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Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) / Chapter Skim
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Pages 407-422

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From page 407...
... - i - A Abelian function, 31, 366 Abramowitz, Milton, 373 Absolute error, 234-235 Academies/societies, universities distinguished from, 30, 57 Adele, 320-321 Adelic quantum mechanics, 320 Airy, George, 225 Alan Turing: The Enigma, 262 Alexander I, Emperor of Russia,121122 Alexanderson, Gerald, 352-353 Algebra, 119, 194,209,225 contrasted with geometry, 385 defined, 17-18,86-87 game theory, 18 Algebraic field theory, 269-271 Algebraic invariants, theory of, 184, 225 Algebraic number theory, 87, 184,185, 194,318-321 Algebraic numbers,173-174,269 American Institute of Mathematics, xi, 351-352 American Mathematical Society, 290 Amplitude of a complex number,180182,333-334 Analysis, 119,130 arithmetic and,18,86-87,91,8990,96 calculus in, 87-88 classic texts, 15, 226 in complex plane,182- 183 continuity concept, 90-91 defined,l5,16, 17, 18, 87 functional, 195 invention, 87-88 limit concept, 16,17, 88, 90-91 Analytic number theory, 18, 86-87, 96, 97-98,153,156,198,231-232, 238-239,322 Anna, Empress of Russia, 59 Anti-Semitism, 163, 254-255 Apery, Roger, 371 Apery's number,371,399 Apostol, Tom, 393-394 Argand, Jean-Robert, 92 Argument of a function, 36 Argument plane,210-216,218,219,221 407
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From page 408...
... , 369 Boole, George, 18, 225 Borchardt, 135 Borel, Emile,92 Bourienne, Louis de, 60 Branges, Louis de, 383 Breaking the Code (Whitemore) , 262 Brent, Richard P.,258 British mathematics and mathematicians, 224-226 Brothers Grimm, 26 Brouwer, Luitzen, 170 Brunswick, Dukes of, 49-50; pl.1 Brunswick Polytechnic, 193 Buckley, William F., Jr., 85 Bump, Daniel, 352, 391 Bump-Ng Theorem, 352 Burkill, J.C., 376 C Calcul des Residus (Lindelof)
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From page 409...
... , 206 Complex plane, 180- 181 analysis in, 182- 183 non-trivial zeros on, 190-192 squaring function on, 209-210 Congress of Vienna, 20-21,61,92 Congruences arithmetic of, 97 Conjugate of a complex number, 181 Connes, Alain, 317-321,384,388; pl.6 Conrey, Brian, 353 Continuity, 88,90-91,119 Continuum Hypothesis, 170,188-189 Contour integration, 402 Convergence absolute, 150 of Basel series, 64 and completeness of a number system, 173 of a complex series, 182-183 conditional, 149-150,339 of the eta series, 145 as illustrated by rulers, 10-15,173 in Riemann's formula for J(x) , 338339,342 of a sequence for Em, 16,67,179 of sequences for ~ and e, 16,173 of series for 1/~1-x)
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From page 410...
... , 9 3, 97 Divergence and convergence compared, l l- 15 of harmonic series, 9-lO, 18,63-64, 338,399 of series for l/(l-x) , 139 of series of reciprocals of primes, 154 of the zeta series for arguments less than one, 80-81 Doxiadis, Apostolos, 90 Dreyfus Affair, 162-163,164,165 Dreyfus,Alfred, 162-163 Dreyfus, Mathieu, 162 DuBois-Reymond, Emil, 253 Dukas, Paul, 156 Dynamical systems, 281,315 Dyson, Freeman, 282,287,288,290291;pl.7 E e, 40-41,55,69,185,202-203,365-366 Ebell, Charlotte (mother of Bernhard Riemann)
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From page 411...
