A conditional explicit result for the prime number theorem in short intervals

The prime number theorem in short intervals for automorphic L-functions

A Short Introduction to the Prime Number Theorem

We briefly discuss the history of the prime number theorem and summarize Newman’s 1980 proof using complex analysis. I have attempted to convey the flavor of the proof without deviating too much from heuristic arguments; all details are left as exercises to the reader. “There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will b...

The Prime Number Theorem

The Prime Number Theorem asserts that the number of primes less than or equal to x is approximately equal to x log x for large values of x (here and for the rest of these notes, log denotes the natural logarithm). This quantitative statement about the distribution of primes which was conjectured by several mathematicians (including Gauss) early in the nineteenth century, and was finally proved ...

Dirichlet Prime Number Theorem

In number theory, the prime number theory describes the asymptotic distribution of prime numbers. We all know that there are infinitely many primes,but how are they distributed? Dirichlet’s theorem states that for any two positive coprime integers a and d, there are infinitely many primes which are congruent to a modulo d. A stronger form of Dirichlet’s theorem states that the sum of the recipr...

A History of the Prime Number Theorem

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