186zpi: Difference between revisions

Revision as of 14:56, 10 August 2024

186 zeta peak index (abbreviated 186zpi), is the equal-step tuning system obtained from the 186st peak of the Riemann zeta function.

Tuning			Strength			Closest EDO		Integer limit
ZPI	Steps per octave	Step size (cents)	Height	Integral	Gap	EDO	Octave (cents)	Consistent	Distinct
186zpi	41.3438354846780	29.0248832971658	1.876590	0.241233	11.567493	41edo	1190.02021518380	2	2

Theory

Record on the Riemann zeta function with primes 2 and 3 removed

186zpi sets a height record on the Riemann zeta function with primes 2 and 3 removed. The previous record is 125zpi and the next one is 565zpi. It is important to highlight that the optimal equal tunings obtained by excluding the prime numbers 2 and 3 from the Riemann zeta function differs very slightly from the optimal equal tuning corresponding to the same peaks on the unmodified Riemann zeta function.

Unmodified Riemann zeta function						Riemann zeta function with primes 2 and 3 removed
Tuning			Strength	Closest EDO		Tuning		Strength	Closest EDO
ZPI	Steps per octave	Step size (cents)	Height	EDO	Octave (cents)	Steps per octave	Step size (cents)	Height	EDO	Octave (cents)
125zpi	30.6006474885974	39.2148564976330	1.468164	31edo	1215.66055142662	30.5974484926723	39.2189564527704	3.769318	31edo	1215.78765003588
186zpi	41.3438354846780	29.0248832971658	1.876590	41edo	1190.02021518380	41.3477989230936	29.0221010852836	4.469823	41edo	1189.90614449663
565zpi	98.6209462564991	12.1678005084130	2.305330	99edo	1204.61225033289	98.6257548378926	12.1672072570942	4.883729	99edo	1204.55351845233

Harmonic series

As a non-octave, non-tritave scale, 186zpi features a well-balanced harmonic series segment from 5 to 9, and performs exceptionally well across all prime harmonics from 5 to 23, with the exception of 19.

Approximation of harmonics in 186zpi
Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	-10.0	+13.7	+9.1	+0.1	+3.7	-1.9	-0.9	-1.7	-9.9	-0.8	-6.3	+0.3	-11.9	+13.8	-10.9
Error	Relative (%)	-34.4	+47.2	+31.2	+0.3	+12.8	-6.7	-3.2	-5.7	-34.1	-2.6	-21.6	+1.0	-41.1	+47.4	-37.5
Step		41	66	83	96	107	116	124	131	137	143	148	153	157	162	165

Approximation of harmonics in 186zpi
Harmonic		17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
Error	Absolute (¢)	+0.2	-11.6	+10.9	+9.1	+11.7	-10.7	-0.6	+12.8	+0.2	-9.7	+12.0	+7.1	+4.4	+3.8	+5.1	+8.2
Error	Relative (%)	+0.9	-40.1	+37.4	+31.5	+40.5	-37.0	-2.1	+44.0	+0.5	-33.4	+41.5	+24.6	+15.2	+13.0	+17.5	+28.1
Step		169	172	176	179	182	184	187	190	192	194	197	199	201	203	205	207

Approximation of EDONOIs

Based on harmonics with less than 1 cent of error, 186zpi can be approximated by 96ed5, 124ed8 (or every 3 steps of 124edo), 143ed11, 153ed13, 169ed17, 187ed23, and 192ed25.

Intervals and notation

There are several ways to approach notation. The simplest method involves using the notations from 41edo. However, this method does not preserve octave compression when rendered by notation software. To address this issue, consider using the ups and downs notation from 124edo at every 3-degree step (i.e., the edonoi 124ed8).

