Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003)

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, 121-122

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, 180-182, 333-334

Analysis, 119, 130

arithmetic and, 18, 86-87, 91, 89-90, 96

calculus in, 87-88

classic texts, 15, 226

in complex plane, 182-183

continuity concept, 90-91

defined, 15, 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

Apéry, Roger, 371

Apéry’s number, 371, 399

Apostol, Tom, 393-394

Argand, Jean-Robert, 92

Argument of a function, 36

Argument plane, 210-216, 218, 219, 221

Arithmetic, 119

and analysis, 18, 86-87, 89-90, 91, 96

“arithmetic” vs. “number theory,” 371-372

clock, 97, 267

of congruences, 97

defined, 17, 86

of matrices, 272, 273

Artin, Emil, 197, 270, 325; pl. 6

Association for Computing Machinery, 261

AT&T Bell Labs, 291, 357

Atiyah, Sir Michael, 385

B

Babbage, Charles, 225

Bachmann, Paul, 231, 238, 381

Backlund, Ralf Josef, 258, 263, 384

Barnes, E.W., 223, 379

Basel problem, 59, 62, 63-65, 75-76, 214, 332, 370, 399

Basel series, 63-64

Basel University, 58

Battle of Auerstädt, 49-50

Battle of Valmy, 20, 49

Bays, Carter, 126, 236, 380

Bays-Hudson number, 126, 348

Bell, Eric Temple, 55, 59

Berkeley, George, 88

Berlin Academy of Science, 30, 31, 60, 133, 135, 185

Berlin Society of Sciences, 60

Berlin University, 29, 383

Bernoulli, Daniel, 58, 59

Bernoulli, Jakob, 10, 63, 322, 370

Bernoulli, Johann, 10, 58, 63

Bernoulli, Nicholas, 58

Bernstein, Felix, 255, 383

Berry, Sir Michael, 291, 312, 316-317, 342, 356, 386-387, 390; pl. 7

Bertrand, Joseph, 124

Bertrand’s postulate, 124

Bieberbach Conjecture, 383

Bierberbach, Ludwig, 383

Big oh, 237, 238-245, 383, 396

Biron, Ernst Johann, 59

Bohr, Harald, 228, 325, 394

Bohr, Niels, 228

Bohr-Landau Theorem, 394, 396

Bollobás, Béla, 229-230, 378

Bolyai, Farkas, 92

Bolyai Prize, 377

Bolzano, Bernard, 92

Bombieri, Enrico, xiv

Book of Numbers, The (Conway and Guy), 369

Boole, George, 18, 225

Borchardt, 135

Borel, Émile, 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 Résidus (Lindelöf), 379

