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Major Oral - A discussion of an Elementary Proof of the Prime Number Theorem

HyeJin Yeon, Department of Mathematics, Washington University in St. Louis

March 29, 2012 - 2:00 pm to 3:00 pm
Location: Cupples I, Room 199 | Host: Prof. Quo-Shin Chi

Undergraduate Senior Honors Thesis Presentation
Abstract: The prime number theorem (PNT) was first conjectured by Gauss and was finally proved in 1896. This theorem states that 

\lim_{x → ∞} \dfrac {π(x)} {x / log(x)} = 1,

where  π(x) is the prime-counting function which counts the number of primes less than or equal to x. The theorem relates two seemingly unrelated functions π(x) and log(x) in a remarkably simple way. The first proof of 1896 uses analytic function theory and "elementary" proof was considered to be inaccessible until it was finally found in 1948 and 1949 by Atel Selberg and Paul Erdos. This elementary proof entirely avoids the use of analytic function theory and provides a fundamental view on the problem. This paper presents elementary proof of the prime number theorem as it was originally done by Selberg who was awarded the Fields Medal in 1950. The three fundamental asymptotic formulas in Selberg's paper are thoroughly proved in this paper. The final step of the proof uses mathematical induction which ties all the complexities down to a simple formula.

 

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