It is important to note that 124edo provides two possible fifths (3/2). The closest one, from the val <124 197] (i.e. the patent val), is the fifth mapped to 73 steps of 124edo with a relative error of +46.465%. The second closest, from the val <124 196] (i.e. the val 124b), is mapped to 72 steps of 124edo with a relative error of -53.535%. This second fifth, which appears in 124ed8, also corresponds to the fifth of 31edo. Therefore, we choose to use the ups and downs notation of the 124b temperament, denoted as <124 196].

Todo: complete table

Incorporate 3 new columns for ups and downs notation from 124b at every 3-degree. column 1 = ups and downs notation in full, column 2 = ups and downs notation abbreviated, column 3 = octave

Step	Cents	Ratios	Ups and Downs Notation	Step
JI ratios are comprised of 32-integer limit ratios, and are stylized as follows to indicate their accuracy: Bold Underlined: relative error < 8.333 % Bold: relative error < 16.667 % Normal: relative error < 25 % Small: relative error < 33.333 % Small Small: relative error < 41.667 % Small Small Small: relative error < 50 %			124b at every 3-degree limma = 12 steps whole tone = 20 steps
0	0.000		P1	0
1	29.025		^^^1	3
2	58.050	32/31, 31/30, 30/29, 29/28, 28/27, 27/26, 26/25, 25/24	vvA1, ^^d2	6
3	87.075	24/23, 23/22, 22/21, 21/20, 20/19, 19/18, 18/17	vvvm2	9
4	116.100	17/16, 16/15, 31/29, 15/14, 29/27, 14/13	m2	12
5	145.124	27/25, 13/12, 25/23, 12/11, 23/21	^^^m2	15
6	174.149	11/10, 32/29, 21/19, 31/28, 10/9	vvM2	18
7	203.174	29/26, 19/17, 28/25, 9/8, 26/23, 17/15	^M2	21
8	232.199	25/22, 8/7, 31/27, 23/20	^⁴M2	24
9	261.224	15/13, 22/19, 29/25, 7/6	^^^d3	27
10	290.249	27/23, 20/17, 13/11, 32/27, 19/16, 25/21, 31/26	vvm3	30
11	319.274	6/5, 29/24, 23/19	^m3	33
12	348.299	17/14, 28/23, 11/9, 27/22, 16/13	~3	36
13	377.323	21/17, 26/21, 31/25, 5/4	vM3	39
14	406.348	29/23, 24/19, 19/15, 14/11	^^M3	42
15	435.373	23/18, 32/25, 9/7, 31/24, 22/17	vvvA3	45
16	464.398	13/10, 30/23, 17/13, 21/16, 25/19, 29/22	v⁴4	48
17	493.423	4/3	v4	51
18	522.448	31/23, 27/20, 23/17, 19/14, 15/11	^^4	54
19	551.473	26/19, 11/8, 29/21, 18/13	vvvA4	57
20	580.498	25/18, 32/23, 7/5, 31/22	A4	60
