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Distribution of the zeros

Here we present some statistics concerning Rosser blocks (similar to [4]) in the interval [g₀, g_{202,929,369,002}[ where g_m 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 g_j instead of g_j 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)
100	100
200	194	3
300	286	7
400	377	12
500	474	13
600	562	19
700	648	26
800	734	33
900	822	39
1000	916	42
2000	1766	117
3000	2619	189	1
4000	3448	272	3
5000	4283	348	7
6000	5111	428	11
7000	5933	511	15
8000	6756	595	18
9000	7573	687	18
10000	8374	780	22
20000	16404	1680	76	2
30000	24293	2621	151	3
40000	32138	3584	226	4
50000	39911	4545	325	6
60000	47747	5502	407	7
70000	55557	6463	492	9	1
80000	63255	7468	584	13	1
90000	70941	8461	688	17	1
100000	78694	9445	779	19	1
200000	155327	19338	1928	52	1
300000	231468	29239	3212	102	2
400000	307029	39277	4568	172	5
500000	382162	49374	6040	230	10
600000	457117	59460	7543	316	14
700000	532035	69538	9048	415	17
800000	606477	79760	10621	510	20
900000	680914	89958	12226	593	24
1000000	755132	100203	13822	709	32
2000000	1493597	202964	30659	2018	84	1
3000000	2226947	305878	48558	3720	144	4
4000000	2955896	409684	67054	5604	222	8
5000000	3683812	513502	85804	7559	294	11
6000000	4408593	617583	105170	9663	399	14
7000000	5132780	721517	124711	11863	495	21
8000000	5855461	825777	144378	14182	596	24
9000000	6577169	930105	164168	16611	700	29
10000000	7297808	1034545	184107	19115	821	36
20000000	14468637	2079339	390565	46990	2422	151	2
30000000	21596794	3126672	604104	78371	4491	264	4
40000000	28697660	4176593	821277	112051	6951	387	6
50000000	35780081	5227667	1041205	147420	9623	514	13
60000000	42844350	6280942	1263392	184291	12450	668	24
70000000	49898901	7333130	1487915	222035	15530	849	30
80000000	56942022	8386070	1714272	260638	18807	1033	34
90000000	63977024	9439916	1941455	299933	22257	1240	46
100000000	71004702	10493485	2169607	340362	25813	1436	54
200000000	140956168	21047448	4497851	771612	68636	4036	213	6
300000000	210558033	31617277	6873303	1235332	119881	7259	447	11
400000000	279932491	42200618	9277752	1719718	176656	10966	707	15
500000000	349149251	52784640	11704717	2219404	238345	15128	1006	21
600000000	418241465	63378588	14144726	2730872	303303	19612	1319	33	1
700000000	487223473	73972149	16600658	3252376	372321	24508	1671	49	1
800000000	556125003	84569213	19065780	3781754	443883	29641	2055	70	1
900000000	624951974	95167296	21540650	4318869	517462	35173	2414	93	2
1000000000	693705579	105772067	24023917	4861253	593617	40796	2825	108	3
2000000000	1378703049	211862337	49169230	10535585	1441324	109596	7799	412	18
3000000000	2060739152	318014576	74674157	16486073	2398220	194725	14060	837	41
4000000000	2740899415	424213483	100390347	22615616	3426927	290955	21222	1324	67
5000000000	3419703673	530433810	126268863	28870453	4507467	397010	29215	1954	86
6000000000	4097452935	636672187	152260340	35225809	5632643	511452	37725	2658	130
7000000000	4774324731	742922585	178356702	41657955	6795145	632791	46686	3410	180
8000000000	5450425206	849199974	204531439	48163023	7987496	760086	56390	4213	223
9000000000	6125886927	955481244	230774142	54728312	9209407	893119	66640	5028	272	1
10000000000	6800807306	1061761781	257082304	61342210	10454501	1031509	77384	5895	327	3
20000000000	13527867886	2124865522	522577717	129487695	23875914	2626327	205153	16728	1058	19
30000000000	20229502262	3188199629	790769473	199940953	38448725	4497699	361734	30630	2068	64
40000000000	26915097324	4251614288	1060639474	271828674	53773228	6565415	540041	46475	3225	124
50000000000	33588889921	5315098750	1331712161	344763667	69660345	8785661	735743	64046	4602	175
60000000000	40253506211	6378627274	1603722934	418503966	85984646	11128430	948422	83013	6126	254	3
70000000000	46910569976	7442125618	1876507268	492918191	102677377	13578182	1175258	103466	7796	349	5
80000000000	53561142139	8505633204	2149940711	567904763	119687960	16121316	1413239	124770	9667	465	8
90000000000	60206204739	9569151456	2423916525	643373318	136968837	18745835	1663141	147116	11716	586	8
100000000000	66846343112	10632689067	2698377244	719267535	154490362	21445494	1922667	170262	13789	711	12
200000000000	133058013095	21267502139	5461556480	1494697226	339158120	51480797	4984698	446545	39824	2438	77	1

