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II. Clarkson, James A. (1966) On the series of prime reciprocals. Proc. Amer. Math. Soc., I7: 541; M R 32, # 5573. 12. Davenport, Harold ( 1967) Multiplicative Number Theory. Lectures in Advanced Mathematics, No. I. Chicago: Markham Publishing Co. 13. Dickson, Leonard Eugene (1919) History of the Theory of Numbers. (3 volumes). Washington, D. C.: Carnegie Institution of Washington. Reprinted by Chelsea Publishing Co., New York, 1966. 14. Dickson, Leonard Eugene (1930) Studies in the Theory of Numbers. Chicago: The University of Chicago Press. 15. Dirichlet, P. G. Lejeune (1837) Beweis des Satzes, dass jede unbegrenzte arith• metische Progression, deren erstes Glied und Differenz ganze Zahlen ohne geme• inschlaftichen Factor sind, unendliche viele Primzahlen enthalt. Abhand. Ak. Wiss. Berlin: 45-81. [Werke, I: 315-342.] 16. Dirichlet, P. G. Lejeune (1840) Ueber eine Eigenschaft der quadratischen Formen. Bericht Ak. Wiss. Berlin: 49-52. [Werke, I: 497-502.] 17. Edwards, H. M. (1974) Riemann's Zeta Function. New York and London: Academic Press. 18. Ellison, W. J. (1971) Waring's problem. Amer. Math. Monthly, 78: 10-36. 19. Erdos, Paul (1949) On a new method in elementary number theory which leads to an elementary proof of the theorem. Proc. Nat. Acad. Sci. U.S.A., 35: 374-384: MR 10, 595. 20. Euler, Leonhard (1737) Variae observationes circa series infinitas. Commentarii Academiae Scientiarum Jmperialis Petropolitanae, Y: 160-188. [Opera Omnia (1), /4; 216-244.] 21. Euler, Leonhard (1748) lntroductio in Analysin Jnfinitorum, Vol. I. Lausanne: Bousquet. [Opera Omnia (1), 8.] 22. Franklin, F. (1881) Sur lc developpement du produit infini (I - x)(l - x 2 ) (I - x 3 )(1 - x4 ). • .. Comptes Rendus Acad. Sci. (Paris), 92: 448-450. 23. Gauss, C. F. (1801) Disquisitiones Arithmeticae. Lipsiae. [English translation: Arthur A. Clarke (1966) New Haven: Yale University Press. 24. Gauss, C. F. (1849) Letter to Encke, dated 24 December. [Werke, Vol. II, 444-447.] 25. Gerstenhaber, Murray (1963) The 152nd proof of the law of . Amer. Math. Monthly, 70: 397--398; MR 27, # 100. 26. Goldbach, C. (1742) Letter to Euler, dated 7 June. 27. Grosswald, Emil (1966) Topics from the Theory of Numbers. New York: The Macmillan Co. 28. Hadamard, J. ( 1896) Sur Ia distribution des zeros de Ia fonction ((s) et ses conse• quences arithmetiques. Bull. Soc. Math. France, 24: 199-220. 29. Hagis, Peter, Jr. (1973) A lower bound for the set of odd perfect numbers. Math. Comp., 27: 951-953; MR 48, #3854. 30. Hardy, G. H. (1940) Ramanujan. Twelve Lectures on Subjects Suyyested by His Life and Work. Cambridge: The University Press. 31. Hardy, G. H. and Wright, E. M. (1960) An Introduction to the Theory of Numbers, 4th ed. Oxford: Clarendon Press. 32. Herner, Ove (1954) Notes on the Diophantine equation y 2 - k = x3. Ark. Mat., 3: 67-77; MR 15,776. 33. Ingham, A. E. (1932) The Distribution of Prime Numbers. Cambridge Tracts in Mathematics and Mathematical Physics, No. 30. Cambridge: The University Press.