21	609.523	24/17, 17/12, 27/19, 10/7	vd5	63
22	638.547	23/16, 13/9, 29/20, 16/11	^^d5	66
23	667.572	19/13, 22/15, 25/17, 28/19, 31/21	vvv5	69
24	696.597	3/2	P5	72
25	725.622	32/21, 29/19, 26/17, 23/15	^^^5	75
26	754.647	20/13, 17/11, 31/20, 14/9	vvA5, ^^d6	78
27	783.672	25/16, 11/7, 30/19, 19/12	vvvm6	81
28	812.697	27/17, 8/5, 29/18	m6	84
29	841.722	21/13, 13/8, 31/19, 18/11	^^^m6	87
30	870.746	23/14, 28/17, 5/3	vvM6	90
31	899.771	32/19, 27/16, 22/13	^M6	93
32	928.796	17/10, 29/17, 12/7, 31/18	^⁴M6	96
33	957.821	19/11, 26/15, 7/4	^^^d7	99
34	986.846	30/17, 23/13, 16/9	vvm7	102
35	1015.871	25/14, 9/5, 29/16	^m7	105
36	1044.896	20/11, 31/17, 11/6	~7	108
37	1073.921	24/13, 13/7, 28/15, 15/8	vM7	111
38	1102.946	32/17, 17/9, 19/10	^^M7	114
39	1131.970	21/11, 23/12, 25/13, 27/14, 29/15, 31/16	vvvA7	117
40	1160.995		v⁴1 +1oct	120
41	1190.020	2/1	v1 +1oct	123
42	1219.045		^^1 +1oct	126
43	1248.070	31/15, 29/14	vvvA1 +1oct	129
44	1277.095	27/13, 25/12, 23/11, 21/10	v⁴m2 +1oct	132
45	1306.120	19/9, 17/8, 32/15, 15/7	vm2 +1oct	135
46	1335.145	28/13, 13/6	^^m2 +1oct	138
47	1364.170	24/11, 11/5, 31/14	vvvM2 +1oct	141
48	1393.194	20/9, 29/13, 9/4	M2 +1oct	144
49	1422.219	25/11, 16/7	^^^M2 +1oct	147
50	1451.244	23/10, 30/13	vvA2 +1oct, ^^d3 +1oct	150
51	1480.269	7/3, 26/11	vvvm3 +1oct	153
52	1509.294	19/8, 31/13, 12/5	m3 +1oct	156
53	1538.319	29/12, 17/7, 22/9	^^^m3 +1oct	159
54	1567.344	27/11, 32/13	vvM3 +1oct	162
55	1596.369	5/2	^M3 +1oct	165
56	1625.393	28/11, 23/9, 18/7	^⁴M3 +1oct	168
57	1654.418	31/12, 13/5	^^^d4 +1oct	171
58	1683.443	21/8, 29/11	vv4 +1oct	174
59	1712.468	8/3, 27/10	^4 +1oct	177
60	1741.493	19/7, 30/11, 11/4	~4 +1oct	180
61	1770.518	25/9, 14/5	vA4 +1oct	183
62	1799.543	31/11, 17/6	^^A4 +1oct, vvd5 +1oct	186
63	1828.568	20/7, 23/8, 26/9	^d5 +1oct	189
64	1857.593	29/10, 32/11	~5 +1oct	192
65	1886.617		v5 +1oct	195
66	1915.642	3/1	^^5 +1oct	198
67	1944.667	31/10	vvvA5 +1oct	201
68	1973.692	28/9, 25/8, 22/7	v⁴m6 +1oct	204
69	2002.717	19/6, 16/5	vm6 +1oct	207
70	2031.742	29/9, 13/4	^^m6 +1oct	210
71	2060.767	23/7	vvvM6 +1oct	213
72	2089.792	10/3	M6 +1oct	216
73	2118.816	27/8, 17/5, 24/7	^^^M6 +1oct	219