Calculus, 119

in analysis, 87-88

invention, 87-88, 313

limit concept, 16, 175

and PNT, 107-113

Cambridge spy ring, 226, 380

Canonical form, 40-41

Cantor, Georg, 18, 92, 179

Carathéodory, Constantin, 92, 372

Card trick exercise, 3-8

Carl Wilhelm Ferdinand, Duke of Brunswick, 49-50; pl. 1

Caroline of Brunswick, 368

Carroll, Lewis, 395

Casti, John L., 377

Catherine, Empress of Russia (wife of Peter the Great), 58

Catherine the Great, Empress of Russia, 60-61, 121

Cauchy, Augustin-Louis, 92, 119

Cauchy-Riemann equations, 121

Cayley, Arthur, 225, 226, 277

Chaos theory, 312-317, 387-388

Characteristic of a field, 268, 395, 402

Characteristic polynomial of a matrix, 272-273, 274, 276

Charles of Sweden, 56-57

Chebyshev bias, 125-126, 380

Chebyshev limits, 154

Chebyshev, Pafnuty Lvovich, 122-124, 125, 154; pl. 3

Chiliads, 54

Chinese culture and language, 82-84

Chowla, Sarvadaman, 288, 386

Chrystal, George, 364

Church, Alonzo, 195

“Clariton,” 356

Class number problem, 386

Clay, Landon T., xi, 353-354

Clay Mathematics Institute, xi, 353-354

Clebsch, Alfred, 364

Clock arithmetic, 97, 267

Closed forms, 63, 64-65, 75, 171, 176, 314

Collège de France, 159, 317, 354

Columbia University, 164

Complete number system, 173, 320

Complex function theory, 30-31, 121, 124, 159, 198, 206-208, 379

Complex numbers, 169-170, 171, 172, 173, 175-177, 180, 190

amplitude, 180-182, 215

to complex powers, 204-205

conjugate, 181, 191, 274, 336

functions of, 201-204, 206-208, 216-217

modulus, 180-182

Complex Numbers and Functions (Estermann), 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), 338-339, 342

of a sequence for , 16, 67, 179

of sequences for π and e, 16, 173

of series for 1/(1–x), 138

of a series for e, 202-203

of series via sequences, 16-17

of the zeta series for complex arguments, 205

of the zeta series for real arguments, 79

Conway, John, 369

Counting logic vs. measuring logic, 82-86

Courant Institute conference, 245, 351-352, 355, 359

Courant, Richard, 187, 255

Cours d’Analyse (Jordan), 226

Course of Pure Mathematics, A, (Hardy), 226

Coxeter, H.S.M. “Donald,” 196, 378

Cramér, Harald, 323, 388, 394; pl. 8

Cramér model, 323-324

Creasy, Sir Edward, 20

Critical line, 191, 192, 198-199, 217, 221-222, 337, 346-347

height up, 258-260

number of zeros on, 258, 259, 289-290

Critical strip, 191, 216, 306-309

Critique of Pure Reason (Kant), 130

D

Data vs. datum, 85-86

Davenport, Harold, 372, 383

Davis, Martin, 187

Davis, Philip J., 122, 123

Decembrists, 122

Dedekind, Richard, 19, 25, 27, 30, 92, 120, 131, 132, 133, 134, 193, 362-363, 366; pl. 2

Deléglise, Marc, 380

Deligne, Pierre, 270, 325, 355, 384; pl. 6

Denjoy’s probabilistic interpretation, 325, 388

Density Hypothesis, xiv

“Derbyshire function,” 242-243

Derbyshire, John, pl. 8

Derivatives, 109, 110

Desargue’s Theorem, 196

Descartes, René, 18, 164

Diamond, Harold, 287, 386

Differential geometry, 128

Differentiation, 41-42, 109-110

Dirichlet, Lejuene, 91, 93, 94-95, 96-97, 119, 126-127, 133, 134, 194, 232, 372, 374; pl. 2

Disquisitiones Arithmeticae (Gauss), 93, 97

Divergence

and convergence compared, 11-15

of harmonic series, 9-10, 18, 63-64, 338, 399

of series for 1/(1–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, 290-291; pl. 7

E

e, 40-41, 55, 69, 185, 202-203, 365-366

Ebell, Charlotte (mother of Bernhard Riemann), 22

Edwards, Harold, 153, 217, 263, 298, 375, 377, 384

Eigenfunction, 318

Eigenvalues, 273, 274, 276, 283, 284, 285, 295

Einstein, Albert, 128, 165, 313, 386

Eisenstein, Gotthold, 119, 129

“Elementary” methods, 124-125, 198

Elizabeth, Empress of Russia (daughter of Peter the Great), 58

Encke, Johann Franz, 53-54

Entire functions, 332-333

ε, 74, 371

Eratosthenes of Cyrene, 100-101, 372

Erdős, Paul, 125, 378

Ernest Augustus, King of Hanover, 26, 366

Error term, 190, 234-235, 236-237, 241, 243-244, 326, 327-349

Erwin Schrödinger Institute, 352

Esterhazy, Major Count Ferdinand Walsin-, 162

Estermann, Theodor, 206, 226, 379

Eta function, 145-146

Euclid of Alexandria, 18, 34, 232

Euler, Catherine (née Gsell), 59, 62

Euler, Leonhard, 9, 15, 40, 55-56, 58, 59, 60, 61-62, 65, 75-76, 87, 88, 95, 97, 98, 100, 106, 121, 146-147, 374; pl. 1

