crc standard mathematical tables and formulae

xiii Chapter 1 Numbers and Elementary Mathematics.. Chapter 1Numbers and Elementary Mathematics 1.1 PROOFS WITHOUT WORDS.. NUMBERS AND ELEMENTARY MATHEMATICS1.4.9 Prime numbers of specia

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Table of Contents

Preface xi

Editorial Board xiii Chapter 1 Numbers and Elementary Mathematics 1

1.1 Proofs without words 3

1.2 Constants 5

1.3 Special numbers 12

1.4 Number theory 24

1.5 Series and products 42

Chapter 2 Algebra 61

2.1 Elementary algebra 63

2.2 Polynomials 67

2.3 Vector algebra 72

2.4 Linear and matrix algebra 77

2.5 Abstract algebra 99

Chapter 3 Discrete Mathematics 127

3.1 Set theory 129

3.2 Combinatorics 133

3.3 Graphs 144

3.4 Combinatorial design theory 164

3.5 Difference equations 177

Chapter 4 Geometry 185

4.1 Euclidean geometry 187

4.2 Coordinate systems in the plane 188

4.3 Plane symmetries or isometries 194

4.4 Other transformations of the plane 201

4.5 Lines 203

4.6 Polygons 205

4.7 Surfaces of revolution: the torus 213

4.8 Quadrics 213

4.9 Spherical geometry and trigonometry 218

4.10 Conics 222

4.11 Special plane curves 233

4.12 Coordinate systems in space 242

4.13 Space symmetries or isometries 245

4.14 Other transformations of space 248

4.15 Direction angles and direction cosines 249

4.16 Planes 250

4.17 Lines in space 251

4.18 Polyhedra 253

4.19 Cylinders 257

4.20 Cones 257

4.21 Differential geometry 259

Chapter 5 Analysis 267

5.1 Differential calculus 269

5.2 Differential forms 279

5.3 Integration 282

5.4 Table of indefinite integrals 294

5.5 Table of definite integrals 330

5.6 Ordinary differential equations 337

5.7 Partial differential equations 349

5.8 Integral equations 358

5.9 Tensor analysis 361

5.10 Orthogonal coordinate systems 370

5.11 Interval analysis 375

5.12 Real analysis 376

5.13 Generalized functions 386

5.14 Complex analysis 388

Chapter 6 Special Functions 399

6.1 Ceiling and floor functions 401

6.2 Exponentiation 401

6.3 Logarithmic functions 402

6.4 Exponential function 403

6.5 Trigonometric functions 404

6.6 Circular functions and planar triangles 412

6.7 Tables of trigonometric functions 416

6.8 Angle conversion 419

6.9 Inverse circular functions 420

6.10 Hyperbolic functions 422

6.11 Inverse hyperbolic functions 426

6.12 Gudermannian function 428

6.13 Orthogonal polynomials 430

6.14 Gamma function 437

6.15 Beta function 441

6.16 Error functions 442

6.17 Fresnel integrals 443

6.18 Sine, cosine, and exponential integrals 445

6.19 Polylogarithms 447

6.20 Hypergeometric functions 448

6.21 Legendre functions 449

6.22 Bessel functions 454

6.23 Elliptic integrals 463

6.24 Jacobian elliptic functions 466

6.25 Clebsch–Gordan coefficients 468

6.26 Integral transforms: Preliminaries 470

6.27 Fourier integral transform 470

6.28 Discrete Fourier transform (DFT) 476

6.29 Fast Fourier transform (FFT) 478

6.30 Multidimensional Fourier transforms 478

6.31 Laplace transform 479

6.32 Hankel transform 483

6.33 Hartley transform 484

6.34 Mellin transform 484

6.35 Hilbert transform 485

6.36 Z-Transform 488

6.37 Tables of transforms 492

Chapter 7 Probability and Statistics 507

7.1 Probability theory 509

7.2 Classical probability problems 519

7.3 Probability distributions 524

7.4 Queuing theory 533

7.5 Markov chains 536

7.6 Random number generation 539

7.7 Control charts and reliability 545

7.8 Statistics 550

7.9 Confidence intervals 558

7.10 Tests of hypotheses 565

7.11 Linear regression 579

7.12 Analysis of variance (ANOVA) 583

7.13 Sample size 590

7.14 Contingency tables 595