74	2147.841	31/9	vvA6 +1oct, ^^d7 +1oct	222
75	2176.866	7/2	vvvm7 +1oct	225
76	2205.891	32/9, 25/7, 18/5	m7 +1oct	228
77	2234.916	29/8, 11/3	^^^m7 +1oct	231
78	2263.941	26/7	vvM7 +1oct	234
79	2292.966	15/4	^M7 +1oct	237
80	2321.991	19/5, 23/6	^⁴M7 +1oct	240
81	2351.016	27/7, 31/8	^^^d1 +2oct	243
82	2380.040		vv1 +2oct	246
83	2409.065	4/1	^1 +2oct	249
84	2438.090		^⁴1 +2oct	252
85	2467.115	29/7, 25/6	^^^d2 +2oct	255
86	2496.140	21/5, 17/4	vvm2 +2oct	258
87	2525.165	30/7, 13/3	^m2 +2oct	261
88	2554.190	22/5	~2 +2oct	264
89	2583.215	31/7	vM2 +2oct	267
90	2612.239	9/2	^^M2 +2oct	270
91	2641.264	32/7, 23/5	vvvA2 +2oct	273
92	2670.289	14/3	v⁴m3 +2oct	276
93	2699.314	19/4	vm3 +2oct	279
94	2728.339	24/5, 29/6	^^m3 +2oct	282
95	2757.364		vvvM3 +2oct	285
96	2786.389	5/1	M3 +2oct	288
97	2815.414		^^^M3 +2oct	291
98	2844.439	31/6, 26/5	vvA3 +2oct, ^^d4 +2oct	294
99	2873.463	21/4	vvv4 +2oct	297
100	2902.488	16/3	P4 +2oct	300
101	2931.513	27/5	^^^4 +2oct	303
102	2960.538	11/2	vvA4 +2oct	306
103	2989.563	28/5, 17/3	^A4 +2oct	309
104	3018.588	23/4	d5 +2oct	312
105	3047.613	29/5	^^^d5 +2oct	315
106	3076.638		vv5 +2oct	318
107	3105.663	6/1	^5 +2oct	321
108	3134.687		^⁴5 +2oct	324
109	3163.712	31/5, 25/4	^^^d6 +2oct	327
110	3192.737	19/3	vvm6 +2oct	330
111	3221.762	32/5	^m6 +2oct	333
112	3250.787	13/2	~6 +2oct	336
113	3279.812	20/3	vM6 +2oct	339
114	3308.837	27/4	^^M6 +2oct	342
115	3337.862		vvvA6 +2oct	345
116	3366.886	7/1	v⁴m7 +2oct	348
117	3395.911		vm7 +2oct	351
118	3424.936	29/4	^^m7 +2oct	354
119	3453.961	22/3	vvvM7 +2oct	357
120	3482.986	15/2	M7 +2oct	360
121	3512.011	23/3	^^^M7 +2oct	363
122	3541.036	31/4	vvA7 +2oct, ^^d1 +3oct	366
123	3570.061		vvv1 +3oct	369
124	3599.086	8/1	P1 +3oct	372
125	3628.110		^^^1 +3oct	375
126	3657.135	25/3	vvA1 +3oct, ^^d2 +3oct	378
127	3686.160		vvvm2 +3oct	381
128	3715.185	17/2	m2 +3oct	384
129	3744.210	26/3	^^^m2 +3oct	387
130	3773.235		vvM2 +3oct	390
131	3802.260	9/1	^M2 +3oct	393
132	3831.285		^⁴M2 +3oct	396
133	3860.309	28/3	^^^d3 +3oct	399
134	3889.334	19/2	vvm3 +3oct	402
135	3918.359	29/3	^m3 +3oct	405
136	3947.384		~3 +3oct	408
137	3976.409	10/1	vM3 +3oct	411
138	4005.434		^^M3 +3oct	414