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332 Index of Special Symbols

din. d ,( n, divides (does not divide), 14

(a, b), (a 1, ••• , an), greatest common divisor (gcd),15, 21 [a, b], least common multiple (lcm), 22 Jl(n), Mobius function, 24 cp(n), Euler totient, 25 J * g, , 29 /(n) = GJ. identity function, 30 f-1, Dirichlet inverse, 30 u(n) = 1, unit function, 31 A(n), Mangoldt function, 32 A.(n), , 37 a:x(n), a(n), d(n), divisor functions, 38 IX ° F, generalized convolution, 39 fp(x), Bell series off modulo p, 43 f'(n) = f(n)log n, derivative, 45 C, Euler's constant, 53 0, big oh notation, 53 asymptotic equality, 53 -' ((s), Riemann zeta function, 55 n(x), number of primes 5: x, 74 r/l(x), Chebyshev r/1-function, 75

333 Index of special symbols

.9(x ), Chebyshev .9-function, 75 M(x), partial sums of Mobius function, 91 o, little oh notation, 94 a= b (mod m), congruence, 106 a, residue class a modulo m, 109 a,' reciprocal of a modulo m, 111 x(n), Dirichlet character, 138 L(l, x), sum of series L x(n)jn, 141 L'(l, x), sum of series - L x(n)log njn, 148 ck(n), Ramanujan sum, 160 G(n, x), associated with x. 165 G(k; n), quadratic Gauss sum, 177 nRp, nRp, quadratic residue (nonresidue) mod p, 178 (nip), , 179 (niP), Jacobi symbol, 188 expm(a), exponent of a modulo m, 204 ind9 a, index of a to base g, 213 L(s, x), Dirichlet L-function, 224 abscissa of absolute convergence, 225 abscissa of convergence, 233 r(s), gamma function, 250 ((s, a), Hurwitz zeta function, 251 F(x, s), periodic zeta function, 257 Bn(x), Bn, Bernoulli polynomials, (numbers), 264 Bn(x), periodic Bernoulli functions, 267 p(n), partition function, 307 w(n), w(- n), pentagonal numbers, 311

334 Chebyshev, Pafnuti Liwowich, 9, 75. 104 Index Chebyshev function 9(x), 75 Chebyshev function 1/!(x), 75 Chen, Jing-run, II, 304, 329 Chinese remainder theorem, 117 Clarkson, James A.. 18. 330 Classes of residues, I09 Clausen, Thomas, 275 Common divisor, 14 Commutative group, 129 Complete residue system, 110 Completely , 33 Conductor of a character, 171 Congruence. 106 Convolution. Dirichlet. 29 generalized, 39 van der Corput, 1. G., 59 Critical line, 293 Critical strip, 293 Cross-classification principle, 123 Cyclic group, 131

