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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