139	4034.459	31/3	vvvA3 +3oct	417
140	4063.484	21/2	v⁴4 +3oct	420
141	4092.509	32/3	v4 +3oct	423
142	4121.533		^^4 +3oct	426
143	4150.558	11/1	vvvA4 +3oct	429
144	4179.583		A4 +3oct	432
145	4208.608		vd5 +3oct	435
146	4237.633	23/2	^^d5 +3oct	438
147	4266.658		vvv5 +3oct	441
148	4295.683	12/1	P5 +3oct	444
149	4324.708		^^^5 +3oct	447
150	4353.732		vvA5 +3oct, ^^d6 +3oct	450
151	4382.757	25/2	vvvm6 +3oct	453
152	4411.782		m6 +3oct	456
153	4440.807	13/1	^^^m6 +3oct	459
154	4469.832		vvM6 +3oct	462
155	4498.857	27/2	^M6 +3oct	465
156	4527.882		^⁴M6 +3oct	468
157	4556.907	14/1	^^^d7 +3oct	471
158	4585.932		vvm7 +3oct	474
159	4614.956		^m7 +3oct	477
160	4643.981	29/2	~7 +3oct	480
161	4673.006		vM7 +3oct	483
162	4702.031	15/1	^^M7 +3oct	486
163	4731.056	31/2	vvvA7 +3oct	489
164	4760.081		v⁴1 +4oct	492
165	4789.106	16/1	v1 +4oct	495
166	4818.131		^^1 +4oct	498
167	4847.156		vvvA1 +4oct	501
168	4876.180		v⁴m2 +4oct	504
169	4905.205	17/1	vm2 +4oct	507
170	4934.230		^^m2 +4oct	510
171	4963.255		vvvM2 +4oct	513
172	4992.280	18/1	M2 +4oct	516
173	5021.305		^^^M2 +4oct	519
174	5050.330		vvA2 +4oct, ^^d3 +4oct	522
175	5079.355		vvvm3 +4oct	525
176	5108.379	19/1	m3 +4oct	528
177	5137.404		^^^m3 +4oct	531
178	5166.429		vvM3 +4oct	534
179	5195.454	20/1	^M3 +4oct	537
180	5224.479		^⁴M3 +4oct	540
181	5253.504		^^^d4 +4oct	543
182	5282.529	21/1	vv4 +4oct	546
183	5311.554		^4 +4oct	549
184	5340.579	22/1	~4 +4oct	552
185	5369.603		vA4 +4oct	555
186	5398.628		^^A4 +4oct, vvd5 +4oct	558
187	5427.653	23/1	^d5 +4oct	561
188	5456.678		~5 +4oct	564
189	5485.703		v5 +4oct	567
190	5514.728	24/1	^^5 +4oct	570
191	5543.753		vvvA5 +4oct	573
192	5572.778	25/1	v⁴m6 +4oct	576
193	5601.802		vm6 +4oct	579
194	5630.827	26/1	^^m6 +4oct	582
195	5659.852		vvvM6 +4oct	585
196	5688.877		M6 +4oct	588
197	5717.902	27/1	^^^M6 +4oct	591
198	5746.927		vvA6 +4oct, ^^d7 +4oct	594
199	5775.952	28/1	vvvm7 +4oct	597
200	5804.977		m7 +4oct	600
201	5834.002	29/1	^^^m7 +4oct	603
202	5863.026		vvM7 +4oct	606
203	5892.051	30/1	^M7 +4oct	609
204	5921.076		^⁴M7 +4oct	612
205	5950.101	31/1	^^^d1 +5oct	615
206	5979.126		vv1 +5oct	618
207	6008.151	32/1	^1 +5oct	621

