Theory of Numbers
Dr. Krishnaswami Alladi
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Lecture 1 This lecture focuses on the Euclid's proof of the infinitude of primes, and a consequent very rough upper bound of the prime counting function. Error notation is introduced and defined. Additionally, the inclusion-exclusion principle is utilized and demonstrated.
Lecture 2 In this lecture, the bounds on the the prime counting function from the previous lecture are significantly sharpened and some estimates that hint at the prime number theory theorem make their appearance.
Lecture 3 Tchebychev's inequality is proved using a Riemann-Stieljes integral and this then sufficient to prove Bertrand's postulate.
Lecture 4 The proof outlined in the prior lecture is completed in detail and a definition for the Euler-Masceroni constant is given.
Lecture 5 This lecture showcases proofs by Paul Erdos and Srinivasa Ramanujan that demonstrate both methods for proving Bertrand's postulate and means to improve upon it.
Lecture 6 We finish the proof of Ramanujan and begin proving Merten's Formula.
Lecture 7 We complete Merten's formula.
Lecture 10 We begin a proof of the prime number theorem.
Lecture 11 We continue the proof of the prime number theorem.