Darling, H. B. C., 324 Abel, Niels Henrik, 77 Davenport, Harold, 330 Abelian group, 129 Decomposition property of reduced residue Abel's identity, 77 systems, 125 Abscissa, of absolute convergence, 225 Derivative of arithmetical functions, 45 of convergence. 233 Dickson, Leonard Eugene, 12, 330 Additive number theory, 304 Diophantine equation, 5, 190 Algorithm, division, 19 Dirichlet, Peter Gustav Lejeune, 5, 7, 29, 53, Euclidean, 20 138. 146. 224 Analytic continuation, of Dirichlet Dirichlet. character x!n). 138 L-functions, 255 convolution (product). 29 of Hurwitz zeta function, 254 divisor problem, 59 of Riemann zeta function, 255 estimate for d(n), 53, 57 Apostol, Tom M., 329 inverse, 30 Arithmetic, fundamental theorem of, 17 L-function, 224 Arithmetical function, 24 series, 224 Asymptotic equality, 53 theorem on primes in arithmetic Average (arithmetic mean) of an progressions, 7, 146, 154 arithmetical function. 52 Disquisitiones arithmeticae, 5 Average order, 52 Divisibility, 14 Ayoub, Raymond G., 329 Division algorithm, 19 Divisor, 14 Bell, Eric Temple, 29, 42, 329 u.(n), 38 Bell series. 43. 231 Bernoulli, numbers, 265 periodic functions, 267 Edwards, H. M., 330 polynomials, 264 Elementary proof of prime number Binomial congruence, 214 theorem. 9, 98 Borozdkin, K. G., 10, 329 Ellison, W. J., 306, 330 Erdos, Paul, 9, 330 Buh~tab, A. A., II, 329 Euclidean algorithm, 20 Cauchy, Augustin-Louis, 44, 144, 198 Euclid's lemma, 16 Cauchy pn1duct, 44 Euler, Leonhard, 4, 5, 7, 9, 19, 25, 53, 54, 113, Chandrasekharan, Komaravolu, 329 180, 185, 230, 308, 312, 315 Character, Dirichlet, 138 Euler-Fermat theorem, 113 of an abelian group, 133 Euler product, 230 primitive, 168 Euler's constant, 5.3, 250 principal, 138 Euler's criterion, 180

335 Index

Euler's pentagonal-number theorem, 312 Group. definition of. 129 Euler's summation formula, 54 abelian. 129 Euler totient function cp(n l. 25 cyclic, 131 Evaluation, of( -lip), 181 Group character. 133 of(21pl, 181 of ((- n, u), 264 Hadamard. Jacques, 9. 74, 330 of ((2n), 266 Hagis, Peter, Jr.. 5, 330 of L(O, xl. 268 Half-plane. of absolute convergence. 225 Exponent of u modulo m, 204 of convergence. 233 Exponential congruence, 215 Hardy, Godfrey Harold. 59, 293, 305, 330 Herner, Ove. 330 Factor, 14 Hilbert. David, 293 Fermat, Pierre de, 5, 7, II, 113, 114 Hurwitz. Adolf. 249 Fermat conjecture. II Hurwitz formula for ((s, a). 257 Fermat prime, 7, Hurwitz zeta function ((s. a), 249, 251, 253, Fermat theorem (little), 114 255 Finite Fourier expansion, 160 Formal power series, 41 Identity clement. 30. 129 Fourier coefficient, 160 Identity function l(n ), 30 Franklin, Fabian, 313,330 Index. 213 Function, arithmetical, 24 Index calculus, 214 Bernoulli periodic B.(x), 267 Indices (table of).216. 217 Chebyshev 9(x), 75 Induced modulus. 167 Chebyshev 1/J(x), 75 Induction, principle of, 13 completely multiplicative, 33 Infinitude of primes, 16. 19 Dirichlet L(s, xl. 224 Inequalities, for l((s, u)l. 270 divisor d(n), a,(n), 38 for i((s) 1. 270, 287. 291 Euler totient cp(n), 25 for I L(s, xl 1. 272 Hurwitz zeta ((s, u), 249 for rr(n), 82 Liouville ).(n), 37 for nth primer•. 84 Mangoldt A(n), 32 for d(n), 294 Mobius Jl(n), 24 for cp(n ). 298 periodic zeta F(x, s), 257 for p(n). 316,318 Riemann zeta ((s), 9, 249 Ingham, A. E., 330 K(n), 247 Inverse, Dirichlet, 30 M(x),91 of completely multiplicative function. 36 v(n), 247 Inversion formula, Mobius. 32 rr(x), 8, 74 generalized, 40 1/1 1(x), 278 lseki, Kaneshiro. 99. 332 Functional equation, for r(s), 250 for L(s, xl, 263 for ((s), 259 Jacobi, Carl Gustav Jacob, 187. 305. 313, 319 for ((s, h/k), 261 Jacobi symbol (niP), 188 Jacobi triple product identity, 319 Fundamental theorem of arithmetic, 17 Jordan totient J ,(n), 48 Gamma function. 250 Gauss, Carl Friedrich, 5, 7, 8, I 06. 165, 177, Kloosterman, H. D., 176 182, 185, 306, 326 Kloosterman sum, 176 Gauss sum, associated with x. 165 Kolberg, Oddmund, 324 quadratic, 177. 306 Kolesnik, G. A., 59 Gauss' lemma, 182 Kruyswijk, D., 324, 327, 331 Gauss' triangular-number theorem, 326 , 308 Lagrange, Joseph Louis, 5, 115, 144, 158 Geometric sum, 157, 158 Lagrange interpolation formula, 158 Gerstenhaber, Murray, 186, 330 Lagrange's theorem on polynomial Goldbach, C., 6, 9, 304 congruences, 115 Goldbach conjecture, 9, 304 Landau, Edmund, 59, 237, 248, 301, 331 Greatest common divisor, 15, 20, 21 Landau's theorem, 237, 248 Greatest integer symbol, 8, 25, 54, 72 Lattice points, 57, 62 Gordon, Basil, 104 visibility of, 62

