The prime number theorem via Fourier transformation

Quanhai Chen

doi:10.1117/12.2649456

27 September 2022 The prime number theorem via Fourier transformation

Quanhai Chen

Proceedings Volume 12345, International Conference on Applied Statistics, Computational Mathematics, and Software Engineering (ASCMSE 2022); 123450F (2022) https://doi.org/10.1117/12.2649456
Event: 2022 International Conference on Applied Statistics, Computational Mathematics, and Software Engineering (ASCMSE 2022), 2022, Qingdao, China

Abstract

For many centuries, people have done every possibility to find the number of prime numbers. During the 300 BC, Euclid had written a proof for there exist infinitely many prime numbers. However, the number of primes must be less than the number of integers. Hence, mathematicians started to consider precisely how many prime numbers are there comparing with all the integers. Until the end of 18th century, Gauss conjectured the total number of primes, and it was proven in the end of 19th century. The following paper will present a proof adapted from Ikehara in 1931. More precisely, the following paper converts the original problem into a better-behavior equation and then constructs another function to show the Prime Number Theorem with Function Theory and Fourier transforms. Notice that there are many more elegant proofs have been done during 20th century, and the following paper is for general understanding.

Citation Download Citation

Quanhai Chen "The prime number theorem via Fourier transformation", Proc. SPIE 12345, International Conference on Applied Statistics, Computational Mathematics, and Software Engineering (ASCMSE 2022), 123450F (27 September 2022); https://doi.org/10.1117/12.2649456

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