Euler-Maclaurin summation, 263

Euler-Mascheroni number, 55, 369-370

Euler product formula, 105-106, 373

Euler, Salome (née 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

Fejér, Lipót, 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

Dreyfus Affair, 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 Académie des Sciences, 154

French Revolution, 19

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

Functions.

See also Prime counting function;

other specific functions

area under, 113

argument of, 36, 207-208

of complex numbers, 201-204, 206-208, 216-217

constant, 67

defined, 35-36, 129

domain of, 36-37, 70, 138-142, 201, 331-332

entire, 332-333

gradient of, 111

graphing, 37

limit on size of, 239

mapping, 36

value of, 36, 67, 207-208, 212, 214, 216

zero of, 139, 148, 154, 385

zeta, 37

G

Galois, Évariste, 369

Game theory, 18, 372

Gardner, Martin, 367

Gathorne-Hardy, Jonathan, 25

Gaudin, Michel, 387

Gauss, Carl Friedrich, 27, 29, 31, 48-49, 50-54, 87, 90, 92, 93, 96, 120-121, 126, 128, 131, 132-133, 134, 135, 159, 193, 194, 201, 235, 369, 374, 375; pl. 1

Gaussian-normal random number, 283

Gaussian Orthogonal Ensemble (GOE), 316

Gaussian Unitary Ensemble (GUE), 286-287, 291, 294, 315, 387

Gel’fond, Alexander, 354

Generalized Riemann Hypothesis, xiv

Geometric number theory, 87

Geometry, 119

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

Gödel, 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 Möbius function, 245-246

proof of, 102-104, 107

sieve of Eratosthenes and, 100-101

turning, 303-311

Gonek, Steve, xiv

Gordan, Paul Albert, 185

Gordan’s Problem, 184

Göttingen, city of, 255-256, 383

Göttingen Seven, 26-27, 119, 120

Göttingen University, 26-27, 29, 30-31, 51, 93, 94, 119, 120, 130, 133, 134, 166, 185, 230, 252, 254-256, 257, 264, 363-364

Gradient, 108-109, 110, 111, 114

Gram, Jørgen Pedersen, 154, 198-199, 257, 258, 263; pl. 5

Grand Riemann Hypothesis, xiv

Gray, Jeremy J., 377

Griffiths, Phillip A., x

Grünbaum, Branko, 378

Gsell, Catherine, 59, 62

Gsell, Salome, 62

Gutzwiller, Martin, 316, 321

Guy, Richard, 369

H

Habilitation, 119-120

Hadamard, Jacques, x, 92, 153, 154-156, 158-159, 160-161, 163-166, 189, 194, 223, 230, 232, 352, 356, 359, 361, 376; pl. 3

Hadamard, Lucie, 163

Hadamard, Mathieu-Georges, 164

Hadamard, Pierre, 158-159

Hadamard’s Three Circles Theorem, 159, 376

Hamiltonian operator, 224, 318

Handbook of Mathematical Functions (Abramowitz and Stegun), 373

Handbuch der Lehre von der Verteilung der Primzahlen (Landau), 231-232, 238-239

Hanoverian kings, 21, 368

Hardy, G.H., 52-53, 92, 224, 226, 227-229, 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

infinity in, 15-16

Harvard University, xi, 166, 353

Haselgrove, Brian, 259

Hasse, Helmut, 270

Hebrew University of Jerusalem, 164-165, 230

Heilbronn, Hans, 232

Hejhal, Dennis, 322

Heliotrope, 128

Hensel, Fanny (née 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-Pólya 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

