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thesisETDMichaelHenrypdfa.pdf (375 KB)
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Various Old and New Results in Classical Arithmetic by Special Functions
Author Info
Henry, Michael A
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=kent1524583992694218
Abstract Details
Year and Degree
2018, MS, Kent State University, College of Arts and Sciences / Department of Mathematical Sciences.
Abstract
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.
Committee
Gang Yu (Advisor)
Ulrike Vorhauer (Committee Member)
Oma De la Cruz Cabrera (Committee Member)
Pages
58 p.
Subject Headings
Mathematics
Keywords
number theory, classical analysis, special functions, special constants, continued fractions, irrational numbers, transcendental numbers, classical arithmetic
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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)
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Document number:
kent1524583992694218
Download Count:
366
Copyright Info
© 2018, all rights reserved.
This open access ETD is published by Kent State University and OhioLINK.