A Primer of Analytic Number Theory: from Pythagoras to Riemann Jeffrey Stopple Index More Information

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Cambridge University Press 0521813093 - A Primer of Analytic Number Theory: From Pythagoras to Riemann Jeffrey Stopple Index More information

Index

#, number of elements in a set, 101 and ␨(2n), 154–157 A =, Abel summation, 202, 204, 213 and Euler-Maclaurin summation, 220 definition, 201 definition, 149 Abel’s Theorem Bernoulli, Jacob, 146, 150, 152 I, 140, 145, 201, 267, 270, 272 Bessarion, Cardinal, 282 II, 143, 145, 198, 267, 270, 272, 318 binary quadratic forms, 270 absolute convergence definition, 296 applications, 133, 134, 139, 157, 167, 194, equivalence relation ∼, 296 198, 208, 215, 227, 236, 237, 266 reduced, 302 definition, 133 Birch Swinnerton-Dyer conjecture, xi, xii, abundant numbers, 27, 29, 31, 43, 54, 60, 61, 291–294, 326 82, 177, 334, 341, 353 Black Death, 43, 127 definition, 27 Blake, William, 216 Achilles, 125 Boccaccio’s Decameron, 281 aliquot parts, 27 Boethius, 28, 29, 43, 278 aliquot sequences, 335 Bombelli, Raphael, 282 amicable pairs, 32–35, 39, 43, 335 de Bouvelles, Charles, 61 definition, 334 Bradwardine, Thomas, 43, 127 ibn Qurra’s algorithm, 33 in Book of Genesis, 33 C2, twin prime constant, 182 amplitude, 237, 253 Cambyses, Persian emperor, 5 Analytic Class Number Formula, 273, 293, Cardano, Girolamo, 25, 282 311–315 Catalan-Dickson conjecture, 336 analytic continuation, 196 Cataldi, Pietro, 30, 333 Anderson, Laurie, ix cattle problem, 261–263 Apollonius, 261, 278 Chebyshev, Pafnuty, 105, 108 Archimedes, 20, 32, 92, 125, 180, 260, 285 Chinese Remainder Theorem, 259, 266, 307, area, basic properties, 89–91, 95, 137, 138, 308, 317 198, 205, 346 Cicero, 21 Aristotle’s Metaphysics,5,127 class number, see also h arithmetical function, 39 Clay Mathematics Institute, xi ∼, asymptotic, 64 comparison test Athena, 28 infinite series, 133 St. Augustine, 28, 326 integrals, 90 automorphism, 305 congruent, 254 average order, 67 congruent number, 279 connections to elliptic curves, 280–281, Bachet de Méziriac,Claude-Gaspar, 61, 283 293 Bachet equation, 284, 289 Constantine X, Byzantine emperor, 279 Basel problem, 146 Constantinople Bk , Bernoulli numbers, 150–153, 210 fall to Ottoman Turks, 282 and ␨(1 − 2n), 212 sacked by crusaders, 279

379

380 Index

continuity in terms of “Big Oh,” 87 Fundamental Theorem of Finite Calculus convergent series, 130 I, 15, 141, 330–332 ∗, convolution, 39–42 II, 19, 141 definition, 39 critical strip, 195 Galileo, 31, 127 cryptography, 148, 283 ␥, Euler’s constant, 44, 49–52, 74, 77, 103, St. Cyril, 278 129, 168, 174, 175, 179, 184, 229, 347, 353 d, discriminant, 296 definition, 49 D, derivative operator, 245 (s), Gamma function, 148 Dante’s Inferno, 281 analytic continuation, 208 deficient numbers, 27, 29, 31, 43, 60, 82, and sin(x), 207 334, 345 and factorials, 206 definition, 27 definition, 204 derivative in terms of “Big Oh,” 87 partial fractions expansion, 208 Descartes, René,38 product formula, 229 , difference operator, 11–23, 95, 328–329 Gauss, Carl Friedrich, 2, 6, 193, 254, 265, 273, definition, 11 290, 296, 315, 322 Diophantine equations, 277 Geometric series, 20, 116, 123, 125–126, 130, examples, 279, 284–286 131, 134, 135, 140, 151, 157, 163, 167, Diophantus, x, 278 203, 210, 227, 231, 247, 266, 312, 331, Dirichlet, Lejeune, 76, 273, 307 358 divergent series, 130 Geometrical Number, x Gerbert d’Aurillac, 29 Egyptian hieroglyphic inscription, 24 Germain, Marie-Sophie, 180 elliptic curve, definition, 274 gnomon, 1, 10, 11, 328 elliptic integrals, 148, 274 Great Year, 5, 127 entire functions, 229 Gregory’s series, xii, 132, 144, 149, 273, 311 envelope, 237 GRH, generalized Riemann hypothesis, 315 equivalence relation, 46, 47, 67, 81, 254, group structure 296 2 × 2 integer matrices, 297 Eratosthenes, 261 arithmetic functions under convolution ∗,40 Erf(x), error function, 95, 118 classes modulo n, 255 Euclid, 27, 30, 37, 38, 282 classes of binary quadratic forms, 298 Euclidean algorithm, 255, 259 points on an elliptic curve, 274 Eudemus, 6 solutions to Pell’s equation, 262 E, Euler operator, 202, 245–247, 355 definition, 202 h, class number, 273 Euler, Leonhard, 34, 38, 146, 159, 180, 193, algorithm for, 302 201, 204, 206, 207, 213, 225, 254, 260, and Siegel zero, 320–326 265, 285, 286 definition, 299 Euler-Maclaurin summation, 220 Gauss’ Conjecture, 315, 322, 326 exp(x), xiii upper bound, 320 Hadamard, Jacques, 187, 230 factorial powers, 13–18 Hardy, G.H., 76, 228, 284 definition, 13 Harmonic series, 130, 133, 146, 160, 161, Faulhaber, Johann, 152 249 Fermat’s Last Theorem, x, 148, 179, 180, definition, 128 285, 286, 293 heat equation, 221, 225 Fermat’s Little Theorem, 256 Hecke, Erich, 322 de Fermat, Pierre, x, xii, 30, 31, 33, 254, 257, Hensel’s Lemma, 258, 307 260, 280, 285, 286, 295, 333 Hercules, 29 Frederick II, Holy Roman Emperor, 280 Herodotus, The Histories,5 fundamental discriminant, 306 hexagonal numbers, 12 Fundamental Theorem of Calculus Hillsboro Mathematical Club, 262 I, 93, 94, 95, 99, 101, 136, 239, 242 Hn, Harmonic numbers, 19, 23, 56, 61, 70, II, 94 73, 92, 95, 96, 127, 129, 168, 220