7.15 Probability tables 598

Chapter 8 Scientific Computing 615

8.1 Basic numerical analysis 616

8.2 Numerical linear algebra 629

8.3 Numerical integration and differentiation 638

Chapter 9

Mathematical Formulas from the Sciences 659

9.1 Acoustics 661

9.2 Astrophysics 662

9.3 Atmospheric physics 664

9.4 Atomic Physics 665

9.5 Basic mechanics 666

9.6 Beam dynamics 668

9.7 Classical mechanics 669

9.8 Coordinate systems – Astronomical 670

9.9 Coordinate systems – Terrestrial 671

9.10 Earthquake engineering 672

9.11 Electromagnetic Transmission 673

9.12 Electrostatics and magnetism 674

9.13 Electronic circuits 675

9.14 Epidemiology 676

9.15 Finance 677

9.16 Fluid mechanics 678

9.17 Fuzzy logic 679

9.18 Human body 680

9.19 Image processing matrices 681

9.20 Macroeconomics 682

9.21 Modeling physical systems 683

9.22 Optics 684

9.23 Population genetics 685

9.24 Quantum mechanics 686

9.25 Quaternions 688

9.26 Relativistic mechanics 689

9.27 Solid mechanics 690

9.28 Statistical mechanics 691

9.29 Thermodynamics 692

Chapter 10 Miscellaneous 693

10.1 Calendar computations 695

10.2 Cellular automata 696

10.3 Communication theory 697

10.4 Control theory 702

10.5 Computer languages 704

10.6 Cryptography 705

10.7 Discrete dynamical systems and chaos 706

10.8 Electronic resources 709

10.9 Elliptic curves 711

10.10 Financial formulas 714

10.11 Game theory 719

10.12 Knot theory 722

10.13 Lattices 724

10.14 Moments of inertia 726

10.15 Music 727

10.16 Operations research 729

10.17 Recreational mathematics 741

10.18 Risk analysis and decision rules 742

10.19 Signal processing 744

10.20 Symbolic logic 750

10.21 Units 753

10.22 Voting power 760

10.23 Greek alphabet 762

10.24 Braille code 762

10.25 Morse code 762

List of References 763

List of Figures 767

List of Notations 769

Index 777

It has long been the established policy of CRC Press to publish, in handbook form,the most up-to-date, authoritative, logically arranged, and readily usable referencematerial available

Just as pocket calculators obviated the need for tables of square roots andtrigonometric functions; the internet has made many other tables and formulas un-

necessary Prior to the preparation of this 32 nd Edition of the CRC Standard

Mathe-matical Tables and Formulae, the content has been reconsidered The criteria lished for inclusion in this edition are:

estab-• information that is immediately useful as a reference (e.g., names of powers of

• information that cannot be found on the internet due to the difficulty of entering

a query (e.g., integral tables);

• illustrations of how mathematical information is interpreted

Using these criteria, the previous edition has been carefully analyzed by ers from mathematics, engineering, and the sciences As a result, numerous changeshave been made in several sections, and several new areas were added These im-provements include:

practition-• There is a new chapter entitled “Mathematical Formulas from the Sciences.” Itcontains, in concise form, the most important formulas from a variety of fields(including: acoustics, astrophysics, ); a total of 26 topics

• New material on contingency tables, estimators, process capability, runs test,and sample sizes has been added to the statistics chapter

• New material on cellular automata, knot theory, music, quaternions, and nal trigonometry has been added

ratio-• In many places, tables have been updated and streamlined For example, theprime number table now only goes to 8,000 Also, many of the tables in thesection on financial computations have been updated (while the examples illus-trating those tables remained)

