 arXiv   pre-print
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.  ...  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]

N. A. Carella
2018 arXiv   pre-print
It is shown that 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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