Stationary points of the Hardy Z-function

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Numbers

$j$

$c_j$

2.475726622637559869083102189686627974809365039250354701980211123811607681753880273110553606668961604

10.21207484523579443110737968509429508806037165961428131089715209510683045223255695742082709112095488

17.88258207693668271918356373443223959312714910404430603519476670626348376687811269581041394357308563

23.10465065128401126480759481457411820834290932669379366518301876524499249566751816402358421622904998

27.73588329071004203611426659762777897951590250028040288457273806777968878927006370133110602813814624

31.69793013788870280122718325934600485706178402822302651624985044335522198100881330971269394808562618

35.39273010706615233314685599810339391217947988852519862238106085180421084935404727324840343677728720

39.17856631916394018454250518783722635238037293158584576234986483858831112641924648432967660148474482

42.17702005358910245397156871886654084304616794581689402500574408612430694118318716686567645204114288

10:

45.63611307622311419900525992309194603733286231076140394980248952548187057557125261898076764528827424

11:

48.87000415928118683987988603764683796812980907727512398276803498359320713618230170336149695756372997

12:

51.51858765436405084806093588416691592805461283521512041707723108793990599426147375361423425112091268

13:

54.70489516462072294778187860432144098732067356520047486936830607346267450663691706158660204624269657

14:

57.77046518807109637317530008503447375371720257091925257449993392208855903283988752338669159597257922

15:

60.11884001748145123207806491899689374027533568852175255838980399335048646236796702848255676624866562

16:

63.06042770772062531339074578361181594342689692803805234472464605448745620101887015264187750673331710

17:

66.04487053682631198505745600288328119053659876659970129116747932443621421851114934201318634040734359

18:

68.36015656890296006485194283169086353541979978331018252579614011358701828987244890148284116271552256

19:

70.88139960302482023609855692092049458013156851659654854205502884715197029480357193868118913356275541

20:

73.76348442201623242682363362073578414703434145361913743966424804302328980683415641221848004379590629

21:

76.39295015056153456237165369831608917329051516865525352971063805123586441515124759303342961250084972

22:

78.36224922807260644254569481929294092087212344490510909987402270753273466271431906457538079131520405

23:

81.12917376195677956547197154815844296964169905287847281867644251281737146543345946281111965810609730

24:

83.78516778931814889391078143592442968589843452779510084634090632779053248606895013761100002783763664

25:

86.04475133078160060933395680249430107741524066197453639277226457407565493337646901275443722503854468

26:

88.14703139294968203446387198590989169168530794555332103923049799674640290429091248640056260096811570

27:

90.72385716842208318799131965996345546060516964204641638565513605460422002336266220622837611110023161

28:

93.43535207484366559133209116019972087292087662500823541955214042943235968722967319159977129825479677

29:

95.27966062997123646034991845312720764572880731259135990289847924971351349552815920008606073922754453

30:

97.51136659309703029929263717033754289889009500346063314135897042168362916519904611445708494749664940

31:

100.0506251059990293637317708946317406982068072503124239864873049249164849831333636451664923433528636

32:

102.4421490546672989477757371417119172316590199899327372953910173854137208457127187375216653578862482

33:

104.5616301296148261610576472622781832465797964610579957073071944516268447320440022145456296238648390

34:

106.3897646228004933527144606817880166227226728273846860543092498384560931115721427318991375220944938

35:

108.9867908802936487144947439383618208091997824390930921755994619956183938699976332021147763476541253

36:

111.4363788873588056633576982205562954719097967786790949296312414900253500107086272701273244981140292

37:

113.2051850103112019442800427579138756368733176880522688044009779096357516400402345122868426679285577

38:

115.3131862213136942013367326205818991963564839030753171496657273436961558535571166571523303110238279

39:

117.5572600912922846861879143609018036762924491220505761894516861136653462287420785722313031557537550

40:

119.9686456144628523582669387038508516996861904503228345379909376115351555865818480225295215279439211

41:

122.0984275737177647119794528426035075671625087120291409359490532241919672398776996934784193310587780

42:

123.6578797253751261281838180832710449347322140148389279263991197281023038050129485284044500832459665

43:

126.0019010810881728195587031274562019160451276726317281787881237843617661398820172522902287558300330

44:

128.4489625720640338353019190262739636745219105006745490027340749604765508891971022716060817530210853

45:

130.3311278302163300263765013477259575012695177332327643897664466684988451622226613548704537007861854

46:

132.2944678703488173289663701580525022794758530163306854496064273089959434542617236005047407249143829

47:

134.1566490685228628115343964992304034792990429596655062305659339361517384851394341717018307867725598

48:

136.4556749433270364819825399738028726865204081294655554725573190631983590437003235002519645789676282

49:

138.8355693905703149838783532797371570728257858525571995298103285305538401811445237005495728200550597