336 Index

Law of quadratic reciprocity. 185. 189. Periodic arithmetical function, 157 193. 200 Periodic zeta function. 257 Least common multiple, 22 Polya. G .. 173.299 Leech, John. 10, 331 Polya inequality for character sums. 173. Legendre, Adrien-Marie, 5. 6 7, 179, 185, 331 176.299 Legendre's identity. 67 Polygonal numbers. 2. 5 Legendre symbol (nl p). 179 Polynomial congruence. 115 Lehmer. Derrick Henry. 6, 293, 316.331 Prachar, Karl. 331 Lehmer. Derrick Norman. 6. 331 Prime number theorem, 9, 65, 74, 79, 92, 94, Lemma of Gauss, 182 98. 278. 289 LeVeque, William Judson, 2, 191. 331 Primes. 2. 16. Levinson, Norman, 293, 331 contained in a factorial. 6 7 L-function L(s. xl. 224 m arithmetic progressions, 7, 146, 154 Linear congruence. Ill, 112. 114, 214 Fermat. 7 van Lint, Jacobus Hendricus, 318, 331 infinitude of. 16. 19 Liouville. Joseph. 37 Mersenne. 4 Liouville function J.(n),)7 Primitive character, 168 Little Fermat theorem, 114 Primitive root, 204 Littlewood, John Edensor, 10, 305, 331 Principal character, 134, 138 Logarithmic integral Li(x), 102 Product, of arithmetical functions, 29 Lucas, Edouard, 275 of Dirichlet series, 228 Pythagorean triple. 2 MacMahon, Percy A .. 316 von Mangoldt. H .. 32 Quadratic, congruence. 178 von Mangold! function i\(n), 32 Gauss sum. 177. 195 Mean value formulas for Dirichlet series, 240 nonresidue, 178 Mersenne. P .. 4 reciprocity law. 185, 189, 193, 200 Mersenne numbers, 4 residue. 178 Mertens, Franz. 91 Mertens' conjecture. 91 Rademacher. Hans. 316, 324.331 Mills, W. H., 331 Ramanujan. Srinivasa. 160, 305. 324, 328 Mobius. Augustus Ferdinand, 24 Ramanujan partition identities, 324, 328 Mobius function p(n), 24 Ramanujan sum, 160, 176 Mobius function Pk(n) of order k, 50 Reciprocity law, for Jacobi symbols, 189 Mobius inversion formula, 32 for Legendre symbols, 185, 193, 200 product form. 47 for quadratic Gauss sums, 200 generalized. 40 Reduced fraction. 21 Mordell. Louis Joel, 305, 306, 324 Relatively prime, 15, 21 Multiplication, Dirichlet. 29 in pairs, 21 of residue classes, 138 Renyi. Alfred, 10, 332 Multiplicative function, 33 Residue. quadratic. 178 Multiplicative number theory, 304 Residue class, 109 Residue system, complete, 110 Nevanlinna, Veikko, 331 reduced, 113, 125 Niven, Ivan. 331 Riemann, Georg Friedrich Bernhard, 9, 225 Nonresidue, 178 293, 332 Number-theoretic function, 24 Riemann hypothesis, 293, 301 Riemann-Lebesgue lemma, 279 0, big oh notation, 53 Riemann-Stieltjes integral, 77 o, little oh notation, 94 Riemann zeta function, 249 Order of a group, 130 Ring of formal power series, 42 Orthogonality relation, for group characters, Robinson, Raphael M., 7, 332 137 Rosser, 1. Barkley, 293, 332 for Dirichlet characters, 140 Schnirelmann, L., 10, 332 Partition, 304 Selberg, Atle, 9, 46, 100, 332 Partition function p(n), 307 Selberg asymptotic formula, 100, 103, 104 Pentagonal numbers, 2, 5, 311 Selberg identity, 46 Pentagonal-number theorem, 312 Separable Gauss sums, 165, 171 Perfect numbers, 4 Shanks, Daniel, 312, 326. 332

