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Distribution of the zeros
Here we present some statistics concerning Rosser blocks (similar to [4]) in the interval [g0, g202,929,369,002[ where gm is the mth Gram point. This report presents specific results which were obtained during the numerical verification of the Riemann Hypothesis. Therefore, we recommend the reader to study the four papers by Brent [1], Brent et al. [2], and van de Lune et al. [2], [3] for an easy understanding of this report and the notation.

A previous version of my program was organized in such a way that in case the value of Z(t), obtained with method A, was too small for a rigorous sign determination, a few small shifts of the arguments were tried before method B was involved. Therefore, my program uses for j < 58,095,202,838.671, in relatively few cases, an approximation to the Gram point gj instead of gj itself.
Consequently, the statistics presented in this report cannot be accumulated to the statistics found by [1].

More details about the distribution of the verified zeros in form of statistical data which contains interesting zero-patterns and at moment unproved heuristics can be found in the draft Results connected with the first 100 billion zeros of the Riemann zeta function.

Number of Rosser blocks of given length

n J(1, n) J(2, n) J(3, n) J(4, n) J(5, n) J(6, n) J(7, n) J(8, n) J(9, n) J(10, n) J(11, n) J(12, n)
100100            
2001943           
3002867           
40037712           
50047413           
60056219           
70064826           
80073433           
90082239           
100091642           
20001766117           
300026191891          
400034482723          
500042833487          
6000511142811          
7000593351115          
8000675659518          
9000757368718          
10000837478022          
20000164041680762         
300002429326211513         
400003213835842264         
500003991145453256         
600004774755024077         
7000055557646349291        
80000632557468584131        
90000709418461688171        
100000786949445779191        
200000155327193381928521        
3000002314682923932121022        
4000003070293927745681725        
50000038216249374604023010        
60000045711759460754331614        
70000053203569538904841517        
800000606477797601062151020        
900000680914899581222659324        
10000007551321002031382270932        
20000001493597202964306592018841       
300000022269473058784855837201444       
400000029558964096846705456042228       
5000000368381251350285804755929411       
60000004408593617583105170966339914       
700000051327807215171247111186349521       
800000058554618257771443781418259624       
900000065771699301051641681661170029       
10000000729780810345451841071911582136       
200000001446863720793393905654699024221512      
300000002159679431266726041047837144912644      
4000000028697660417659382127711205169513876      
500000003578008152276671041205147420962351413      
6000000042844350628094212633921842911245066824      
7000000049898901733313014879152220351553084930      
80000000569420228386070171427226063818807103334      
90000000639770249439916194145529993322257124046      
1000000007100470210493485216960734036225813143654      
2000000001409561682104744844978517716126863640362136     
3000000002105580333161727768733031235332119881725944711     
40000000027993249142200618927775217197181766561096670715     
5000000003491492515278464011704717221940423834515128100621     
60000000041824146563378588141447262730872303303196121319331    
70000000048722347373972149166006583252376372321245081671491    
80000000055612500384569213190657803781754443883296412055701    
90000000062495197495167296215406504318869517462351732414932    
10000000006937055791057720672402391748612535936174079628251083    
2000000000137870304921186233749169230105355851441324109596779941218    
30000000002060739152318014576746741571648607323982201947251406083741    
4000000000274089941542421348310039034722615616342692729095521222132467    
5000000000341970367353043381012626886328870453450746739701029215195486    
60000000004097452935636672187152260340352258095632643511452377252658130    
70000000004774324731742922585178356702416579556795145632791466863410180    
80000000005450425206849199974204531439481630237987496760086563904213223    
900000000061258869279554812442307741425472831292094078931196664050282721   
1000000000068008073061061761781257082304613422101045450110315097738458953273   
2000000000013527867886212486552252257771712948769523875914262632720515316728105819   
3000000000020229502262318819962979076947319994095338448725449769936173430630206864   
400000000002691509732442516142881060639474271828674537732286565415540041464753225124   
500000000003358888992153150987501331712161344763667696603458785661735743640464602175   
60000000000402535062116378627274160372293441850396685984646111284309484228301361262543  
70000000000469105699767442125618187650726849291819110267737713578182117525810346677963495  
80000000000535611421398505633204214994071156790476311968796016121316141323912477096674658  
900000000006020620473995691514562423916525643373318136968837187458351663141147116117165868  
100000000000668463431121063268906726983772447192675351544903622144549419226671702621378971112  
200000000000133058013095212675021395461556480149469722633915812051480797498469844654539824243877 1
Number of Gram intervals containing excatly m zeros