50:

140.4462865389453528988086022098282686007267980027326309890361976884267333871870135390188983489355172

51:

142.2305740848544695726768204490572874078139255638395720074793036119769257934634242055229869752324798

52:

144.5312835431472608261303936365005561665876811026514991084619553103884154243837789661289203690863489

53:

146.6896253405539323534647358794158487174785161810225933860197158795865107197272956616216915687377268

54:

148.6428311641690380813378509487745252729354213716367898071563035223614554403122296240794005914808578

55:

150.4870571994692705766553514947693980481339225677476464270173690983656825619066809430281425356648281

56:

152.1433738050032632105851830667667196435742395041291622719371002781100100870428886849132184630328834

57:

154.5044285588978839722651412014185231992047696191907360604379588249037673091779479253270399201607095

58:

156.7697607010804916689161044236557591247683913214942752240109862652746718413120556205167560343742918

59:

158.2506283608140403648893683041794145875379862971665098128862614518658008246242922075985578369793397

60:

160.1173090399730525107564299981690685805730563798294253213006897780539452707350688283312186712741720

61:

162.1352837804149532063055519018277539507110192272965015537646161109321431602065868593025826348054774

62:

164.2445245333153302489472139748914339534212644203896919842107264621688621578477212601743659475680840

63:

166.3068693591654700077221634009961435285199781198389769876619185468946354479477758154347471421050551

64:

168.0717077990873451569775357299286936929231866556224401158673728310597679260626091502995026776671436

65:

169.5276140747020772582821406584827949259985144086971585841676298028904761764975847925491880981361439

66:

171.7591059972980732187266342979784089914789097679386443033212015734379191550059277156126898147250331

67:

174.0264857887995908328066827274446625404706849147968474440517071845885935401164712562062641388680256

68:

175.6154570147481121060891259184624876639549524470315924189166092063143399160622951054940817457810129

69:

177.4212103162778226763103242044736485753876907493441082927649228744461152578522912915787219682323890

70:

179.1819155171033517570644179201352975779224667853569290355381417481547867163265953109752406325979347

71:

181.1303765333652522762323095569469032824797777503590821918243348610694537478602709576487822899671076

72:

183.3325111447201802235372454453528510536087657706848876785312352038293195591096914400801270799397514

73:

185.2202454765203051105694325380856455654489916912235673411165901677686213249912816987270572678257256

74:

186.5288895455454787197467205748189687324369513595567621930672353911954341137578490650240906780337266

75:

188.4322315394036888447347708944728857327526410610265438578879880552383906418611624172454973876364932

76:

190.6123790665753302028426804677123613766279988936313162180000824855957377843453932257925066843112032

77:

192.5354865889346655736087515457685694759445722811367569674037582217127856422587838840180887697939241

78:

194.1976087950703491789799892637967200595792825350066077369924506359475412661597396686803504526744201

79:

196.0328287018461079792373982864116984089918561375963330893690799230170848122897224810459931263299054

80:

197.4923745017324839105350393276468177870710033925193106783782866232041827512111005100599327774467312

81:

199.6517944228043405240443665468614062975095838877147330347369697338302531264621473607949886455099167

82:

201.8238437375989368555216326799331617287277607100304409400839750911053495949174954054282365077023085

83:

203.3301297083866258846601568991097580073467235059887640391122399960632025355637966726908829383463693

84:

204.8312778544412130969733383352994785353919129542635180477186616708632366118790343591619102680706886

85:

206.7402195968745174809076418813617035576171488891780590008964875315992597060470948361431305188180904

86:

208.7259845996788435069323061193625272720219377805796607977098966072142215065368240166813311791191388

87:

210.5923444481639361462974878090118730359795935263823150295933789374941335150906921649848545092931528

88:

212.4487595509784562439334318290955937377050019373653124909109581817391225809487168371259379388689290

89:

213.9551725910949524897339006884294345012874535352440176708117925524077954129697205252808866206361073

90:

215.4650238565073500014178239979812727159412792803762716610749521183160798095686153268523140715225775

91:

217.6053380911909600754143954152289793672705726980920010994857692215381136509635789742885740149142323

92:

219.7470519013136830590126450525756255770712777921735888318199174142748818095701692191226881383000399

93:

221.0805386986390044080481556548347605355288018755858099070756546870450410823716572075991089059733926

94:

222.7950072398109919775624510242726038324276449949273383484908426686032135596540519653480805191473465

95:

224.5050035430703080254296108980050141037054405373280962164966631693559272219357619755214174389249161

96:

226.3015470518484034655497810932934971540255047181992880536405053584707937015084447195528257981677138

97:

228.3211158519460470284985895828682517196723584439119158046811306074944408652761888626429949099026109

98:

230.1577080569243193019192713527529849622046907476533768098577925664661480226658113816396607279489346

99:

231.6180680772027146712739714258782132180245875383961292633158613814076747419693460112362542423231655

100:

232.9949195177573290634928955103177035511421233758250535310006921576552344929516906904457480917171279

Definition

The table holds $c_j$, the $j$th distinct positive real root of $Z^{\prime}(t)=0$, in increasing order, where $Z(t)=e^{i\theta(t)}\zeta(1/2+it)$ is the Hardy Z-function [1].