337 Index

Shapiro, Harold N., 85, 146, 332 Vallee- Poussin, C. J. de Ia, 9, 74, 332 Shapiro's Tauberian theorem, 85 Vinogradov, A. 1., II, 332 Shen, Mok-Kong, 9, 332 Vinogradov, I. M., 10, 304, 332 Sierpinski, Waclaw, II, 148, 332 Visibility of lattice points, 62 Smallest primitive roots (table of), 213 Voronoi, G., 59 Squarefree, 21 von Staudt, Karl Georg Christian, 275 Walfisz, Arnold, 91. 332 Subgroup, 130 Waring, Edward, 306 Summation formula of Euler, 54 Waring's problem, 306 Symbol, Jacobi In I P), 188 Williams, H. C.. 6, 332 Legendre (nl p), 179 Wilson, John, 116 System of residues, complete, 110 Wilson's theorem, 116 reduced, 113 Wrathall, Claude P., 7, 332 Wright, E. M., 330 Wolstenholme's theorem, 116 Tatuzawa, Tikao, 99, 332 Tauberian theorem, 85 Yin, Wen-lin, 10, 332 Theta function, 306 Titchmarsh, Edward Charles, 301, 332 Zarnke, C. R., 6, 332 Totient function rp(n), 25 Zero-free regions of ((s), 291, 292 Triangular numbers, 2, 326 Zeros, of L-function L(s, 274 Triangular-number theorem, 326 x). of Riemann's zeta function Trivial zeros of ((s), 259 ((s), 259, 274, 293 Twin primes, 6 Zeta function, Hurwitz, 249 periodic, 257 Unique factorization theorem, 17 Riemann. 9, 249 Uspensky, J. V., 332 Zuckerman, Herbert S., 324, 331

