Improved Riemann Ladder Function

Dan Liu

doi:10.4236/jamp.2021.94042

Abstract

In Riemann’s prime distribution formula, there is a key Riemann ladder function. This is the basis for Riemann to study the distribution of prime numbers. But the calculation of Riemann’s ladder function is very complicated. According to the definition of Riemannian ladder function, we greatly improve the Riemannian ladder function, obtain new ladder function and improved Riemannian prime distribution formula, and prove the improved Riemannian prime distribution formula. We use the improved Riemannian prime distribution formula and merdens theorem to obtain a strong prime theorem.

Highlights

The distribution of prime numbers is a difficult problem in number theory
We improve the Riemann ladder function and prove a strong prime number theorem by using the metens theorem [3]
From (6.2), we prove the prime theorem in a simple way π

Summary

Introduction

The distribution of prime numbers is a difficult problem in number theory. The proof of prime theorem attracts many scholars and puzzles many wise people. Mathematicians have proved thousands of theorems on the premise of Riemann conjecture. In 1737, the Swiss mathematician Leonhard Euler published a formula:. Where n is an integer, p is a prime and s is a real number. From 1792 to 1793, Gauss, a German mathematician, studied the number of primes in 1000 adjacent integers around x. Log x where, the number of primes not exceeding x is π(x). Represents the number of primes in r adjacent integers near x. We improve the Riemann ladder function and prove a strong prime number theorem by using the metens theorem [3].

The Elementary Proof of Prime Number Theorem

Liu DOI

Merdens’ Theorem

Prime Theorem with Remainder

Conclusions