The Riemann Hypothesis

157 Pages·2017·2.45 MB·english

by  Roland van der Veen

The Riemann Hypothesis
A Million Dollar Problem
1 2 3 4 5 6 7 8 9
2.5 10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
2 30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
1.5
50 51 52 53 54 55 56 57 58 59
1 60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
0.5
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
0
5 10 15 20 25 30
y
Thehalftitlepagefigureshowsthegraphofthefunction
1
f(y)=|ζ( + iy)|
2
for 0≤y ≤32 against a background in which the primes less than 100
aremarked(seealsofigure3.7onpage54).
Originaltitle:DeRiemann-hypothese–Eenmiljoenenprobleem
EpsilonUitgaven,Utrecht,2011
Translatedbytheauthors.
(cid:2)c 2015by
TheMathematicalAssociationofAmerica(Incorporated)
LibraryofCongressCatalogCardNumber2015959833
PrinteditionISBN978-0-88385-650-5
ElectroniceditionISBN978-0-88385-989-6
PrintedintheUnitedStatesofAmerica
CurrentPrinting(lastdigit):
10987654321
The Riemann Hypothesis
A Million Dollar Problem
Roland van der Veen
Leiden University
and
Jan van de Craats
University of Amsterdam
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Contents
Preface ix
1 Primenumbers 1
1.1 Primesaselementarybuildingblocks..................... 1
1.2 Countingprimes....................................... 3
1.3 Usingthelogarithmtocountpowers..................... 7
1.4 Approximationsforπ(x)................................ 9
1.5 Theprimenumbertheorem.............................. 11
1.6 Countingprimepowerslogarithmically................... 11
1.7 TheRiemannhypothesis—alookahead.................. 14
1.8 Additionalexercises.................................... 16
2 Thezetafunction 21
2.1 Infinitesums........................................... 21
2.2 Seriesforwell-knownfunctions......................... 26
2.3 Computationofζ(2).................................... 29
2.4 Euler’sproductformula................................. 32
2.5 Lookingbackandaglimpseofwhatistocome ........... 34
2.6 Additionalexercises.................................... 34
3 TheRiemannhypothesis 41
3.1 Euler’sdiscoveryoftheproductformula.................. 41
3.2 Extendingthedomainofthezetafunction................ 43
3.3 Acrashcourseoncomplexnumbers ..................... 45
3.4 Complexfunctionsandpowers.......................... 47
3.5 Thecomplexzetafunction.............................. 50
3.6 Thezeroesofthezetafunction........................... 51
3.7 Thehuntforzetazeroes................................. 54
3.8 Additionalexercises.................................... 55
4 PrimesandtheRiemannhypothesis 59
4.1 Riemann’sfunctionalequation........................... 60
4.2 Thezeroesofthezetafunction........................... 63
4.3 Theexplicitformulaforψ(x)............................ 66
4.4 Pairingupthenon-trivialzeroes......................... 69
vii
viii Contents
4.5 Theprimenumbertheorem.............................. 72
4.6 Aproofoftheprimenumbertheorem.................... 73
4.7 Themusicoftheprimes ................................ 76
4.8 Lookingback.......................................... 78
4.9 Additionalexercises.................................... 81
AppendixA.Whybigprimesareuseful 87
AppendixB.Computersupport 91
AppendixC.Furtherreadingandinternetsurfing 99
AppendixD.Solutionstotheexercises 101
Index 143
Preface
Mathematicsisfullofunsolvedproblemsandothermysteries,butnonemore
importantandintriguingthantheRiemannhypothesis.Bafflingthegreatest
minds for more than a hundred and fifty years, the Riemann hypothesis
isattheverycoreofmathematics.Aproofofitwouldmeananenormous
advance.Inaddition,theRiemannhypothesiswaschosenasoneoftheseven
Millennium Problems1 by the Clay Mathematics Institute. This means that
provingtheRiemannhypothesiswillnotonlymakeyouworldfamous,but
alsoearnsyouaonemilliondollarprize.
TheRiemannhypothesisconcernstheprimenumbers
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,...
i.e.,theintegernumbersgreaterthan1thatarenotdivisiblebyanysmaller
number(except1).Ubiquitousandfundamentalinmathematicsastheyare,
it is important and interesting to know as much as possible about these
numbers.Simplequestionswouldbe:howaretheprimenumbersdistributed
among the positive integers? How many prime numbers are there? What
is the number of prime numbers of one hundred digits? Of one thousand
digits? These questions were the starting point of a groundbreaking paper
by Bernhard Riemann written in 1859. As an aside in his article Riemann
formulatedhisnowfamoushypothesis,thatsofarnobodyhascomecloseto
proving:
TheRiemannHypothesis. Allnontrivialzeroesofthezetafunctionlieon
thecriticalline.
Hidden behind this at first mysterious statement, lies a whole mathe-
maticaluniverseofprimenumbers,infinitesequences,infiniteproductsand
complexfunctions.Thepresentbookisafirstexplorationofthisfascinating
world.
1seehttp://www.claymath.org/millennium/RiemannHypothesis/
ix