Of course, the same successful format which has characterized earlier editions of the

Handbookhas been retained, while its presentation has been updated and made moreconsistent from page to page Material is presented in a multi-sectional format, witheach section containing a valuable collection of fundamental reference material—tabular and expository

xi

In line with the established policy of CRC Press, the Handbook will be updated

in as current and timely a manner as is possible Suggestions for the inclusion of newmaterial in subsequent editions and comments regarding the present edition are wel-comed The home page for this book, which will include errata, will be maintained

athttp://smtf.mathtable.com

This new edition of the Handbook will continue to support the needs of

practi-tioners of mathematics in the mathematical and scientific fields, as it has for over 80years Even as the internet becomes more ubiquitous, it is this editor’s opinion thatthe new edition will continue to be a valued reference

Updating this edition and making it a useful tool has been exciting It would not havebeen possible without the loving support of my family, Janet Taylor and Kent TaylorZwillinger

Daniel Zwillinger

ZwillingerBooks@gmail.com

Daniel Zwillinger

Rensselaer Polytechnic Institute

Troy, New York

Editorial Advisory Board

AT&T Bell Labs

Murray Hill, New Jersey

Chapter 1

Numbers and Elementary Mathematics

1.1 PROOFS WITHOUT WORDS 3

1.2 CONSTANTS 5

1.2.1 Binary prefixes . 5

1.2.2 Decimal multiples and prefixes . 6

1.2.3 Interpretations of powers of 10 . 6

1.2.4 Roman numerals . 7

1.2.5 Types of numbers . 8

1.2.6 DeMoivre’s theorem . 9

1.2.7 Representation of numbers . 9

1.2.8 Symmetric base three representation . 9

1.2.9 Hexadecimal addition and subtraction table . 10

1.2.10 Hexadecimal multiplication table . 10

1.2.11 Hexadecimal–decimal fraction conversion . 11

1.3 SPECIAL NUMBERS 12

1.3.1 Powers of 2 . 12

1.3.2 Powers of 16 in decimal scale . 13

1.3.3 Powers of 10 in hexadecimal scale . 13

1.3.4 Special constants . 14

1.3.5 Factorials . 16

1.3.6 Bernoulli polynomials and numbers . 17

1.3.7 Euler polynomials and numbers . 18

1.3.8 Fibonacci numbers . 18

1.3.9 Sums of powers of integers . 19

1.3.10 Negative integer powers . 20

1.3.11 Integer sequences . 21

1.3.12 p-adic Numbers . 23

1.3.13 de Bruijn sequences . 23

1.4 NUMBER THEORY 24

1.4.1 Congruences . 24

1.4.2 Chinese remainder theorem . 25

1.4.3 Continued fractions . 26

1.4.4 Diophantine equations . 28

1.4.5 Greatest common divisor . 31

1.4.6 Least common multiple . 31

1.4.7 M¨obius function . 32

1.4.8 Prime numbers . 33

1

Openmirrors.com

2 CHAPTER 1 NUMBERS AND ELEMENTARY MATHEMATICS

1.4.9 Prime numbers of special forms . 35

1.4.10 Prime numbers less than 8,000 . 38

1.4.11 Factorization table . 40

1.4.12 Euler totient function . 41

1.5 SERIES AND PRODUCTS 42

1.5.1 Definitions . 42

1.5.2 General properties . 42

1.5.3 Convergence tests . 44

1.5.4 Types of series . 45

1.5.5 Fourier series . 49

1.5.6 Series expansions of special functions . 54

1.5.7 Summation formulas . 58

1.5.8 Faster convergence: Shanks transformation . 58

1.5.9 Summability methods . 59

1.5.10 Operations with power series . 59

1.5.11 Miscellaneous sums . 59

1.5.12 Infinite products . 60

1.5.13 Infinite products and infinite series . 60

Openmirrors.com

1.1 PROOFS WITHOUT WORDS

—the Chou pei suan ching

(author unknown, circa B.C. 200?)

