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Various Old and New Results in Classical Arithmetic by Special Functions

Henry, Michael A

Abstract Details

2018, MS, Kent State University, College of Arts and Sciences / Department of Mathematical Sciences.
Beginning with the essentials from the theory of simple continued fractions, we review some early results in Diophantine approximation. We proceed to prove that Liouville’s number L is transcendental; in addition we discuss the transcendence measure of $e$, known as Davis’ theorem, which also gives us e is a transcendental number by Roth’s theorem. Having reviewed these results we generalize a result of Landau, giving rise to what we will call Landau sums. These sums generate a countable spectrum of irrational numbers that are computable in quadratic time due to their representation as theta functions; it is still open as to whether these numbers are transcendental. Altering the Landau sums we also get another class of sums that we call lateral Landau sums. These sums also give a spectrum of numbers that we speculate are irrational. These results sit nicely within the intersection of classical analysis and classical number theory. We say a few words about this relationship and end with some conjectures.
Gang Yu (Advisor)
Ulrike Vorhauer (Committee Member)
Oma De la Cruz Cabrera (Committee Member)
58 p.

Recommended Citations

Citations

  • Henry, M. A. (2018). Various Old and New Results in Classical Arithmetic by Special Functions [Master's thesis, Kent State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=kent1524583992694218

    APA Style (7th edition)

  • Henry, Michael . Various Old and New Results in Classical Arithmetic by Special Functions. 2018. Kent State University, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=kent1524583992694218.

    MLA Style (8th edition)

  • Henry, Michael . "Various Old and New Results in Classical Arithmetic by Special Functions." Master's thesis, Kent State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=kent1524583992694218

    Chicago Manual of Style (17th edition)