Index 381

asymptotics, 64 mathematician, ix, xii, 21, 28, 193, bounds, 47–52 359 definition, 19 Mean Value Theorem, 321 (2) Hn , Second-order harmonic numbers, 129 M, Mellin transform, 235–236, 244–247 definition, 76 definition, 234 (3) Hn , Third-order harmonic numbers, 78, 129 Mengoli, Pietro, 131, 146 Holzmann, Wilhelm, 283 Mersenne numbers, 39, 43, 45, 180, 186, 353, Homer 354, 365 Iliad, 125 definition, 31 Odyssey,29 distribution of prime, 182–185 Hypatia, 278 divisors of, 257 Hypothesis I, 97, 100, 102, 104, 347 Mersenne, Marin, 31, 33–35, 38, 43 Mertens’ Formula, 103, 104, 129, 168, 174, i, definition, 188 177, 178, 182, 184, 293 Iamblichus, 5, 28, 29, 32, 43, 279 Millenium Prize Problems, xi Im(z), imaginary part, 189 modulo, definition, 254 independent events, 96 ␮(n), Möbiusfunction, 41, 337 induction, 3, 4, 8, 9, 17, 18, 106, 108, definition, 40 136, 155, 206, 328 inversion formula, 41, 248 infinite series, 130 Muhammad ibn Alhocain, 280 injective, 234, 289 music, see also Riemann Hypothesis integral test, 139 de Nemore, Jordanus, 60, 334 Justinian, Byzantine emperor, 279 Newton, Isaac, 24, 92, 131, 276 Nicholas V, Pope, 282 L-function, 260 Nicomachus, 9, 27, 29, 43, 186, 278 Dirichlet, 266–273, 310–326 elliptic curve, 291–294 O, “Big Oh” Lagrange, Joseph-Louis, xi, 263, 273 definition, 46, 84 (n), Von Mangoldt function, 231 improved definition, 69 (s), 225 Oresme, Nicole, 20, 127, 130, 133 (s, d), 314 Otto III, Holy Roman Emperor, 29 (s, F), 314 Lambert series, 204 PARI, xiii, 28 lattice point, 66 de Parthenay, Catherine, x, 283 Laurent series, 123 Pazzi Antonio, 282 Legendre symbol, 288 Pell’s equation, 5, 148, 260–273 Leonardo of Pisa, 20, 280, 282 pentagonal numbers, 11–12 level curves, 190, 191, 213 Samuel Pepys’ Diary, 148 Library of Alexandria, 279, 282 perfect numbers, 27, 29–31, 39, 43, 332–335, limits in terms of “Big Oh,” 87 337 linear, 235 definition, 27 Livy, 20 Euclid’s theorem, 37 , “less than less than,” definition, 44, 84 Euler’s theorem, 38, 39, 43 log(x) Nicomachus’ claim, 43, 186 definition as an integral, 95 odd, 38 means natural logarithm, xiii period, 237, 253 logarithmic derivative, 157, 230 phase shift, 237 definition, 124 ␾(s), 195 Li(x), Logarithmic integral, 239–242 Pi notation for products, 97, 159 definition, 101 Pink Floyd, 126 L(x), Euler’s dilogarithm, 121, 135 ␲(x) definition, 101 Maple, xiii, 32, 35, 192, 199, 202 Gram’s series, 248 Mathematica, xiii, 32, 35, 192, 199, 202, lower bounds, 108–110 238, 253, 294, 300, 303, 306, 354, 360, Riemann’s explicit formula, 248 361, 371 upper bounds, 105–108