Approximation to JI

The following table illustrates the representation of the 32-integer limit intervals in 186zpi. Prime harmonics are in bold; inconsistent intervals are in italic.

Intervals by direct approximation (even if inconsistent)
Ratio	Error (abs, ¢)	Error (rel, %)
17/13	0.030	0.102
5/1	0.075	0.259
25/17	0.100	0.344
25/13	0.129	0.446
23/11	0.138	0.477
25/1	0.150	0.517
11/8	0.155	0.533
17/5	0.175	0.602
13/5	0.204	0.704
17/1	0.250	0.861
13/1	0.279	0.963
9/7	0.289	0.996
23/8	0.293	1.011
23/1	0.621	2.140
31/29	0.641	2.209
30/29	0.642	2.211
23/5	0.696	2.399
29/6	0.717	2.470
9/8	0.736	2.535
11/1	0.760	2.617
25/23	0.771	2.657
11/5	0.835	2.876
23/17	0.871	3.001
21/19	0.881	3.037
11/9	0.891	3.069
23/13	0.901	3.103
25/11	0.910	3.135
8/1	0.914	3.151
8/5	0.990	3.409
17/11	1.009	3.478
8/7	1.025	3.531
23/9	1.029	3.546
13/11	1.039	3.580
25/8	1.065	3.668
17/8	1.164	4.012
27/19	1.171	4.033
11/7	1.180	4.065
13/8	1.194	4.114
31/30	1.283	4.420
23/7	1.318	4.542
31/6	1.358	4.679
9/1	1.650	5.686
9/5	1.725	5.944
20/19	1.726	5.947
25/9	1.800	6.203
19/4	1.801	6.205
17/9	1.900	6.547
24/19	1.906	6.568
13/9	1.930	6.649
7/1	1.939	6.682
7/5	2.015	6.941
31/28	2.060	7.099
25/7	2.090	7.199
17/7	2.189	7.543
13/7	2.219	7.645
21/20	2.607	8.984
21/4	2.683	9.242
29/28	2.702	9.308
32/19	2.716	9.356
19/3	2.821	9.719
19/15	2.896	9.977
27/20	2.897	9.980
27/4	2.972	10.238
32/31	3.085	10.630
15/14	3.343	11.519
14/3	3.418	11.777
13/6	3.428	11.811
17/6	3.458	11.913
30/13	3.503	12.069
30/17	3.533	12.171
25/6	3.557	12.256
32/21	3.597	12.393
6/5	3.632	12.515
6/1	3.708	12.774
32/29	3.726	12.839
28/19	3.741	12.887
30/1	3.783	13.032
32/27	3.886	13.389
31/4	4.000	13.781
31/20	4.075	14.039
29/13	4.145	14.280
29/17	4.174	14.382
29/25	4.274	14.726
23/6	4.329	14.914
12/7	4.333	14.928
29/5	4.349	14.985
16/15	4.368	15.050
30/23	4.404	15.172
29/1	4.424	15.243
16/3	4.443	15.309
11/6	4.467	15.391
22/15	4.523	15.583
30/11	4.542	15.649
20/3	4.547	15.666
22/3	4.598	15.842
4/3	4.622	15.924
29/4	4.641	15.990
15/4	4.697	16.183
29/20	4.716	16.248
31/13	4.786	16.489
31/17	4.816	16.591
28/27	4.911	16.920
31/25	4.915	16.935
31/5	4.990	17.194
29/23	5.046	17.383
31/1	5.066	17.452
27/14	5.069	17.463
29/11	5.184	17.860
15/2	5.283	18.201
29/8	5.339	18.394
3/2	5.358	18.459
10/3	5.433	18.718
12/11	5.513	18.993
32/3	5.536	19.075
26/15	5.562	19.164
32/15	5.612	19.334
26/3	5.637	19.422
7/6	5.647	19.456
23/12	5.651	19.470
31/23	5.687	19.592
30/7	5.722	19.714
31/19	5.801	19.986
31/11	5.825	20.069
31/8	5.980	20.603
29/9	6.075	20.929
27/16	6.094	20.994
19/14	6.239	21.496
27/22	6.248	21.528
12/1	6.272	21.610
12/5	6.347	21.869
29/7	6.364	21.925
21/16	6.383	21.991
25/12	6.422	22.127
29/19	6.442	22.195
17/12	6.522	22.471
19/18	6.528	22.492
22/21	6.538	22.524
13/12	6.552	22.573
28/3	6.561	22.606
28/15	6.637	22.865
31/21	6.682	23.023
31/9	6.716	23.138
28/13	6.846	23.588
28/17	6.876	23.690
31/27	6.972	24.019
28/25	6.976	24.034
31/7	7.005	24.134
27/2	7.008	24.145
28/5	7.051	24.292
27/10	7.083	24.404
30/19	7.084	24.406
28/1	7.126	24.551
19/6	7.159	24.665
19/16	7.264	25.027
27/26	7.288	25.108
21/2	7.297	25.141
29/21	7.324	25.232
21/10	7.372	25.400
22/19	7.419	25.561
