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# Experimental Number Theory.

Publisher: Oxford : Oxford University Press, Incorporated, 2007Copyright date: ©2007Description: 1 online resource (231 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9780191523731Genre/Form: Electronic books. Additional physical formats: Print version:: Experimental Number TheoryDDC classification: 512.70285 LOC classification: QA241.V43 2007Online resources: Click to View
Contents:
Intro -- Contents -- 1 Basic examples -- 1.1 How things vary with p -- 1.1.1 Quadratic Reciprocity Law -- 1.1.2 Sign functions -- 1.1.3 Checking modularity of sign functions -- 1.1.4 Examples -- 1.2 Recognizing numbers -- 1.2.1 Rational numbers in R -- 1.2.2 Rational numbers modulo m -- 1.2.3 Algebraic numbers -- 1.3 Bernoulli polynomials -- 1.3.1 Definition and properties -- 1.3.2 Calculation -- 1.3.3 Related questions -- 1.3.4 Proofs -- 1.4 Sums of squares -- 1.4.1 k = 1 -- 1.4.2 k = 2 -- 1.4.3 k ≥ 3 -- 1.4.4 The numbers r[sub(k)](n) -- 1.4.5 Theta functions -- 1.5 Exercises -- 2 Reciprocity -- 2.1 More variation with p -- 2.1.1 Local zeta functions -- 2.1.2 Formulation of reciprocity -- 2.1.3 Global zeta functions -- 2.2 The cubic case -- 2.2.1 Two examples -- 2.2.2 A modular case -- 2.2.3 A non-modular case -- 2.3 The Artin map -- 2.3.1 A Galois example -- 2.3.2 A non-Galois example -- 2.4 Quantitative version -- 2.4.1 Application of a theorem of Tchebotarev -- 2.4.2 Computing densities numerically -- 2.5 Galois groups -- 2.5.1 Tchebotarev's theorem -- 2.5.2 Trink's example -- 2.5.3 An example related to Trink's -- 2.6 Exercises -- 3 Positive definite binary quadratic forms -- 3.1 Basic facts -- 3.1.1 Reduction -- 3.1.2 Cornachia's algorithm -- 3.1.3 Class number -- 3.1.4 Composition -- 3.2 Examples of reciprocity for imaginary quadratic fields -- 3.2.1 Dihedral group of order 6 -- 3.2.2 Theta functions again -- 3.2.3 Dihedral group of order 10 -- 3.2.4 An example of F. Voloch -- 3.2.5 Final comments -- 3.3 Exercises -- 4 Sequences -- 4.1 Trinomial numbers -- 4.1.1 Formula -- 4.1.2 Differential equation and linear recurrence -- 4.1.3 Algebraic equation -- 4.1.4 Hensel's lemma and Newton's method -- 4.1.5 Continued fractions -- 4.1.6 Asymptotics -- 4.1.7 More coefficients in the asymptotic expansion -- 4.1.8 Can we sum the asymptotic series?.
4.2 Recognizing sequences -- 4.2.1 Values of a polynomial -- 4.2.2 Values of a rational function -- 4.2.3 Constant term recursion -- 4.2.4 A simple example -- 4.3 Exercises -- 5 Combinatorics -- 5.1 Description of the basic algorithm -- 5.2 Partitions -- 5.2.1 The number of partitions -- 5.2.2 Dual partition -- 5.3 Irreducible representations of S[sub(n)] -- 5.3.1 Hook formula -- 5.3.2 The Murnaghan-Nakayama rule -- 5.3.3 Counting solutions to equations in S[sub(n)] -- 5.3.4 Counting homomorphism and subgroups -- 5.4 Cyclotomic polynomials -- 5.4.1 Values of ø below a given bound -- 5.4.2 Computing cyclotomic polynomials -- 5.5 Exercises -- 6 p-adic numbers -- 6.1 Basic functions -- 6.1.1 Mahler's expansion -- 6.1.2 Hensel's lemma and Newton's method (again) -- 6.2 The p-adic gamma function -- 6.2.1 The multiplication formula -- 6.3 The logarithmic derivative of Γ[sub(p)] -- 6.3.1 Application to harmonic sums -- 6.3.2 A formula of J. Diamond -- 6.3.3 Power series expansion of ψ[sub(p)](x) -- 6.3.4 Application to congruences -- 6.4 Analytic continuation -- 6.4.1 An example of Dwork -- 6.4.2 A generalization -- 6.4.3 Dwork's exponential -- 6.5 Gauss sums and the Gross-Koblitz formula -- 6.5.1 The case of F[sub(p)] -- 6.5.2 An example -- 6.6 Exercises -- 7 Polynomials -- 7.1 Mahler's measure -- 7.1.1 Simple search -- 7.1.2 Refining the search -- 7.1.3 Counting roots on the unit circle -- 7.2 Applications of the Graeffe map -- 7.2.1 Detecting cyclotomic polynomials -- 7.2.2 Detecting cyclotomic factors -- 7.2.3 Wedge product polynomial -- 7.2.4 Interlacing roots of unity -- 7.3 Exercises -- 8 Remarks on selected exercises -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- P -- R -- S -- T -- V -- W -- Z.
Summary: This graduate text shows how the computer can be used as a tool for research in number theory through numerical experimentation. Examples of experiments in binary quadratic forms, zeta functions of varieties over finite fields, elementary class field theory, elliptic units, modular forms, are provided along with exercises and selected solutions.
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Intro -- Contents -- 1 Basic examples -- 1.1 How things vary with p -- 1.1.1 Quadratic Reciprocity Law -- 1.1.2 Sign functions -- 1.1.3 Checking modularity of sign functions -- 1.1.4 Examples -- 1.2 Recognizing numbers -- 1.2.1 Rational numbers in R -- 1.2.2 Rational numbers modulo m -- 1.2.3 Algebraic numbers -- 1.3 Bernoulli polynomials -- 1.3.1 Definition and properties -- 1.3.2 Calculation -- 1.3.3 Related questions -- 1.3.4 Proofs -- 1.4 Sums of squares -- 1.4.1 k = 1 -- 1.4.2 k = 2 -- 1.4.3 k ≥ 3 -- 1.4.4 The numbers r[sub(k)](n) -- 1.4.5 Theta functions -- 1.5 Exercises -- 2 Reciprocity -- 2.1 More variation with p -- 2.1.1 Local zeta functions -- 2.1.2 Formulation of reciprocity -- 2.1.3 Global zeta functions -- 2.2 The cubic case -- 2.2.1 Two examples -- 2.2.2 A modular case -- 2.2.3 A non-modular case -- 2.3 The Artin map -- 2.3.1 A Galois example -- 2.3.2 A non-Galois example -- 2.4 Quantitative version -- 2.4.1 Application of a theorem of Tchebotarev -- 2.4.2 Computing densities numerically -- 2.5 Galois groups -- 2.5.1 Tchebotarev's theorem -- 2.5.2 Trink's example -- 2.5.3 An example related to Trink's -- 2.6 Exercises -- 3 Positive definite binary quadratic forms -- 3.1 Basic facts -- 3.1.1 Reduction -- 3.1.2 Cornachia's algorithm -- 3.1.3 Class number -- 3.1.4 Composition -- 3.2 Examples of reciprocity for imaginary quadratic fields -- 3.2.1 Dihedral group of order 6 -- 3.2.2 Theta functions again -- 3.2.3 Dihedral group of order 10 -- 3.2.4 An example of F. Voloch -- 3.2.5 Final comments -- 3.3 Exercises -- 4 Sequences -- 4.1 Trinomial numbers -- 4.1.1 Formula -- 4.1.2 Differential equation and linear recurrence -- 4.1.3 Algebraic equation -- 4.1.4 Hensel's lemma and Newton's method -- 4.1.5 Continued fractions -- 4.1.6 Asymptotics -- 4.1.7 More coefficients in the asymptotic expansion -- 4.1.8 Can we sum the asymptotic series?.

