The second research problem discussed in this book is distribution of prime numbers. Elementary derivation of the explicit formulae for both the prime-counting function and Riemann’s weighted-prime counting function, based on the technique of Mellin’s transform, is presented in some detail. As the third research problem, the binomial regularization of Riemann’s zeta function is introduced. Finally, as the additional research problem, the derivation of the number of partitions of a positive integer into a finite number of positive integer summands, based on Hardy–Ramanujan–Rademacher’s formula, is presented in some detail.
The monograph is helpful for the mathematicians working in number theory, combinatorics, mathematical analysis, function theory, and their applications. Also, it can be a higher-level introduction for students starting their work in these branches of mathematics.
1 Brief Historical Overview
1 Riemann’s Zeta Function
2 Mertens’s Conjecture
3 Prime Number Theorem
2 Prime Number Theorem & Riemann’s Hypothesis
3 Distribution of Prime Numbers
4 Binomial Regularization
5 Addendum: Integer Partitions
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