Imaginary numbers, 169-170, 175-178, 180

Improper fraction, 171

Incomplete number system, 173

Industrial Revolution, 118

Infinite field, 266

Infinite product, 373

Infinite series, 59, 63, 75, 138, 145, 149-150, 304-305

Infinity, 15

of irrational numbers, 179

point at, 214

of prime numbers, 34, 95-97, 105

of rational numbers, 179

Ingham, Albert, 125

Institute for Advanced Study (Princeton), x, 125, 264, 287, 291

Integers, 171, 172, 173, 174

Integrable problems, 314

Integrals, 88, 110-112, 127, 160, 305, 306

Integration, 42, 110, 111, 113, 149, 335

contour, 394

International Congress of Philosophy, 225

International Congresses of Mathematicians, x, 165-166, 184, 188, 225

Introduction to the Theory of the Riemann Zeta-Function, An (Patterson), 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

J

Jacobi, Carl, 119

Jacoby, Johanna, 230

Johns Hopkins University, 154

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, 244-245, 381, 397

König, Samuel, 370

Kronecker, Leopold, 135, 170, 185, 188, 376-377

Kulik, Yakov, 153

Kummer, Eduard, 135, 372

Kummer, Ottilie (née Mendelssohn), 372

L

Lagrange, Joseph-Louis, 92

Landau, Edmund, 38, 224, 230-232, 238-239, 255-256, 276, 278, 325, 394; pl. 4

Laplace, Pierre-Simon, 92, 93

Lead diagonal of a matrix, 272

League for Human Rights, 164

League of Nations, 164

Least squares method, 53, 54

Lebesgue, Henri, 33, 88, 92

Legendre, Adrien-Marie, 53, 54-55, 92, 93, 232, 369

Lehman, R. Sherman, 236, 258, 259

Lehman’s Theorem, 236

Lehmer, Derrick, 258

Lehrer, Tom, 374

Leibnitz, Gottfried, 22, 88, 112, 370

Lermontov, Mikhail, 122

Letters to a German Princess (Euler), 62

LH. See Lindelöf Hypothesis

Liddell, Alice, 395

Limit

analysis as the study of, 16-18, 87-88, 90-91

and continuity, 91

as a fundamental concept in calculus, 88

harmonic series has no, 9

irrational powers defined via, 67

of a sequence, 16, 175

of a series, 17

Lindelöf, Ernst, 223, 379, 384, 395; pl. 8

Lindelöf Hypothesis

diagram, 401

interesting mainly in critical strip, 216

Lindelöf and, 379

RH and, 393, 401-402

stated, 399, 401-402

Lindelöf mu function, 394, 400-402

Lindemann, Ferdinand von, 174, 185

Listing, Johann, 374, 381

Littlewood, Ann, 229-230

Littlewood, J.E., 193, 223-224, 225, 227, 229, 230, 231, 233, 235, 349, 357, 375, 394; pl. 4

Littlewood violation, 235-236, 326, 345, 348, 356, 380

Li(x), 113-117, 328, 333, 335-336, 373, 394, 396-397

ln, 75

Lobachevsky, Nikolai, 122, 130

Log

defined, 69

natural (base e), 69, 75

“taking a,” 71-72

Log function, 43-44, 69, 70-75, 107- 109, 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