4.2 Recognizing sequences -- 4.2.1 Values of a polynomial -- 4.2.2 Values of a rational function -- 4.2.3 Constant term recursion -- 4.2.4 A simple example -- 4.3 Exercises -- 5 Combinatorics -- 5.1 Description of the basic algorithm -- 5.2 Partitions -- 5.2.1 The number of partitions -- 5.2.2 Dual partition -- 5.3 Irreducible representations of S[sub(n)] -- 5.3.1 Hook formula -- 5.3.2 The Murnaghan-Nakayama rule -- 5.3.3 Counting solutions to equations in S[sub(n)] -- 5.3.4 Counting homomorphism and subgroups -- 5.4 Cyclotomic polynomials -- 5.4.1 Values of ø below a given bound -- 5.4.2 Computing cyclotomic polynomials -- 5.5 Exercises -- 6 p-adic numbers -- 6.1 Basic functions -- 6.1.1 Mahler's expansion -- 6.1.2 Hensel's lemma and Newton's method (again) -- 6.2 The p-adic gamma function -- 6.2.1 The multiplication formula -- 6.3 The logarithmic derivative of Γ[sub(p)] -- 6.3.1 Application to harmonic sums -- 6.3.2 A formula of J. Diamond -- 6.3.3 Power series expansion of ψ[sub(p)](x) -- 6.3.4 Application to congruences -- 6.4 Analytic continuation -- 6.4.1 An example of Dwork -- 6.4.2 A generalization -- 6.4.3 Dwork's exponential -- 6.5 Gauss sums and the Gross-Koblitz formula -- 6.5.1 The case of F[sub(p)] -- 6.5.2 An example -- 6.6 Exercises -- 7 Polynomials -- 7.1 Mahler's measure -- 7.1.1 Simple search -- 7.1.2 Refining the search -- 7.1.3 Counting roots on the unit circle -- 7.2 Applications of the Graeffe map -- 7.2.1 Detecting cyclotomic polynomials -- 7.2.2 Detecting cyclotomic factors -- 7.2.3 Wedge product polynomial -- 7.2.4 Interlacing roots of unity -- 7.3 Exercises -- 8 Remarks on selected exercises -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- P -- R -- S -- T -- V -- W -- Z.

This graduate text shows how the computer can be used as a tool for research in number theory through numerical experimentation. Examples of experiments in binary quadratic forms, zeta functions of varieties over finite fields, elementary class field theory, elliptic units, modular forms, are provided along with exercises and selected solutions.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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