The Pythagorean Theorem

A Property of the Sequence of Odd Integers (Galileo, 1615)

13

1+35+7

1+3+57+9+11

=

1+3+ +(2n–1) (2n+1)+(2n+3)+ +(4n–1)

13

4 CHAPTER 1 NUMBERS AND ELEMENTARY MATHEMATICS

sin(x + y) = sinxcosy + cosxsiny

cos(x + y) = cosxcosy – sinxsiny

d (a,b) (a,ma + c)

x y

The Arithmetic Mean-Geometric Mean

c d

a

b

c

d a d

Reprinted from “Proofs Without Words: Exercises in Visual Thinking,” byRoger B Nelsen, 1997, MAA, pages: 3, 40, 49, 60, 70, 72, 115, 120 CopyrightThe Mathematical Association of America All rights reserved

Reprinted from “Proofs Without Words II: More Exercises in Visual Thinking,”

by Roger B Nelsen, 2001, MAA, pages 46, 111 Copyright The Mathematical sociation of America All rights reserved

A byte is 8 bits A kibibyte is210= 1024 bytes Other prefixes for power of 2 are:

Factor Prefix Symbol

6 CHAPTER 1 NUMBERS AND ELEMENTARY MATHEMATICS

The prefix names and symbols below are taken from Conference G´en´erale des Poids

et Mesures, 1991 The common names are for the United States

Factor Prefix Symbol Common name

0.000 000 000 001 = 10−12 pico p trillionth

0.000 000 000 000 001 = 10−15 femto f quadrillionth0.000 000 000 000 000 001 = 10−18 atto a quintillionth

10−21 zepto z hexillionth

10−24 yocto y heptillionth

10−15 the radius of the hydrogen nucleus (a proton) in meters

10−11 the likelihood of being dealt 13 top honors in bridge

10−10 the radius of a hydrogen atom in meters

10−9 the number of seconds it takes light to travel one foot

10−6 the likelihood of being dealt a royal flush in poker

100 the density of water is 1 gram per milliliter

101 the number of fingers that people have

102 the number of stable elements in the periodic table

105 the number of hairs on a human scalp

106 the number of words in the English language

107 the number of seconds in a year

108 the speed of light in meters per second

109 the number of heartbeats in a lifetime for most mammals

1010 the number of people on the earth

1015 the surface area of the earth in square meters

1016 the age of the universe in seconds

1018 the volume of water in the earth’s oceans in cubic meters

1019 the number of possible positions of Rubik’s cube

1021 the volume of the earth in cubic meters

1024 the number of grains of sand in the Sahara desert

1028 the mass of the earth in grams

1033 the mass of the solar system in grams

1050 the number of atoms in the earth

1078 the volume of the universe in cubic meters

(Note: these numbers have been rounded to the nearest power of ten.)

The major symbols in Roman numerals are I= 1, V = 5, X = 10, L = 50, C = 100,

D= 500, and M = 1,000 The rules for constructing Roman numerals are:

1 A symbol following one of equal or greater value adds its value (For example,

II= 2, XI = 11, and DV = 505.)

2 A symbol following one of lesser value has the lesser value subtracted fromthe larger value An I is only allowed to precede a V or an X, an X is onlyallowed to precede an L or a C, and a C is only allowed to precede a D or

an M (For example IV= 4, IX = 9, and XL = 40.)

3 When a symbol stands between two of greater value, its value is subtractedfrom the second and the result is added to the first (For example, XIV=10+(5−1) = 14, CIX= 100+(10−1) = 109, DXL= 500+(50−10) = 540.)

4 When two ways exist for representing a number, the one in which the symbol

of larger value occurs earlier in the string is preferred (For example, 14 isrepresented as XIV, not as VIX.)

