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196 TheTheoryofHardy’sZ-Function
CAMBRIDGE TRACTS IN MATHEMATICS
GENERAL EDITORS
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G.H.Hardy,1877-1947
The Theory of Hardy’s Z-Function
ALEKSANDAR IVIC´
UniverzitetuBeogradu,Serbia
cambridge university press
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Informationonthistitle:www.cambridge.org/9781107028838
(cid:2)C AleksandarIvic´2013
Frontispiece:G.H.Hardy.KindlysuppliedbyTrinityCollege,Cambridge
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Firstpublished2013
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Ivic,A.,1949–author.
ThetheoryofHardy’sZ-function/AleksandarIvic.
pages cm.–(Cambridgetractsinmathematics;196)
Includesbibliographicalreferencesandindex.
ISBN978-1-107-02883-8(hardback)
1.Numbertheory. I.Title.
QA241.I83 2012
512.7–dc23 2012024804
ISBN978-1-107-02883-8Hardback
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Contents
Preface pagexi
Notation xv
1 Definitionofζ(s),Z(t)andbasicnotions 1
1.1 Thebasicnotions 1
1.2 Thefunctionalequationforζ(s) 3
1.3 PropertiesofHardy’sfunction 6
1.4 Thedistributionofzeta-zeros 8
Notes 14
2 Thezerosonthecriticalline 21
2.1 Theinfinityofzerosonthecriticalline 21
2.2 Alowerboundforthemeanvalues 23
2.3 Lehmer’sphenomenon 25
2.4 Gapsbetweenconsecutivezerosonthecriticalline 28
Notes 41
3 TheSelbergclassof L-functions 49
3.1 TheaxiomsofSelberg’sclass 49
3.2 TheanalogsofHardy’sandLindelo¨f’sfunctionforS 51
3.3 Thedegreed andtheinvariantsofS 52
F
3.4 ThezerosoffunctionsinS 56
Notes 57
4 Theapproximatefunctionalequationsforζk(s) 61
4.1 AsimpleAFEforζ(s) 61
4.2 TheRiemann-Siegelformula 63
4.3 TheAFEforthepowersofζ(s) 70
vii
viii Contents
4.4 Thereflectionprinciple 84
4.5 TheAFEswithsmoothweights 87
Notes 94
5 Thederivativesof Z(t) 99
5.1 Theθ and(cid:4)functions 99
5.2 Theformulaforthederivatives 101
Notes 106
6 Grampoints 109
6.1 DefinitionandorderofGrampoints 109
6.2 Gram’slaw 112
6.3 Ameanvalueresult 115
Notes 120
7 ThemomentsofHardy’sfunction 123
7.1 Theasymptoticformulaforthemoments 123
7.2 Remarks 130
Notes 132
8 TheprimitiveofHardy’sfunction 135
8.1 Introduction 135
8.2 TheLaplacetransformofHardy’sfunction 138
8.3 ProofofTheorem8.2 142
8.4 ProofofTheorem8.3 150
Notes 153
9 TheMellintransformsofpowersof Z(t) 157
9.1 Introduction 157
9.2 SomepropertiesofthemodifiedMellintransforms 159
9.3 AnalyticcontinuationofM (s) 164
k
Notes 172
10 FurtherresultsonM (s)andZ (s) 176
k k
10.1 SomerelationsforM (s) 176
k
10.2 MeansquareidentitiesforM (s) 180
k
10.3 EstimatesforM (s) 186
k
10.4 Naturalboundaries 198
Notes 201
11 OnsomeproblemsinvolvingHardy’sfunctionand
zeta-moments 206
11.1 ThedistributionofvaluesofHardy’sfunction 206