... 1 Euler-Maclaurin summation, 263 Euler-Mascheroni number,55, 369-370 Euler product formula,105- 106,373 Euler, Salome (nee Gsell) ,62 Evelyn, John, 56 Exiguus, Dionysus, 84 Existence proofs,184- 185 Exponential function canonical form, 40-41 for complex numbers, 202 defined, 39-40 inverse of, 43-44 Exponentiation, 68-69 Extended Riemann Hypothesis, xiv Extrapolation card trick exercise, 7-8 defined, 7 F Factor defined, 32 proper, 32 trivial, 32, 36 Factorial function, 124, 147 Faure, Felix, 162, 163 411 Fejer, Lipot, 378 Fermat, Pierre de, 371 Fermat's Last Theorem, x, xi,90, 161, 197,271,354,371 Fermi, Enrico, 280 Feynman, Richard, 291 Field characteristic of, 268 defined, 266 finite, 267 infinite, 266 theory, 197, 265-271 Fields Medal, 261, 270,384, 385 Finite fields, 267, 321-322 Form factor, 287,288-289 "Foundations," 18,185,225 Four-Color Theorem, x, xi, 197 Fourier, Joseph, 92,93, 119 Fractions,174- 175,177- 178 improper, 171 mixed, 171 powers, 66-67 proper, 171 vulgar, 171 France anti-Semitism in, 163 DreyfusAffair,162-163,164, 165 nineteenth century culture and politics,157- 158,159 Franklin, James, 325, 326, 389 Frederick the Great, King of Prussia, 59-60,61,92 Frege, Gottlob, 360 French Academie des Sciences, 154 French Revolution, l9 Freudenthal, Hans, 127, 131 Friedrich Wilhelm, Duke of Brunswick, 51 Friedrich Wilhelm IV, King of Prussia, 30 Fry, John, xi, 352-353 Function theory, 121, 133,209-210, 221,225
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From page 412...
... , 316 Gaussian Unitary Ensemble (GUE) , 286-287,291,294,315,387 Gel'fond, Alexander, 354 Generalized Riemann Hypothesis, xiv Geometric number theory, 87 Geometry, ll9 INDEX defined, 17,86 differential, 128 Euclidean, 18 foundations of, 185 non-Euclidean, 122,130-131 topology, 18 George II, King of England, 26 George III, King of England and Hanover, 26,60 George IV, King of England and Hanover, 21,368 George V, King of Hanover, 48,368 Germain, Sophie, 92 Germany Berlin mob, 30 educational system, 24-25,29,30, 93,120 mathematics and mathematicians in, 91-93,185,254-256 Nazi control of, 254-256,264 structure, 21,24 unification of, 160,366,368 Ghosh, Amit, xiv Gleick, James, 387 Godel, Kurt, 195,391 Goffman, Erving, 52,257 Gogol, Nikolai, 122 Goldbach, Christian, 90 Goldbach Conjecture, 90, 197,371, 379 Golden Key, The, 55,59,72,97,135,222 calculus version, 309-311 expression, 105,138,303-304 and Mobius function, 245-246 proof of, 102-104,107 sieve of Eratosthenes and, l00-l0l turning, 303-311 Gonek, Steve, xiv Gordan, Paul Albert, 185 Gordan's Problem, 184 Gottingen, city of, 255-256,383 Gottingen Seven, 26-27,119,120
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From page 413...
... ,231232,238-239 Hanoverian kings,21,368 Hardy, G.H.,52-53,92,224,226,227229,232,287,359-360,361,376; pl.4 Harmonic series, 9, 58 convergence of,11- 15 divergence of, 9-10,12, 63,64, 76, 88,399 infinityin,15-16 413 Harvard University, xi, 166, 353 Haselgrove, Brian, 259 Hasse, Helmut, 270 Hebrew University of Jerusalem,164165,230 Heilbronn, Hans, 232 Hejhal, Dennis, 322 Heliotrope, 128 Hensel, Fanny (nee Mendelssohn) , 388 Hensel, Kurt,319,388 Herglotz, Gustav, 255, 256 Hermite, Charles, 159-160, 174,194, 275 Hermitian matrix, 275-276,277, 282, 283,284-285,286,288-289,295 Hilbert, David, x,92,159,166,170, 184-190,196-197,252,253-254, 256,276,277,279,353,354,377, 391;pl.4 Hilbert, Franz, 186 Hilbert-Polya Conjecture, 277-278, 279 Hindenburg, Paul von Beneckendorf und von, 254 Hirst, Thomas, 94-95 Hitler, Adolph, 254 Hodges, Andrew, 262, 377, 384 Hollond, H.A., 375 Hudson, Richard, 126, 236, 380 Humboldt, Alexander von, 24, 93 Humboldt, Wilhelm von, 24, 29, 92 Hungarians, 377-378 Hutchinson, J.I., 258, 263 Huxley, Martin, 357 Huygens, Christiaan,58 I i,176 Ignorabimus principle, 253 Imaginary axis, 180 Imaginarynumbers,169-170,175-178, 180