8 CHAPTER 1 NUMBERS AND ELEMENTARY MATHEMATICS

1 Natural numbers The set of natural numbers,{0, 1, 2, }, is customarilydenoted by N Many authors do not consider0 to be a natural number

2 Integers The set of integers,{0, ±1, ±2, }, is customarily denoted by Z

3 Rational numbers The set of rational numbers,{pq | p, q ∈ Z, q 6= 0}, iscustomarily denoted by Q

(a) Two fractions pq andrsare equal if and only ifps = qr

(b) Addition of fractions is defined by pq +r

s =ps+qrqs (c) Multiplication of fractions is defined bypq ·r

s =prqs

4 Real numbers Real numbers are defined to be converging sequences of

rational numbers or as decimals that might or might not repeat The set of real

numbersis customarily denoted by R

Real numbers can be divided into two subsets One subset, the algebraic

num-bers, are real numbers which solve a polynomial equation in one variable withinteger coefficients For example;√

2 is an algebraic number because it solvesthe polynomial equationx2

− 2 = 0; and all rational numbers are algebraic

Real numbers that are not algebraic numbers are called transcendental

num-bers Examples of transcendental numbers includeπ and e

5 Definition of infinity The real numbers are extended to include the symbols+∞ and −∞ with the following definitions

(h) −∞ − ∞ = −∞

(i) ∞ · ∞ = ∞(j) −∞ · (−∞) = ∞

6 Complex numbers The set of complex numbers is customarily denoted

by C They are numbers of the forma + bi, where i2 =−1, and a and b arereal numbers

addition (a + bi) + (c + di) (a + c) + i(b + d)

multiplication (a + bi)(c + di) (ac− bd) + (ad + bc)i

a + bi

a

a2+ b2



−

b

a2+ b2

icomplex conjugate z = a + bi z = a− bi

Properties include:z + w = z + w and zw = z w

When writing a number in baseb, the digits used range from 0 to b− 1 If

b > 10, then the digit A stands for 10, B for 11, etc When a base other than 10 isused, it is indicated by a subscript:

as the previous digit Continue this process until a quotient of0 is obtained

remainder of 9; hence, “9” is the last digit Divide 47 by 12, yielding a quotient of 3 and

a remainder of 11 (which we represent with a “B”) Divide 3 by 12 yielding a quotient

of 0 and a remainder of 3 Therefore,57310= 3B912

Converting from baseb to base r can be done by converting to and from base

10 However, it is simple to convert from baseb to base bn For example, to vert1101111012to base 16, group the digits in fours (because 16 is24), yielding

con-1 con-10con-1con-1 con-1con-10con-12, and then convert each group of 4 to base 16 directly, yielding1BD16

In this representation, powers of 3 are added and subtracted to represent numbers.The symbols{↓, 0, ↑} are used for {−1, 0, 1} For example “5” is written as ↑↓↓since5 = 9− 3 − 1 To negate a number, turn its symbol upside down: “−5” iswritten as↓↑↑ Basic arithmetic operations are simple in this representation

10 CHAPTER 1 NUMBERS AND ELEMENTARY MATHEMATICS

A= 10, B = 11, C = 12, D = 13, E = 14, F = 15

Example:6 + 2 = 8; hence 8− 6 = 2 and 8 − 2 = 6

Example:4 + E = 12; hence 12− 4 = E and 12 − E = 4

1.2.11 HEXADECIMAL–DECIMAL FRACTION CONVERSION

Hex Decimal.90 0.5625.91 0.5664.92 0.5703.93 0.5742.94 0.5781.95 0.5820.96 0.5859.97 0.5898.98 0.5938.99 0.5977.9A 0.6016.9B 0.6055.9C 0.6094.9D 0.6133.9E 0.6172.9F 0.6211.A0 0.6250.A1 0.6289.A2 0.6328.A3 0.6367.A4 0.6406.A5 0.6445.A6 0.6484.A7 0.6523.A8 0.6563.A9 0.6602.AA 0.6641.AB 0.6680.AC 0.6719.AD 0.6758.AE 0.6797.AF 0.68365.B0 0.6875.B1 0.6914.B2 0.6953.B3 0.6992.B4 0.7031.B5 0.7070.B6 0.7109.B7 0.7148.B8 0.7188.B9 0.7227.BA 0.7266.BB 0.7305.BC 0.7344.BD 0.7383.BE 0.7422.BF 0.7461

