Several problems from number theory ; Kai kurie skaičių teorijos uždaviniai

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VILNIUS UNIVERSITYGiedrius AlkauskasSeveral problems from number theoryDoctoral dissertationPhysical sciences, mathematics (01 P)Vilnius, 20091The work on this dissertation was performed in 2006-2009 at Vilnius University.Scientific supervisor:Prof. Dr. Habil. Antanas Laurinˇcikas (Vilniaus University, Physical sciences, Mathematics- 01 P).2VILNIAUS UNIVERSITETASGiedrius AlkauskasKai kurie skaiˇciu¸ teorijos uˇzdaviniaiDaktaro disertacijaFiziniai mokslai, matematika (01 P)Vilnius, 20093Disertacija rengta 2006-2009 metais Vilniaus Universitete.Mokslinis vadovas:Prof. habil. dr. Antanas Laurinˇcikas (Vilniaus universitetas, fizinai mokslai, matematika -01 P).4ContentsContents 50.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70.1.1 Actuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70.1.2 Aims and problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70.1.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80.1.4 Novelty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80.1.5 Statements presented for the defence . . . . . . . . . . . . . . . . . . 80.1.6 History of the problem and main results . . . . . . . . . . . . . . . . 80.1.7 Approbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110.1.8 Principal publications . . . . . . . . . . . . . . . . . . . . . . . . . .

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Publié le 01 janvier 2009
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Several
VILNIUS UNIVERSITY
Giedrius Alkauskas
problems from number
Doctoral dissertation Physical sciences, mathematics (01 P)
Vilnius, 2009
1
theory
The work on this dissertation was performed in 2006-2009 at Vilnius University.
Scientific supervisor: Prof. Dr. Habil. Antanas - 01 P).
Laurinˇcikas
(Vilniaus
2
University,
Physical
sciences,
Mathematics
Kai
VILNIAUS UNIVERSITETAS
Giedrius Alkauskas
kurieskaicˇi¸uteorijosuzˇdaviniai
Daktaro disertacija Fiziniai mokslai, matematika (01 P)
Vilnius, 2009
3
Disertacija rengta 2006-2009 metais Vilniaus Universitete.
Mokslinis vadovas: Prof. habil. dr. Antanas 01 P).
Laurincˇikas
(Vilniaus
4
universitetas,
fizinai
mokslai,
matematika
-
Contents
Contents 0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1.1 Actuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1.2 Aims and problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1.4 Novelty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1.5 Statements presented for the defence . . . . . . . . . . . . . . . . . . 0.1.6 History of the problem and main results . . . . . . . . . . . . . . . . 0.1.7 Approbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1.8 Principal publications . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Explicit series for the dyadic period function 1.1 Introduction and main result . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2pquestion mark functions andp. . . . . . . . .continued fractions . . . 1.3 Complex case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Properties of integral transforms ofFp(x) .. . . . . . . . . . . . . . . . . . 1.5 Three term functional equation . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The proof: approach throughp . . . . . . . . . . . . . . . . . . . . .= 2 . 1.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Approach throughp= 0 .. . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Auxiliary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.3 Numerical values for the moments . . . . . . . . . . . . . . . . . . . 1.7.4 Rational functionsHn(z) .. . . . . . . . . . . . . . . . . . . . . . . 1.7.5 Rational functionsQn(z) . . . . . . . . . . . . . . . . . . . . . . . . 2 Functional equation related to quadratic and norm forms 2.1 The formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Solution for one Gaussian quadratic form . . . . . . . . . . . . . . . . . . . 2.3 One special type of quadratic forms . . . . . . . . . . . . . . . . . . . . . . 2.4 Outline for a general quadratic form . . . . . . . . . . . . . . . . . . . . . . 2.5 One cubic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Fermat’s
References
little
theorem
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Introduction
0.1 Introduction This thesis consists of three completely independent chapters. The first chapter deals with the Stieltjes transform of the Minkowski question mark function, the second one investigates functional equations associated to various forms in two or more variables, and finally the third gives an amusing proof of celebrated Fermat’s little theorem. Because of independence of chapters, we indulge in being very brief in this introduction, since each chapter has its own self contained introduction.
0.1.1 Actuality In recent decade, the interest in the Minkowski question mark function ?(x) grew signif-icantly. Nevertheless, all previous results concerned ?(x) as a function itself. Chapter 1 establishes the result of a completely new kind, which can be thought as a first step in un-derstanding a deep arithmetic and analytic structure of integral transforms of this function. Further, though results of Chapter 2 are of no exceptional significance, I anticipate that the functional equations associated to certain forms encode a rich arithmetic structure of an underlying variety or field. Chapter 3 is a mathematical joke, though it contains a rigorous and original proof.
0.1.2 Aims and problems The aim of Chapter 1 is to find an expression of the dyadic period function in terms of objects which carry afiniteamount of information, also allowing to use one or several limit processes. By the definition, the dyadic period function is defined via Stieltjes integral, which, in this case, is a rather complicated and ineffective expression. Further, Taylor coefficients of the dyadic period function are real numbers, which, conjecturally, are not arithmetic (by “arithmetic” we mean algebraic numbers, periods, exponential periods, etc). Thus, each of them carries infinite amount of information. Nevertheless, our main result states that there exists a nice expression for this function, which involves only one limit process (infinite sum). The aim of Chapter 2 to is to show that functional equations related to norm forms are in fact rich mathematical objects (at this stage, only algebra of an underlying field or
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