26/21	7.577	26.104
29/27	7.613	26.228
31/24	7.707	26.554
28/23	7.747	26.691
26/7	7.761	26.739
32/13	7.871	27.119
28/11	7.886	27.168
32/17	7.901	27.221
10/7	7.965	27.443
32/25	8.001	27.565
7/2	8.040	27.702
26/9	8.050	27.735
32/5	8.076	27.824
32/1	8.151	28.082
19/2	8.179	28.178
19/10	8.254	28.437
10/9	8.254	28.439
9/2	8.329	28.698
29/24	8.348	28.763
26/19	8.458	29.141
31/3	8.622	29.705
31/15	8.697	29.964
32/23	8.772	30.222
28/9	8.776	30.237
13/4	8.786	30.270
22/7	8.800	30.319
17/4	8.815	30.372
20/13	8.861	30.529
20/17	8.891	30.631
32/11	8.910	30.699
25/4	8.915	30.716
26/11	8.941	30.803
16/7	8.955	30.852
5/4	8.990	30.974
4/1	9.065	31.233
26/23	9.079	31.281
22/9	9.089	31.315
20/1	9.140	31.492
11/10	9.145	31.508
11/2	9.220	31.766
16/9	9.244	31.848
29/3	9.263	31.914
23/10	9.284	31.985
29/15	9.338	32.173
23/2	9.359	32.243
23/4	9.686	33.373
18/7	9.691	33.387
26/1	9.700	33.421
23/20	9.762	33.632
26/5	9.775	33.679
32/9	9.801	33.768
11/4	9.825	33.850
26/25	9.850	33.938
20/11	9.900	34.109
10/1	9.905	34.125
26/17	9.950	34.282
2/1	9.980	34.384
5/2	10.055	34.642
32/7	10.090	34.764
23/22	10.118	34.861
25/2	10.130	34.901
16/11	10.135	34.917
17/10	10.155	34.986
13/10	10.184	35.088
17/2	10.230	35.244
13/2	10.259	35.346
14/9	10.269	35.380
23/16	10.273	35.394
19/13	10.587	36.475
19/17	10.617	36.577
29/12	10.697	36.853
9/4	10.716	36.919
25/19	10.716	36.921
22/1	10.739	37.001
20/9	10.791	37.177
19/5	10.791	37.180
22/5	10.814	37.259
19/1	10.866	37.438
18/11	10.870	37.452
25/22	10.890	37.518
16/1	10.894	37.534
16/5	10.969	37.793
22/17	10.989	37.862
7/4	11.005	37.915
23/18	11.009	37.929
22/13	11.019	37.964
25/16	11.044	38.052
20/7	11.080	38.174
17/16	11.144	38.395
14/11	11.160	38.448
16/13	11.174	38.497
23/14	11.298	38.925
31/12	11.338	39.062
21/13	11.468	39.512
23/19	11.488	39.579
21/17	11.498	39.614
25/21	11.598	39.958
19/11	11.626	40.056
18/1	11.630	40.069
21/5	11.673	40.216
18/5	11.705	40.328
21/1	11.748	40.475
27/13	11.758	40.508
25/18	11.780	40.587
19/8	11.781	40.589
27/17	11.787	40.610
18/17	11.880	40.930
19/12	11.886	40.952
27/25	11.887	40.954
18/13	11.910	41.032
14/1	11.919	41.066
27/5	11.962	41.213
14/5	11.994	41.324
27/1	12.037	41.471
31/14	12.040	41.482
25/14	12.069	41.583
17/14	12.169	41.926
14/13	12.199	42.028
31/18	12.329	42.478
23/21	12.369	42.615
24/13	12.493	43.044
21/11	12.507	43.092
19/9	12.517	43.124
24/17	12.523	43.146
25/24	12.623	43.489
27/23	12.658	43.611
21/8	12.662	43.626
29/14	12.681	43.691
24/5	12.698	43.748
24/1	12.773	44.006
27/11	12.797	44.089
19/7	12.806	44.120
27/8	12.951	44.622
29/18	12.970	44.687
31/16	13.065	45.014
31/22	13.220	45.547
15/7	13.323	45.902
24/23	13.394	46.147
7/3	13.398	46.161
13/3	13.408	46.194
17/3	13.437	46.296
15/13	13.483	46.453
17/15	13.513	46.555
24/11	13.532	46.624
25/3	13.537	46.640
5/3	13.612	46.898
3/1	13.687	47.157
29/16	13.706	47.223
15/1	13.762	47.416
29/22	13.861	47.756
27/7	13.976	48.153
31/2	13.980	48.164
31/10	14.055	48.423
29/26	14.125	48.664
31/26	14.259	49.127
23/3	14.308	49.297
24/7	14.313	49.312
29/10	14.329	49.368
15/8	14.348	49.434
23/15	14.384	49.556
29/2	14.404	49.627
8/3	14.423	49.692
11/3	14.447	49.774
15/11	14.503	49.967