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From page 414...
... , 217, 385 Inverse function, 41-42,43, 44, 221 Irrational numbers, 40, 69, 76, 170, 171,172,173,174,175,179,266, 367 Irrational powers, 67 Iwaniec, Henryk, xiv Jacobi, Carl, ll9 Jacoby, Johanna, 230 Johns Hopkins University, 154 INDEX Johnson, Dr., 53 Johnson, Paul, 61 Jordan, Camille, 226 Jordan's Theorem, 226 J(x) , 299-302, 305-307, 328-330 K Kanigel, Robert, 227, 228-229 Kant, Immanuel, 130, 252 Katz, Nicholas, 245, 368, 387 Keating, Jonathan, 316, 350-351, 390 Kepler's laws,314 King's College, Cambridge, 261, 380 Klein, Felix, 92, 159 Koch, Elise, 31, 362, 363-364 Koch, Helge von,237,240,242,244245,381,397 Konig, Samuel, 370 Kronecker, Leopold, 135, 170,185, 188, 376-377 Kulik,Yakov,153 Kummer, Eduard, 135, 372 Kummer, Ottilie (nee Mendelssohn)
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From page 415...
... , 62 LH. See Lindelof Hypothesis Liddell, Alice, 395 Limit analysis as the study of,16-18,8788,90-91 and continuity, 91 as a fundamental concept in calculus, 88 harmonic series has no, 9 irrational powers defined vie, 67 of a sequence, 16, 175 of a series, 17 415 Log function,43-44,69,70-75,107109,110,111,149,203-204,244, 328 Log integral function, 113-117,332, 333,335-336,337,340,356-357 Lorenz, Edward,314-315 Lower bound, 380 Lune, Jan van de, 257-258 M Maclaurin, Colin, 263 Mater, Helmut, 324 Mallory, George, 90 Man Who Knew Infinity, The (Kanigel)
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From page 416...
... See Modulo and Modulus of a complex number Modified Generalized Riemann Hypothesis, xiv Modified Grand Riemann Hypothesis, XiV Modulo, 97, 395, 403 Modulus of a complex number,180182,333-334,396-399 Moments of zeta function, xiv Monge, Gaspard,92 Montgomery, Hugh, 193,231-232, 287-288,290-291,352,356; pl.7 Montgomery-Odlyzko Law, 292-294, 312,352,355,387 Montgomery Pair Correlation Conjecture Moon and Sixpence, The (Maugham) , 28 Morgan, Augustus de, 226 Bent.
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From page 417...
... calculus and,106- 113 Chebyshevand,123-124 consequences of, 45-47, 323-324, 359-360 equivalents, 47 expressions, 45, 116 first published work, 54 Gauss and, 51, 53-54 graph, 117 improvedversion,116 log integral function and, l 13- 117 logarithmic sense, 45-46 proofs, 124-125,153-155,159-160, 190,198,233-234,237,356 Prime numbers Chebyshevbias, 125-126 defined, 32 frequency of, see Prime Number Theorem infinity of, 34, 95-97, 105 probabilist model for distribution of, 198 series of reciprocals of, 154 sieve method for finding, 100-101 tables,33,153-154 thinning out of, 34-35 and zeros of zeta function, 154 Princeton University, 245 Principia Mathematica (Whitehead and Russell) , 89, 225,361, 391 Product sign (ll)
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From page 418...