Hex Decimal.C0 0.7500.C1 0.7539.C2 0.7578.C3 0.7617.C4 0.7656.C5 0.7695.C6 0.7734.C7 0.7773.C8 0.7813.C9 0.7852.CA 0.7891.CB 0.7930.CC 0.7969.CD 0.8008.CE 0.8047.CF 0.8086.D0 0.8125.D1 0.8164.D2 0.8203.D3 0.8242.D4 0.8281.D5 0.8320.D6 0.8359.D7 0.8398.D8 0.8438.D9 0.8477.DA 0.8516.DB 0.8555.DC 0.8594.DD 0.8633.DE 0.8672.DF 0.8711.E0 0.8750.E1 0.8789.E2 0.8828.E3 0.8867.E4 0.8906.E5 0.8945.E6 0.8984.E7 0.9023.E8 0.9063.E9 0.9102.EA 0.9141.EB 0.9180.EC 0.9219.ED 0.9258.EE 0.9297.EF 0.9336

Hex Decimal.F0 0.9375.F1 0.9414.F2 0.9453.F3 0.9492.F4 0.9531.F5 0.9570.F6 0.9609.F7 0.9648.F8 0.9688.F9 0.9727.FA 0.9766.FB 0.9805.FC 0.9844.FD 0.9883.FE 0.9922.FF 0.9961

12 CHAPTER 1 NUMBERS AND ELEMENTARY MATHEMATICS

1.3.2 POWERS OF 16 IN DECIMAL SCALE

14 CHAPTER 1 NUMBERS AND ELEMENTARY MATHEMATICS

1.3.4.1 The constant π

The transcendental numberπ is defined as the ratio of the circumference of a circle

to the diameter It is also the ratio of the area of a circle to the square of the radius(r) and appears in several formulas in geometry and trigonometry

circumference of a circle= 2πr, volume of a sphere= 4

3πr

3,area of a circle= πr2, surface area of a sphere= 4πr2.One method of computingπ is to use the infinite series for the function tan−1x andone of the identities



π = lim

k→∞2k

vuut2 −

vuut2 +

vu

t2 +

vu

1 + 1n

n

=

∞Xn=0

xnn! Euler’s formula relatese and π: e

πi=−1

1.3.4.3 The constant γ

Euler’s constantγ is defined by

γ = limn→∞

nXk=1

16 CHAPTER 1 NUMBERS AND ELEMENTARY MATHEMATICS

and the representation in square roots

The factorial ofn, denoted n!, is the product of all positive integers less than or equal

ton; n! = n·(n−1)·(n−2) · · · 2·1 By definition, 0! = 1 If n is a negative integer(n =−1, −2, ) then n! = ±∞ The generalization of the factorial function tonon-integer arguments is the gamma function (see page 437); whenn is an integer,Γ(n) = (n− 1)!