Maier, Helmut, 324

Mallory, George, 90

Man Who Knew Infinity, The (Kanigel), 227

Mangoldt, Hans von, x, 153, 155, 156, 160-161, 189, 192, 232

Many-body problem, 281

Many-valued function, 43, 203

Massachusetts Institute of Technology, 314-315

Mathematica software package, 284-285, 373, 389-390

Mathematical thinking, development of, 69, 152, 170-174, 194-196

Mathematician’s Apology, A (Hardy), 227, 359

Matrices

arithmetic of, 272, 273

characteristic polynomial of, 272-273, 274, 276, 282

defined, 195

eigenvalues of, 273, 274, 276, 283, 284, 285, 295

inventor of, 225, 277

lead diagonal, 272

trace of, 273, 274, 283

Maugham, Somerset, 29

Maupertuis, Pierre de, 370

Maxwell, James Clerk, 226

Measure theory, 88

Measuring logic vs. counting logic, 82-86, 90-91

Median, 387

Mehta, Madan Lal, 288, 386

Meissel, Ernst, 153-154

Meller, N.A., 258

Mendelssohn, Felix, 94, 95

Mendelssohn, Ottilie, 372

Mendelssohn, Rebecca, 94, 95, 133

Mendès-France, Michel, 389

Mengoli, Pietro, 10, 370

Mertens, Franz, 154

Mertens’s function, 250-251, 322

Mittag-Leffler, Gösta, 92, 372

M(k). See Mertens’s function

Möbius, August Ferdinand, 249, 381, 382

Möbius inversion, 302-303

Möbius mu function, 245-251, 302-303, 322, 343-344, 345, 362

Möbius strip, 381-382

Mod. 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, 180-182, 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

μ(n). See Möbius mu function

μ(σ). See Lindelöf mu function

N

Nachlass, 257, 383

Napoleon, 49-50

Napoleonic Wars, 19-20, 24, 49-50, 61, 92, 118

National Science Foundation, 353

Natural numbers, 170, 171, 172, 173, 174

Negative numbers, 65, 70, 80-81, 176

Neuenschwander, Erwin, 24, 365

Neumann, John von, 164, 378, 391

Newman, James R., 128

Newson, Mary Winston, 189

Newton, Sir Isaac, 88, 149, 225, 304, 313

Ng, E.K.-SW., 391

Nicholas I, Emperor of Russia, 122

Noether, Emmy, 186, 231

Non-deductive logic, 325-326

Number theory, 18, 86-87, 96, 97-98, 114, 151, 153, 156, 225, 231, 313, 371-372

Numbers

bogus history of, 174-175

counting vs. measuring, 83-86

historical knowledge of, 174-175, 195

O

O. See Big oh.

Odlyzko, Andrew, 161, 218, 257, 259-261, 263-264, 278, 291, 292, 294, 326, 352, 356, 357-358, 361; pl. 5

Oklahoma State University, 353

Olbers, Heinrich, 90

On the Concept of Number (Kronecker), 185

Open form, 64

Operator theory, 265, 271-279, 351.

See also Matrices

Operators, 273-274, 386.

See also Riemann operators

Order of a zero, 385

Oresme, Nicole d’, 9, 88

P

p-adic numbers, 173, 319-320

Pair correlation function, 287, 288, 290-291

Paphnutius, Bishop, 122-123

Paris Academy of Sciences, 58, 160

Particle physics, 198, 280-281, 295

Pascal, Blaise, 371

Patterson, Samuel J., 217, 385

Paul I, Emperor of Russia, 121

Periodic terms, 328, 330-333, 339-340, 341

Perturbation theory, 351

Peter the Great, Emperor of Russia, 56-57, 58; pl. 1

Petsinis, Tom, 369

π, 185

π(N). See Prime counting function

Picard, Emile, 165, 389-390

Picard’s Theorem, 389-390

Picquart, George, 162, 164

Pietists, 187

Planck’s constant, 316

PNT. See Prime Number Theorem

Poincaré, Henri, 92, 159, 314, 377

Point at infinity, 214

Poisson distribution, 387

Poisson, Siméon-Denis, 92, 93

Pólya, George, 193, 197, 228, 277, 325, 352, 377-378, 380; pl. 7

Polynomial

characteristic of a matrix, 272-273

function, 37, 331-332

zero of, 173

Polytopes, 196, 378

Popular Front, 164

Power functions, derivatives of, 110

Powers

complex, 178, 202-203, 204-205

fractional, 66-67, 68

graphing, 68, 73

irrational, 67

rules, 65-68, 69, 71-72

zero, 65, 66

Prime counting function, 38, 153-154, 160, 297, 298, 299

Prime Number Theorem (PNT)

