Approximation of Prime Number Distribution Functions and Inequalities Involving Primes

Mehdi Hassani
July 2005
 

Abstract: In this thesis, we study the distribution of primes from the  analytic view point. To do this, we study the Riemann zeta function and  the relation of its zeros to the  primes. This connection allows us to estimate Chebychev functions by the Rosser-Schoenfeld method and then we approximate other primes distribution functions. Using these results, we study some inequalities concerning primes and finally, we study various topics in primes distribution theory such as the Prime Number Theorem, intervals containing primes and the Riemann Hypothesis.

Keywords: Primes, Distribution of Primes, Riemann Zeta Function, Riemann Hypothesis, Inequalities.

Whole Thesis: [PDF-in Persian with above English abstract]

Supervisor: Dr. Jamal Rooin

Refereeing Committee: Professor Bahman Mehri, Professor Mehrdad Shahshahani, Dr. Rashid Zaare Nahandi

With Attention of: Professor Michel Waldschmidt

Defending Session Photos: [here]

Acknowledgement: Professor Keith Matthews helped me to betterment this web page. I deem my duty to thank him for his very kind comments.

Note. There is no English version of my thesis. But, I mention that it is a collection of recent results on the approximation of primes. The main source of work was the PhD thesis of Pierre DUSART, which is available here as PDF file. By the way, I have proved some results in it which all of them are published or are preprints.

Published ones are:

I. Equations and Inequalities Involving vp(n!), Journal of Inequalities in Pure and Applied Mathematics (JIPAM), Volume 6, Issue 2, Article 29, 2005, MR2132919. [PDF-reprint], Electronic Journal Page [here]

II. Approximation of π(x) by Ψ(x), Journal of Inequalities in Pure and Applied Mathematics (JIPAM), Volume 7, Issue 1, Article 7, 2006, MR2217170.  [PDF-reprint], Electronic Journal Page [here]

Preprinted ones are:

I. Counting primes in the interval (n^2,(n+1)^2). Available [here]

II. A Remark on the Mandl's Inequality. Available [here]