Number of Gram intervals containing excatly m zeros

n	m = 0	m = 1	m = 2	m = 3	m = 4
100	0	100
200	3	194	3
300	7	286	7
400	12	377	12
500	13	474	13
600	19	562	19
700	26	648	26
800	33	734	33
900	39	822	39
1000	42	916	42
2000	117	1766	117
3000	191	2619	189	1
4000	276	3450	274	1
5000	358	4287	352	3
6000	442	5119	436	3
7000	530	5944	522	4
8000	618	6769	608	5
9000	710	7586	700	5
10000	808	8390	796	6
20000	1770	16472	1746	12
30000	2795	24430	2755	20
40000	3841	32345	3787	27
50000	4915	40209	4837	39
60000	5963	48121	5869	47
70000	7024	56011	6906	59
80000	8136	63798	7996	70
90000	9245	71588	9089	78
100000	10330	79428	10157	86
200000	21528	157153	21110	209
300000	32882	234563	32228	327
400000	44479	311499	43565	457
500000	56236	388110	55072	582
600000	68062	464605	66604	729
700000	79863	541119	78173	845
800000	91889	617200	89933	978
900000	103915	693284	101687	1114
1000000	116055	769179	113477	1289
2000000	238441	1525833	233011	2715
3000000	362437	2279260	354171	4133
4000000	488240	3029188	476904	5668
5000000	614253	3778577	600087	7083
6000000	741478	4525693	724180	8649
7000000	868860	5272533	848354	10253
8000000	996748	6018296	973166	11791
9000000	1124953	6763435	1098273	13340
10000000	1253556	7507820	1223692	14932
20000000	2550785	14929746	2488153	31316
30000000	3861692	22324403	3766120	47786
40000000	5181785	29700950	5052745	64520
50000000	6507746	37065812	6345138	81304
60000000	7839959	44418274	7643575	98192
70000000	9174804	51765708	8944173	115314	1
80000000	10513317	59105831	10248390	132462	1
90000000	11854362	66440792	11555331	149514	1
100000000	13197328	73771916	12864185	166570	1
200000000	26731650	146878550	26047952	341846	2
300000000	40372811	219773682	39334206	519298	3
400000000	54084387	292529187	52688471	697949	6
500000000	67841388	365195359	66085127	878120	7
600000000	81636515	437785038	79520386	1058054	7
700000000	95463311	510312968	92984143	1239566	12
800000000	109313253	582794364	106471527	1420842	14
900000000	123184824	655233236	119979075	1602846	19
1000000000	137079159	727626279	133509989	1784551	23
2000000000	276748876	1450124871	269503678	3622527	48
3000000000	417260233	2170957162	406305061	5477462	83
4000000000	558297647	2890742522	543622125	7337596	110
5000000000	699714727	3609776554	681302850	9205730	139
6000000000	841420450	4328236785	819265260	11077328	178
7000000000	983367988	5046216783	957462695	12952309	225
8000000000	1125535957	5763761470	1095869460	14832842	271
9000000000	1267874445	6480967698	1234441577	16715972	308
10000000000	1410357289	7197887156	1373154180	18601016	359
20000000000	2841209384	14355135417	2766101948	37552317	934
30000000000	4278860063	21498890513	4165640336	56607537	1551
40000000000	5720732751	28634258517	5569286925	75719595	2212
50000000000	7165706899	35763470887	6975940465	94878813	2936
60000000000	8613082084	42887916532	8384924368	114073332	3684
70000000000	10062387755	50008523121	9795794879	133289859	4386
80000000000	11513375271	57125793729	11208291832	152534067	5102
90000000000	12965774927	64240252377	12622176307	171790547	5842
100000000000	14419451849	71352177599	14037295846	191068115	6591
200000000000	29004361000	142375785354	28235361011	384477916	14719

Zero-patterns of Rosser blocks

Zero-pattern	First occurrences at Gram point	Number of Rosser blocks
1	-1	134991683638
3	13999527	2001076
00	13999525	2434579
02	125	10788049143
04	237516723	6970
20	133	10787878419
22	182105255	637642
40	61331768	7050
010	207482540	276571
012	4921	2644970730
014	2920372778	455
030	2144	252737923
032	526196233	6628
210	3356	2644830298
212	1710461477	52389
230	1070232756	6814
410	983377344	415
0110	2828603075	12693
0112	67433	712669524
0114	35787927025	31
0130	18243	46242222
0132	2622913795	255
0310	39889	46225365
0312	5255811310	1090
2110	83701	712597666
2112	6525989682	1503
2130	3624306354	1144
2310	9562776948	298
4110	6123046568	32
01110	17111714889	215
01112	455256	157111518
01114	42986435283	3
01130	68084	11907111
01132	35758065236	4
01310	601944	6759329
01312	43074781029	34
03110	243021	11903587
03112	18677602197	69
21110	1833652	157082042
21112	125998290962	8
21130	10386883653	83
21310	26375964557	44
23110	76818475555	9
41110	68447999759	3
011110	140273759875	1
011112	19986469	21468534
011114	154933087373	1
011130	2842089	3378359
011132	139885755925	1
011310	1181229	1364701
011312	160488027562	1
013110	4718714	1364430
013112	163337090921	1
031110	2656216	3378066
031112	147231070129	3
211110	20046223	21471069
211130	152437579282	2
211310	176376668230	1
0111112	195610937	1329898
0111130	52266282	806098
0111310	20641464	310875
0113110	37091042	189269
0131110	17121221	311178
0311110	13869654	807664
2111110	258779892	1329901
01111112	5987600114	24250
01111130	165152519	107315
01111310	145659810	70095
01113110	454025825	25881
01131110	717574239	25755
01311110	175330804	70825
03111110	112154948	107275
21111110	2894423255	24148
011111112	61261479098	65
011111130	2691284592	4699
011111310	818033831	11717
011113110	978739921	3227
011131110	12459572359	1347
011311110	1331284715	3324
013111110	542964969	11778
031111110	3877899334	4551
211111110	33635792395	60
0111111130	85088120497	21
0111111310	8098448992	834
0111113110	9030602546	358
0111131110	26490347157	45
0111311110	22194047158	41
0113111110	12250043299	348
0131111110	14325990226	829
0311111110	64906424945	31
01111111310	91159692181	12
01111113110	58203743827	31
01113111110	117444552459	1
01131111110	50366441415	22
01311111110	115311453040	14
011111113110	166939438596	1

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.