382 Index

(x) functional equation, 213, 216, 225–228, definition, 242 230, 252, 292 Riemann’s explicit formula, 244 Hadamard product, 194, 230 Plato, x level curves, 213 Plutarch, 4, 20 movie, 199, 354 Poiters, battle of, 127 zeros, 196, 199, 203, 214, 228, 230, 234, pole, 123 237, 239, 247, 249 positive definite, 298 Riemann, G. F. Bernhard, 193, 194, 225, 230, Prime Number Theorem, 97, 102, 105, 128, 232, 244, 251, 252, 308 249, 251 and general relativity, 194 exposition of proof, 249 principal form, 298 s(n), 31, 33, 35, 332, 334–336 properly represent, 303 averages, 81–82, 344–345 Psellus, Michael, 278, 279 bounds, 56, 59, 175, 340 (x) definition, 27 definition, 232 Scot, Michael, 281 Von Magoldt’s explicit formula, 234 Shelley, Percy Bysshe, 1 exposition, 236–238 Siegel zeros movie, 361 and class number, 322–326 Ptolemy, 30, 278, 282 bounds, 320–322 Pyramidal numbers, 8 definition, 316 definition, 8 ␴(n), Sigma function, 25–27, 33, 39, 40, 42, Pythagoras, 5, 28, 32 332 Pythagoreans, ix, 1, 5, 27, 29, 127, 278, 295 averages, 79–81 bounds, 46, 54–56, 59–61, 174–179, 338, Quadratic Reciprocity Law, 228, 264, 272, 353 287, 290, 308, 309, 367 definition, 24 quadrivium, 29, 127, 194, 281 formula, 36–37 quantum mechanics, xii Sigma notation, 2, 6, 24, 97, 115, 117 , Sigma operator, 14–16 r, rank of elliptic curve, 281, 292, 293, 370 definition, 14 definition, 277 simple pole, 123 radius of convergence, 135 singular part, 123 Ramanujan, Srinivasa, 284 Sixtus IV, Pope, 282 rd (n), representation number sn, square numbers, 1, 3–5, 8, 10, 12, 47 algorithm for, 306 definition, 3 definition, 306 reciprocals, 146 product formula, 306 Sophie Germain primes, 34, 353 reduced, 302 and Mersenne numbers, 180 Regiomontanus, 282 and primality testing, 180 relatively prime, 25 definition, 179 represents, 297 distribution of, 180–182 residue, 123 Stanley, Thomas, 1 Re(z), real part, 189 Stirling numbers, 16–18 rF (n), representation number, 306 definition, 17 Rhind Papyrus, 20 Stirling’s formula, 219 Riemann Hypothesis, xi, xii, 196, 199, 228, Summation by Parts, 22, 23, 140, 141, 143, 253, 287, 325, 353 318, 319, 331, 332, 373 and music, 194, 253 surjective, 289 exposition, 252–253 Sylvester II, Pope, see also Gerbert d’Aurillac Riemann zeta function, xii, 77 analytic continuation ␶(n), Divisor function, 24–27, 35, 39, 42, 64, first version, 195–198 332, 336, 337, 342 second version, 209–213 averages, 66–76, 129, 342 third version, 228 bounds, 46, 56–58, 61–63, 338, 340, 341 definition, 146 definition, 24 Euler product, 159 formula, 36–37

Index 383

Taylor polynomial Urban V, Pope, 127 definition, 114 error as an integral, 137 Viète, Fran¸coise,x, 27, 283, 285 Taylor series, 115 Tetrahedral numbers, 6–8 Wallis, John, 146, 148 definition, 6 Wiles, Andrew, x, 285, 291, 293 reciprocals, 21, 131, 331 Thabit ibn Qurra, 20, 32, 33, 39 ␹d (n), Kronecker symbol (t), Jacobi theta function, 220, 228, definition, 308–310 265 examples, 265, 271, 310 (t, F), 314 Xylander, see also Holzmann, Wilhelm Thin Air Equation I, 312 Zeno, 125 II, 313 ␨(s), see also Riemann zeta function Thompson, William (Lord Kelvin), 359 ␨(2), 129 tn, triangular numbers, 2–8, 10–12, 15, definition, 77 39, 46, 72, 152 Euler’s formula, xii, 146–148 asymptotics, 65 ␨(3), 78, 129 bounds, 45, 46 ␨(4), Euler’s formula, 148 definition, 2 ␨(2n), Euler’s formula, 153–157 in Book of Revelations, 152 ␨(s, d), 312 reciprocals, 21, 131, 146 ␨(s, F), Epstein zeta function, 311 Tortoise, 125 ␨(1 − 2n), Euler’s formula, 201–204