370 Index Riemann Hypothesis, 245 Xi function, 230 xi function, 230 canonical factorization of, 240 order of, 240 zeta function, 63 analytic continuation of, 162 critical line, 230 critical strip, 230 functional equation of, 229 local density of zeros of, 309 nontrivial zeros of, 240 trivial zeros of, 230 Riesel, H., 34 Riesz, M., 67 Risager, M. S., 151 Rivat, J., 101 Roth, K. F., 55 Rudnick, Z., 151 Sabbah, C., 223 Saradha, N., 28 sawtooth function, 11 Fourier series of, 210 Sch¨ onhage, A., 30 Schaar, M. Landsberg-Schaar formula, 219 Scheibner, W., 99 Schinzel, A., 101 Schmidt, P. G., 55 Schmidt, W. M., 116, 143 Schur, I., 99, 102 Schur orthogonality theorem, 75 Schur’s Lemma, 74 Schwarz, W., 45 second Frobenius orthogonality thm., 107 second mean value theorem, 35 Selberg, A., 178, 245 Shafarevich, I. R., 116 Shakarchi, R., 216 Shapiro, H. N., 100 Siegel, C. L., 147, 205 Landau-Siegel zeros, 193 Siegel-Walfisz theorem, 204 theorem of, 193 sieve of Eratosthenes, 27 of Eratosthenes-Legendre, 28, 38 singular series, 123, 126, 128 smoothed Chebyshev function, 173, 307 Soundararajan, K., 102 Spilker, J., 45 square root cancellation, 90 squarefree integers counting function of, 42 density of, 43 in short intervals, 44 squarefree kernel, 16 Srinivas, K., 105 Stein, E. M., 216 Stieltjes, T. J., 99 Stirling’s formula, 12, 309 subrepresentation, 72 sum-of-divisors function, 17 Summary, xvi summatory function, 3, 41 of the divisor function, 23 of the Euler phi function, 21 of the logarithm, 3 of the obius function, 50 of the von Mangoldt function, 3 Sylvester, J. J., 29 system of fundamental units, 266 Tao, T. C.-S., 328 Tate, J. T., 247 Tatuzawa, T., 178 Tenenbaum, G., 26, 43, 45, 179 Terras, A., 243 test function, 315 theorem Bertrand’s Postulate, 6 Bohr-Landau, 323 Cauchy-Davenport-Chowla, 141 convolution, 107 Dirichlet approximation, 138 Dirichlet’s, 83 Erd¨ os-Kac, 48 first mean value, 35 Fourier inversion, 79 Frobenius orthogonality first, 77 second, 107 Frobenius reciprocity, 295 Green-Tao, 328 Hilbert-Waring, 143 Ikehara, 281 Ingham zero density, 323 Landau, 157, 190 Landau Prime Ideal, 281 Law of Quadratic Reciprocity, 222 Maschke complete reducibility, 73
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