scholarly journals A new elementary proof of the Prime Number Theorem

2021 ◽  
Author(s):  
Florian K. Richter
Keyword(s):  
Prime Number ◽  
Elementary Proof ◽  
Prime Number Theorem

scholarly journals Pretentious multiplicative functions and the prime number theorem for arithmetic progressions

2013 ◽  
Vol 149 (7) ◽  
pp. 1129-1149 ◽  
Author(s):  
Dimitris Koukoulopoulos
Keyword(s):  
Prime Number ◽  
Elementary Proof ◽  
Counting Function ◽  
Prime Number Theorem ◽  
Arithmetic Progressions ◽  
Multiplicative Functions ◽  
Primes In Arithmetic Progressions

AbstractBuilding on the concept ofpretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

scholarly journals On the Prime Geodesic Theorem for SL4

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Prime Number ◽  
Error Term ◽  
Symmetric Spaces ◽  
Elementary Proof ◽  
Real Variable ◽  
Prime Number Theorem ◽  
Class Numbers ◽  
Prime Geodesic Theorem ◽  
Real Quadratic ◽  
Number Of Authors

In 1949, A. Selberg discovered a real variable (an elementary) proof of the prime number theorem. A number of authors have adapted Selberg’s method to achieve quite a good corresponding error term. The Riemann hypothesis has never been proved or disproved however. Any generalization of the prime number theorem to the more general situations is known in literature as a prime geodesic theorem. In this paper we derive yet another proof of the prime geodesic theorem for compact symmetric spaces formed as quotients of the Lie group SL4 (R). While the first known proof in this setting applies contour integration over square boundaries, our proof relies on an application of modified circular boundaries. Recently, A. Deitmar and M. Pavey applied such prime geodesic theorem to derive an asymptotic formula for class numbers of orders in totally complex quartic fields with no real quadratic subfields.

scholarly journals The Elementary Proof of the Prime Number Theorem

2009 ◽  
Vol 31 (3) ◽  
pp. 18-23 ◽  
Author(s):  
Joel Spencer ◽  
Ronald Graham
Keyword(s):  
Prime Number ◽  
Elementary Proof ◽  
Prime Number Theorem
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