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.