calculus and, 106-113

Chebyshev and, 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

improved version, 116

log integral function and, 113-117

logarithmic sense, 45-46

proofs, 124-125, 153-155, 159-160, 190, 198, 233-234, 237, 356

Prime numbers

Chebyshev bias, 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 (П), 105

Proper factor, 32

Proper fraction, 171

Psychology of Invention in the Mathematical Field (Hadamard), 165, 359

Pushkin, Alexander, 122

Pythagoras of Samos, 175, 367, 379

Pythagoras’s Theorem, 180

Q

Quantum dynamics, 291-292

Quantum factor, 316

Quantum physics, 313, 315-317

Quasi-Riemann Hypothesis, xiv

R

Ramanujan, Srinivasa, 227-228

Random matrix, 282-287, 386, 403

Random numbers

Gaussian-normal, 283

spacings between, 285-286

Random walk, 250

Rational functions, 268-269, 332

Rational numbers, 171, 172, 173, 175, 179, 319

Real axis, 180

Real line, 178-180

Real numbers, 171, 172, 173, 176, 178, 188

Regular Polytopes (Coxeter), 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, 19, 30-31, 131-132, 134, 135

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, 119-121, 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, 30-31, 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 (née Koch), 31, 362, 363-364

Riemann, Friedrich Bernhard (father), 22-23

Riemann Hypothesis

algebraic thread, 197

analytic number theory and, 86

computational thread, 197

consequences of, 358-360

elementary math, xii-xiii

error term, 190, 234-235, 236-237, 244

geometrically stated, 190

harmonic series and, 12

Hilbert’s eighth problem, 189-190, 244, 324

“Hypothesis” vs. “Conjecture,” “Theorem,” etc., 386

mathematicians’ fascination with, 156-157, 186-187, 188, 189-190, 196-198, 200, 232

physical thread, 197-198, 271-272

presentation at Berlin Academy meeting, ix-x, 31

prize for proof/disproof, xi, 154, 354

prospects for proof/disproof, 354-358

in song, 393-403

Stieltjes’ lost proof, 154, 160, 161

statements of, xi-xii, 77, 137, 191

and zeta function, 136, 137

Riemann, Ida (daughter), 362, 364

Riemann, Ida (sister), 22, 31, 363-364

Riemann integral, 127

Riemann, Marie (sister), 134

Riemann operators, 312, 316-319, 320-321

Riemann-Siegel formula, 256-257, 262, 263-264, 292, 316

Riemann surfaces, 121, 209-210

Riemann, Wilhelm (brother), 134

Riemann zeta function. See Zeta function

Riemann’s Zeta Function (Edwards), 153, 217, 375, 384

Ring, 267-268

Rivat, Joel, 380

Romantic Movement, 30, 92, 118

Rosser, J. Barkley, 258

Rubinstein, Michael, 380

Rule of signs, 42, 367-368

Ruler exercises, 10-15

Russell, Bertrand, 225, 226, 360-361

Russia, intellectual life in, 55-57, 58-59, 60-61, 98, 120-122

Rutherford, Ernest, 280

S

S function, 358, 396

St. Pafnuty of Borovsk, 123

St. Petersburg Academy, 30, 57-58, 122

St. Petersburg University, 122

Sarnak, Peter, 245, 278, 352, 380, 387, 388

Scherrer, Paul, 185

Schilling, Carl David, 364

Schneider, Theodor, 354

Schogt, Philibert, 161

Schönhage, Arnold, 263-264

Schrödinger’s wave equation, 313

Schwartz, Hermann, 363-364

Seattle conference on the Riemann Hypothesis, 257, 352, 355

Selberg, Atle, 125, 198, 288, 352, 358, 374, 376, 384; pl. 3

Semiclassical dynamical system, 316

Sequence

defined, 16

of partial sums, 17

series contrasted, 16-17

Series.