The double factorial ofn is denoted by n!! and is defined as n!! = n(n− 2)(n −4)· · · , where the last element in the product is either 2 or 1, depending on whether n

is even or odd The shifted factorial (also called Pochhammer’s symbol) is denoted

by(a)nand is defined as

Approximations to n! for large n include Stirling’s formula (the first term of thefollowing)

n!≈√2πe n

e

n+ 1

1 + 112n+

1288n2+



(1.3.5)and Burnsides’s formula

n!≈√2π

n +1 2e

The Bernoulli polynomialsBn(x) are defined by the generating function

text

et− 1 =

∞Xn=0

Bn(x)tn

These polynomials can be defined recursively byB0(x) = 1, B′

n(x) = nBn−1(x),andR1

0 Bn(x) dx = 0 for n≥ 1 The identity Bk+1(x + 1)− Bk+1(x) = (k + 1)xkmeans that sums of powers can be computed via Bernoulli polynomials

1k+ 2k+· · · + nk =Bk+1(n + 1)− Bk+1(0)

The Bernoulli numbers are the Bernoulli polynomials evaluated at 0:Bn = Bn(0)

A generating function for the Bernoulli numbers is

∞Xn=0

Bn

tnn! =

t

et− 1 In thefollowing table each Bernoulli number is written as a fraction of integers: Bn =

Nn/Dn Note thatB2m+1= 0 for m≥ 1

18 CHAPTER 1 NUMBERS AND ELEMENTARY MATHEMATICS

The Euler polynomialsEn(x) are defined by the generating function

2ext

et+ 1 =

∞Xn=0

En(x)tn

∞Xn=0

En

tnn! =

!n

√52

!n#

(1.3.13)

Note thatFn is the integer nearestφn/√

5 as n→ ∞, where φ is the golden ratio(see page 16) Hence, lim

1.3.9 SUMS OF POWERS OF INTEGERS

1 Define

sk(n) = 1k+ 2k+· · · + nk =

nXm=1

am

k + 3− m

#n

(c) Note the specific values

k2nXk=1

k3nXk=1

k4nXk=1

20 CHAPTER 1 NUMBERS AND ELEMENTARY MATHEMATICS

Riemann’s zeta function is defined to beζ(n) =P∞

k=1 k1n (it is defined for Rek > 1and extended to C) Related functions are

(−1)k(2k + 1)n, γ(n) =

∞Xk=0

1(2k + 1)n

Properties include:

1 α(n) = (1− 21−n)ζ(n)

2 ζ(2k) = (2π)

2k2(2k)! |B2k|

3 γ(n) = (1− 2−n)ζ(n)

4 β(2k + 1) = (π/2)

2k+12(2k)! |E2k|

5 The seriesβ(1) = 1−1

3+15− · · · = π/4 is known as Gregory’s series

6 Catalan’s constant is G= β(2)≈ 0.915966

7 Riemann hypothesis: The non-trivial zeros of the Riemann zeta function (i.e.,

the{zi} that satisfy ζ(zi) = 0) lie on the critical line given by Re zi = 12.(The trivial zeros arez =−2, −4, −6, )

n ζ(n) =

∞Xk=1

1

kn

∞Xk=1

(−1)k+1

kn

∞Xk=0

(−1)k(2k + 1)n

∞Xk=0

1(2k + 1)n

22 CHAPTER 1 NUMBERS AND ELEMENTARY MATHEMATICS

For information on these and hundreds of thousands of other sequences, see “TheOn-Line Encyclopedia of Integer Sequences,” at oeis.org

3 For all non-negative rational numbersx and y

(a) |xy|p=|x|p|y|p

(b) |x + y|p≤ max (|x|p,|y|p)≤ |x|p+|y|p

Note the product formula: |x|Qp∈{2,3,5,7,11, }|x|p= 1

Let Qpbe the topological completion of Q with respect to| · |p Then Qpis thefield ofp-adic numbers The elements of Qp can be viewed as infinite series: theseriesP∞

n=0anconverges to a point in Qpif and only iflimn→∞|an|p= 0

5

= 5−1= 1

5

•

140297

7

= 7−1= 1

7

•

140297

... NUMBERS AND ELEMENTARY MATHEMATICS

That is, the ratioab is an integer

1 If the integersa and b leave the same remainder when divided by the number n,thena and. .. numbers less than106and 105,212 less than1015.The Carmichael numbers less than ten thousand are 561, 1105, 1729, 2465,

2821, 6601, and 8911