... , 196 Reid, Constance, 186, 188 Relative error, 234-235 Relativity, General Theory of,128- 129, 130,318 Rellich, Franz, 383 Renaissance, 175 Repulsion effect, 284, 285-286 RH. See Riemann Hypothesis Riele, Herman te, 161, 236, 258 Riemann, Bernhard, pl.2 academic career, l9,30-31,131132,134,135 INDEX and analytic number theory, 97-98 bereavements, xiv-xv,23, 133, 134 Berlin Academy, ix,119,135 "breakout year," 31 Collected Works, 27,29,131,366 death, 362-364 doctorate and habilitation, l 19- 121, 126,127-129,130 early life and home environment, 22-23 on error term's big oh, 244-245, 381 on error term's sign, 235 friends and colleagues, 27-28, 29, 119,120 health problems, xiv, 23-24,28, 133 honors and awards,31,135 intellectual abilities and interests, 129-130,131,152,194 lecturing style, 132 lunar crater named after, 374 marriage and family,31,362 mentors,94-95,98,119,126- 127, 133 papers and published works,29,3031,127-128,131,133, 135,151 personal characteristics, 27-29, 127 poverty, xv,119- 120 religious faith, 28, 127, 363 schooling and scholarship, 24-25, 27,29,30-31 social awkwardness, xiv-xv, 133 and theory of many-valued functions, 43 Riemann, Clara, 133 Riemann, Elise (nee Koch)
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From page 419...
... See also Harmonic series Basel, 63-64 convergent, 11-15,79 defined, 8 divergent,9-10,81, 139 infinite, 59, 63, 75 of reciprocal squares, 64-65 ruler exercises,10- 15 sequences contrasted, 16-17 Serre, Jean-Pierre, 372, 384 Set theory, 18, 88 Seven Years War, 60
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From page 420...
... , 60 Sorbonne, 159,188,225 Sorcerer's Apprentice (Dukes) , 156 Soundararajan, Kannan, 389 Space nature of, 130,195 operators on, 317-318 Sprague-Grundy Theory, 372 Sprague, Roland Percival, 372 Square roots, 41,43,176,178 Squaring function, 37,42,201-202, 206-209,240 Stegun, Irene A., 373 Steiner, Jakob, 119 Step functions, 124,297-302 Stern, Moritz, 27 Stevens, Wallace, 198 Stieltjes integral, 160 Stieltjes, Thomas, 154,160,161,376 Stirling, James, 123 Strachey, Lytton, 370,380 Summation sign All, 78 "Sweet Betsy from Pike" (tune)
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From page 421...
... , 161 Wiles, Andrew, 90, 161,245,271,354355 421 Wilhelm I, German Kaiser, 160 William IV, King of England and Hanover, 26 Wolfram, Stephen, 389 Wright, Sir Edward, 302 y Yorke, James, 387 z z plane, 379 Zeno, 88 Zeros, 85 in conjugate pairs, 190-191 density of, 396 dividing by, 35 of a function, 139,154,160,169, 190-192,206,211-212,385 gradient, 110 mathematical legitimacy, 89 non-trivial, 77,190- 192,198- 199, 217,221-222,232,289-290,295 number of, 258 order of a, 385 of a polynomial, 173 power, 65,66 spacing in critical strip, 217-218, 232,290 trivial, 148,169,206 Zeta function, 135 Basel problem and, 63-65 on complex plane, 183,213-216 critical line, 221-222 critical strip, 216 decomposition, 358 domain, 142-145,205-206 expression, 77,79,137 graph, 142-144 Mertens's function and, 250-251 Mobius function and, 250-251
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From page 422...
... , 77. See also Zeta function Z. .
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Key Terms

  • zeta function
  • harmonic series
  • critical strip
  • berlin academy
  • counting function

  • prime counting
  • zeta series
  • complex plane
  • integral function
  • error term

  • limit concept
  • function theory
  • card trick
  • log integral
  • characteristic polynomial

  • squaring function
  • lead diagonal
  • generalized riemann
  • friedrich wilhelm
  • sir edward

  • critical line
  • irrational powers
  • pair correlation
  • dreyfus affair
  • ruler exercises

  • basel series
  • canonical form
  • trick exercise
  • grand riemann
  • riemann operators


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