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

Siegel, Carl, 256-257, 263-264, 383; pl. 5

Sieve of Eratosthenes, 100-101, 102-104, 138

Sine function, 147, 332

Skewes, Samuel, 236

Skewes’ number, 236

Snaith, Nina, xiv

Snowflake curve, 381

Society of German Scientists and Physicians, 252

Sommerfeld, Arnold, 256

Sophia Dorothea (mother of Frederick the Great), 60

Sorbonne, 159, 188, 225

Sorcerer’s Apprentice (Dukas), 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 (Σ), 78

“Sweet Betsy from Pike” (tune), 394, 395

Sylvester, James Joseph, 154, 225

T

“Taiye,” 82-83; pl. 8

Teichmüller, Oswald, 255-256, 383

Teichmüller Theory, 383

Telegraph, electric, 120

Tenenbaum, Gérald, 389

Theory of Numbers (Hardy and Wright), 302

Theory of performances, 52

Theory of the Riemann Zeta-function, The (Titchmarsh), 217, 384

Thread, The (Davis), 122

Three-body problem, 314

Time reversal symmetry, 316

Titchmarsh, Edward Charles, 217, 258, 262, 394

Tocqueville, Alexis de, 118

Topology, 18, 121, 209, 374

Trace formula, 321, 388

Transcendental numbers, 174, 185, 354

Trigonometry, 18

Trinity College, Cambridge, 193, 223-224, 225-226, 229, 287, 379, 380

Trinity Hall, Cambridge, 380

Truman, Harry S., 166

Turán, Paul, 238, 239, 378

Turing, Alan, 258, 261-262, 357, 377, 391; pl. 5

Turing machine, 261, 391

Turing Prize, 261

Turing Test, 261

Twiddle principle, 46

Twiddle sign, 45, 368

U

Uncle Petros and Goldbach’s Conjecture (Doxiadis), 90

Universal Computer, The (Davis), 187

Universities, academies distinguished from, 30

University of Bordeaux, 158-159

University of Breslau, 93, 94

University of Bristol, England, 390

University of Cambridge, 259

University of Copenhagen, 228

University of Leipzig, 270

University of Louvain, 161

University of Manchester, 259

University of Marburg, 270

University of Minnesota, 322, 357

University of Wales, Cardiff, 391

University of Washington in Seattle, 352

Upper bound, 235-236

V

Vallée Poussin, Charles de la, x, 153, 155-156, 161, 189, 223, 232, 237, 352, 356, 376; pl. 3

Value plane, 219-221, 335

Victoria, Queen of England, 26

Vienna Academy, 153

“Villikens and his Dinah” (song), 395

Vis viva equation, 313, 315

Volterra, Vito, 92

Vorhauer, Ulrike, 350, 390

W

w plane, 379

Wagon, Stan, 389

Wallace, William, 92

Wave functions, 318

Weber, Heinrich, 29, 119, 257, 366

Weber, Wilhelm, 27, 120, 127, 374

Wedeniwski, Sebastian, 258, 259

Weierstrass, Karl, 135, 164

Weil, André, 270, 325, 385, 395; pl. 6

Weil Conjectures, 270, 355

Wendland, 22, 94

Weyl, Hermann, 170, 255, 385

Whitehead, Alfred North, 225

Whitemore, Hugh, 262

Wigner, Eugene, 282, 387

Wild Numbers, The (Schogt), 161

Wiles, Andrew, 90, 161, 245, 271, 354-355

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

Möbius function and, 250-251

sieve of Eratosthenes and, 102-104, 138

values of, 79-81, 146-147, 263

visualization, 216-218

zeros of, 154, 160, 169, 190-192, 206, 211-212, 217-218, 221-222, 232-233, 234, 259-261, 287-288, 295, 395

ζ(s), 77.

See also Zeta function

Zionism

Dreyfus Affair and, 165

First Congress of, 165

Hadamard, Landau, and, 230

Zola, Émile, 162, 164