n m = 0 m = 1 m = 2 m = 3 m = 4
1000100   
20031943  
30072867  
4001237712  
5001347413  
6001956219  
7002664826  
8003373433  
9003982239  
10004291642  
20001171766117  
300019126191891 
400027634502741 
500035842873523 
600044251194363 
700053059445224 
800061867696085 
900071075867005 
1000080883907966 
20000177016472174612 
30000279524430275520 
40000384132345378727 
50000491540209483739 
60000596348121586947 
70000702456011690659 
80000813663798799670 
90000924571588908978 
10000010330794281015786 
2000002152815715321110209 
3000003288223456332228327 
4000004447931149943565457 
5000005623638811055072582 
6000006806246460566604729 
7000007986354111978173845 
8000009188961720089933978 
9000001039156932841016871114 
10000001160557691791134771289 
200000023844115258332330112715 
300000036243722792603541714133 
400000048824030291884769045668 
500000061425337785776000877083 
600000074147845256937241808649 
7000000868860527253384835410253 
8000000996748601829697316611791 
900000011249536763435109827313340 
1000000012535567507820122369214932 
20000000255078514929746248815331316 
30000000386169222324403376612047786 
40000000518178529700950505274564520 
50000000650774637065812634513881304 
60000000783995944418274764357598192 
7000000091748045176570889441731153141
800000001051331759105831102483901324621
900000001185436266440792115553311495141
1000000001319732873771916128641851665701
20000000026731650146878550260479523418462
30000000040372811219773682393342065192983
40000000054084387292529187526884716979496
50000000067841388365195359660851278781207
600000000816365154377850387952038610580547
7000000009546331151031296892984143123956612
800000000109313253582794364106471527142084214
900000000123184824655233236119979075160284619
1000000000137079159727626279133509989178455123
20000000002767488761450124871269503678362252748
30000000004172602332170957162406305061547746283
400000000055829764728907425225436221257337596110
500000000069971472736097765546813028509205730139
6000000000841420450432823678581926526011077328178
7000000000983367988504621678395746269512952309225
800000000011255359575763761470109586946014832842271
900000000012678744456480967698123444157716715972308
1000000000014103572897197887156137315418018601016359
20000000000284120938414355135417276610194837552317934
300000000004278860063214988905134165640336566075371551
400000000005720732751286342585175569286925757195952212
500000000007165706899357634708876975940465948788132936
6000000000086130820844288791653283849243681140733323684
70000000000100623877555000852312197957948791332898594386
800000000001151337527157125793729112082918321525340675102
900000000001296577492764240252377126221763071717905475842
1000000000001441945184971352177599140372958461910681156591
200000000000290043610001423757853542823536101138447791614719
Zero-patterns of Rosser blocks

Zero-pattern First occurrences at Gram point Number of Rosser blocks
1-1134991683638
3139995272001076
00139995252434579
0212510788049143
042375167236970
2013310787878419
22182105255637642
40613317687050
010207482540276571
01249212644970730
0142920372778455
0302144252737923
0325261962336628
21033562644830298
212171046147752389
23010702327566814
410983377344415
0110282860307512693
011267433712669524
01143578792702531
01301824346242222
01322622913795255
03103988946225365
031252558113101090
211083701712597666
211265259896821503
213036243063541144
23109562776948298
4110612304656832
0111017111714889215
01112455256157111518
01114429864352833
011306808411907111
01132357580652364
013106019446759329
013124307478102934
0311024302111903587
031121867760219769
211101833652157082042
211121259982909628
211301038688365383
213102637596455744
23110768184755559
41110684479997593
0111101402737598751
0111121998646921468534
0111141549330873731
01113028420893378359
0111321398857559251
01131011812291364701
0113121604880275621
01311047187141364430
0131121633370909211
03111026562163378066
0311121472310701293
2111102004622321471069
2111301524375792822
2113101763766682301
01111121956109371329898
011113052266282806098
011131020641464310875
011311037091042189269
013111017121221311178
031111013869654807664
21111102587798921329901
01111112598760011424250
01111130165152519107315
0111131014565981070095
0111311045402582525881
0113111071757423925755
0131111017533080470825
03111110112154948107275
21111110289442325524148
0111111126126147909865
01111113026912845924699
01111131081803383111717
0111131109787399213227
011131110124595723591347
01131111013312847153324
01311111054296496911778
03111111038778993344551
2111111103363579239560
01111111308508812049721
01111113108098448992834
01111131109030602546358
01111311102649034715745
01113111102219404715841
011311111012250043299348
013111111014325990226829
03111111106490642494531
011111113109115969218112
011111131105820374382731
011131111101174445524591
011311111105036644141522
0131111111011531145304014
0111111131101669394385961
Bibliography

[1] R. P. Brent, On the Zeros of the Riemann Zeta Function in the Critical Strip, Mathematics of Computation 33 (1979), 1361-1372.

[2] R. P. Brent, J. van de Lune, H. J. J. te Riele, D. T. Winter, On the Zeros of the Riemann Zeta Function in the Critical Strip II, Mathematics of Computation 39 (1982), 681-688.

[3] J. van de Lune, H. J. J. te Riele, On the Zeros of the Riemann Zeta Function in the Critical Strip III, Mathematics of Computation 41 (1983), 759-767

[4] J. van de Lune, H. J. J. te Riele, D. T. Winter,On the Zeros of the Riemann Zeta Function in the Critical Strip IV,Mathematics of Computation 46 (1986), 667-681.