Parameters

$j$

— positive stationary-point index ($j\geq1$)

Formulas

(1)

$Z^{\prime}(t)=i e^{i\theta(t)}\bigl(\theta^{\prime}(t)\zeta(1/2+it)+\zeta^{\prime}(1/2+it)\bigr)$, where $\theta^{\prime}(t)=\tfrac12\operatorname{Re}\psi(1/4+it/2)-\tfrac12\log\pi$ and $\psi=\Gamma^{\prime}/\Gamma$ is the digamma function.

(2)

$Z(-t)=Z(t)$ and $Z^{\prime}(-t)=-Z^{\prime}(t)$ for real $t$ [1].

Comments

(3)

The phase is $\theta(t)=\operatorname{Im}\log\Gamma(1/4+it/2)-(t/2)\log\pi$, with the analytic log-gamma branch on the right half-plane that is real at $1/4$. Thus $\theta(0)=0$, and $Z(t)$ is real for real $t$ [2]. The function is also called the Riemann-Siegel Z-function. Derivatives are taken with respect to the real height $t$.

(4)

The origin is excluded. Each distinct positive root is counted once, regardless of multiplicity; all 100 roots stored here are simple. The first two, $c_1$ and $c_2$, precede the first zeta-zero ordinate. They correspond to $u_{-1}$ and $u_0$ in [1]; for this simple prefix, $c_j=u_{j-2}$.

(5)

Let $\gamma_n$ denote the $n$th positive zeta-zero ordinate, indexed from one. For $3\leq j\leq100$, $\gamma_{j-2}<c_j<\gamma_{j-1}$. This statement concerns the finite range represented here; it does not assume the Riemann hypothesis or a global interlacing theorem.

(6)

These are stationary locations of the real function $Z(t)$. They are different from the zeros of $\zeta^{\prime}(s)$, whose derivative is taken with respect to the complex variable $s$.

Programs

(P1)

Python

import mpmath as mp
mp.mp.dps = 130
# Approximate c_1; the attached generator certifies the indexed prefix.
c1 = mp.findroot(lambda t: mp.siegelz(t, derivative=1), (2, 3))
print(mp.nstr(c1, 100))

References

[1]

R. R. Hall, On the stationary points of Hardy’s function Z(t), Acta Arithmetica 111 (2004), 125–140. (doi)

Links

[2]

FLINT: Hardy Z-functions and certified zeta-zero isolation

[3]

mpmath: Riemann-Siegel Z-function and its derivatives

Similar tables

Zeros_of_the_Riemann_zeta_function — stores zero ordinates of $Z$, while this table stores its positive stationary locations

Zeros_of_the_derivative_of_the_Riemann_zeta_function — stores zeros of the complex derivative of $\zeta$, a different derivative from $Z^{\prime}$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every stationary-point index $1\leq j\leq100$, including the two roots before the first zeta zero and the subsequent oscillations; $100$ is a noncanonical cutoff for a consecutive reference prefix, chosen to bound the cost of certifying that no roots were missed)

How they were obtained:

FLINT Hardy Z Taylor jets [2] certify a cover of $[0,233]$ at 96 working bits. On each interval, either $Z^{\prime}$ excludes zero, or $Z^{\prime\prime}$ excludes zero and the endpoint signs count exactly zero or one simple root. Near the origin, positivity of $Z^{\prime\prime}$ on $[0,1/1024]$ and evenness of $Z$ exclude every positive root in that interval. The cover proves that the 100 roots are the complete positive prefix through height $233$, independently of the Riemann hypothesis. The attached certification.md gives the proof and the interval counts.

Each isolated root is refined at 493 bits and certified afresh by strict endpoint signs and nonzero $Z^{\prime\prime}$. The enclosures retain at least 383 relative accuracy bits and are serialized to 100 significant digits. A separate product-derivative formula certifies the endpoints of all 100 written decimal intervals at 560 bits. All 200 intervals shifted by ten last-place units are rejected. Independent mpmath computations at 125 and 155 decimal digits [3] agree with every entry; known-answer controls and the parity identities also pass.

A separate isolation of 99 distinct critical-line zeta zeros, together with the total count $N(234)=99$, certifies the zeta-zero numbering needed for the finite gap relations. The checker also obtains $N(233)=98$. The attached checks.py reproduces these tests on the stored entries. The mpmath program provides an approximation; the rigorous error bounds come from the interval certificates.