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Riemann zeta function and prime numbers

Petra Čačkov (2018) Riemann zeta function and prime numbers. MSc thesis.

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    Abstract

    In mathematics still exist a lot of unproven theorems and one of them is Riemann hypothesis about zeroes of zeta function. Although the function was named after Bernhard Riemann, it was studied by Euler about 120 years before him. He showed that there exists a connection between zeta function and prime numbers. But Euler examined the function over real numbers while Riemann expended it to the whole complex plane which enabled the use of other analytical properties for the research of prime numbers. The zeta function can be analytically expanded over the whole complex plane with the help of gamma and Möbius function. The most interesting part of zeta function lies in critical strip of the complex plane. Riemann argued that all zeroes of the zeta function lying in the critical strip have the real part equal to one half. The hypothesis has never been proven and it is part of Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems and is also one of the Clay Mathematics Institute's Millennium Prize Problems. The Riemann hypothesis plays an important part in the distribution as well as at counting the prime numbers. It implies "the best possible" bound for the error of the prime number theorem.

    Item Type: Thesis (MSc thesis)
    Keywords: infinite series, infinite product, prime numbers, prime number theorem, Riemann hypothesis, Riemann zeta function, gamma function, Möbius function
    Number of Pages: 65
    Language of Content: Slovenian
    Mentor / Comentors:
    Mentor / ComentorsIDFunction
    izr. prof. dr. Marko SlaparMentor
    Link to COBISS: http://www.cobiss.si/scripts/cobiss?command=search&base=50126&select=(ID=12180041)
    Institution: University of Ljubljana
    Department: Faculty of Education
    Item ID: 5423
    Date Deposited: 16 Oct 2018 10:07
    Last Modified: 16 Oct 2018 10:07
    URI: http://pefprints.pef.uni-lj.si/id/eprint/5423

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