338 Undergraduate Texts in Mathematics

(continued from page ii) Lang: Linear Algebra. Third edition. Frazier: An Introduction to Wavelets Lang: Short Calculus: The Original Through Linear Algebra Edition of"A First Course in Gamelin: Complex Analysis. Calculus." Gordon: Discrete Probability. Lang: Undergraduate Algebra. Second Hairer/Wanner: Analysis by Its History. edition. Readings in Mathematics. Lang: Undergraduate Algebra. Third Halmos: Finite-Dimensional Vector edition Spaces. Second edition. Lang: Undergraduate Analysis. Halmos: Naive Set Theory. Laubenbacher/Pengelley: Mathematical Hiimmerlin/Hoffmann: Numerical Expeditions. Mathematics. Lax/Burstein!Lax: Calculus with Readings in Mathematics. Applications and Computing. Harris/Hirst/Mossinghoff: Volume I. Combinatorics and Graph Theory. LeCuyer: College Mathematics with Hartshorne: Geometry: Euclid and APL. Beyond. Lidi/Pilz: Applied Abstract Algebra. Hijab: Introduction to Calculus and Second edition. Classical Analjsis. Logan: Applied Partial Differential Hilton/Holton/Pedersen: Mathematical Equations, Second edition. Reflections: In a Room with Many Lovasz!PelikanN esztergom bi: Discrete Mirrors. Mathematics. Hilton/Holton!Pedersen: Mathematical Macki-Strauss: Introduction to Optimal Vistas: From a Room with Many Control Theory. Windows. Malitz: Introduction to Mathematical looss/Joseph: Elementary Stability Logic. and Bifurcation Theory. Second Marsden!Weinstein: Calculus I, II, III. edition. Second edition. Irving: Integers, Polynomials, and Rings: Martin: Countitur The Art of A Course in Algebra Enumerative Combinatories. Isaac: The Pleasures of Probability. Martin: The Foundations of Geometry Readings in Mathematics. and the Non-Euclidean Plane. James: Topological and Uniform Martin: Geometric Constructions. Spaces. Martin: Transformation Geometry: An Jiinich: Linear Algebra. Introduction to Symmetry. Jiinich: Topology. Millman/Parker: Geometry: A Metric Jiinich: Vector Analysis. Approach with Models. Second Kemeny/Snell: Finite Markov Chains. edition. Kinsey: Topology of Surfaces. Moschovakls: Notes on Set Theory. Klambauer: Aspects of Calculus. Owen: A First Course in the Lang: A First Course in Calculus. Mathematical Foundations of Fifth edition. Thermodynamics. Lang: Calculus of Several Variables. Palka: An Introduction to Complex Third edition. Function Theory. Lang: Introduction to Linear Algebra. Pedrick: A First Course in Analysis. Second edition. Peressini!Sullivan!Uhl: The Mathematics of Nonlinear Programming. Undergraduate Texts in Mathematics

Prenowitz/Jantosciak: Join Geometries. Priestley: Calculus: A Liberal Art. Simmonds: A Brief on Tensor Analysis. Second edition. Second edition. Protter/Morrey: A First Course in Real Singer: Geometry: Plane and Fancy. Analysis. Second edition. Singer/Thorpe: Lecture Notes on Protter/Morrey: Intermediate Calculus. Elementary Topology and Second edition. Geometry. Pugh: Real Mathematical Analysis. Smith: Linear Algebra. Third edition. Roman: An Introduction to Coding and Smith: Primer of Modem Analysis. Information Theory. Second edition. Roman: Introduction to the Mathematics Stanton/White: Constructive of Finance: From Risk Management to Combinatorics. Options Pricing. Stillwell: Elements of Algebra: Geometry, Ross: Differential Equations: An Numbers, Equations. Introduction with Mathematica®. Stillwell: Elements of Number Theory. Second edition. Stillwell: Mathematics and Its History. Ross: Elementary Analysis: The Theory Second edition. of Calculus. Stillwell: Numbers and Geometry. Samuel: Projective Geometry. Readings in Mathematics. Readings in Mathematics. Strayer: Linear Programming and Its Saxe: Beginning Functional Analysis Applications. Scharlau/Opolka: From Fermat to Toth: Glimpses of Algebra and Geometry. Minkowski. Second Edition. Schiff: The Laplace Transform: Theory Readings in Mathematics. and Applications. Troutman: Variational Calculus and Sethuraman: Rings, Fields, and Vector Optimal Control. Second edition. Spaces: An Approach to Geometric Valenza: Linear Algebra: An Introduction Constructability. to Abstract Mathematics. Sigler: Algebra. Whyburn/Duda: Dynamic Topology. Silvermanrrate: Rational Points on Wilson: Much Ado About Calculus. Elliptic Curves.