186zpi: Difference between revisions

Revision as of 14:56, 10 August 2024

Contents

Theory

Record on the Riemann zeta function with primes 2 and 3 removed

Harmonic series

Approximation of EDONOIs

Intervals and notation

Approximation to JI

Navigation menu

@@ Line 129: / Line 129: @@
 ! Ratios
 ! Ups and Downs Notation
+! Step
 |-
 | 0
@@ Line 134: / Line 135: @@
 |
 | P1
+| 0
 |-
 | 1
@@ Line 139: / Line 141: @@
 |
 | ^^^1
+| 3
 |-
 | 2
@@ Line 144: / Line 147: @@
 | '''[[32/31]]''', '''<u>[[31/30]]'''</u>, '''<u>[[30/29]]'''</u>, '''[[29/28]]''', [[28/27]], <small>[[27/26]]</small>, <small><small>[[26/25]]</small></small>, <small><small><small>[[25/24]]</small></small></small>
 | vvA1, ^^d2
+| 6
 |-
 | 3
@@ Line 149: / Line 153: @@
 | <small><small><small>[[24/23]]</small></small></small>, <small><small>[[23/22]]</small></small>, [[22/21]], '''[[21/20]]''', '''<u>[[20/19]]'''</u>, [[19/18]], <small><small>[[18/17]]</small></small>
 | vvvm2
+| 9
 |-
 | 4
@@ Line 154: / Line 159: @@
 | <small><small>[[17/16]]</small></small>, '''[[16/15]]''', '''<u>[[31/29]]'''</u>, '''[[15/14]]''', <small>[[29/27]]</small>, <small><small><small>[[14/13]]</small></small></small>
 | m2
+| 12
 |-
 | 5
@@ Line 159: / Line 165: @@
 | <small><small>[[27/25]]</small></small>, [[13/12]], '''<u>[[25/23]]'''</u>, [[12/11]], <small><small><small>[[23/21]]</small></small></small>
 | ^^^m2
+| 15
 |-
 | 6
@@ Line 164: / Line 171: @@
 | <small>[[11/10]]</small>, '''[[32/29]]''', '''<u>[[21/19]]'''</u>, '''<u>[[31/28]]'''</u>, <small>[[10/9]]</small>
 | vvM2
+| 18
 |-
 | 7
@@ Line 169: / Line 177: @@
 | <small><small><small>[[29/26]]</small></small></small>, <small><small>[[19/17]]</small></small>, [[28/25]], '''<u>[[9/8]]'''</u>, <small>[[26/23]]</small>, <small><small><small>[[17/15]]</small></small></small>
 | ^M2
+| 21
 |-
 | 8
@@ Line 174: / Line 183: @@
 | <small><small>[[25/22]]</small></small>, '''<u>[[8/7]]'''</u>, [[31/27]], <small><small>[[23/20]]</small></small>
 | ^<sup>4</sup>M2
+| 24
 |-
 | 9
@@ Line 179: / Line 189: @@
 | <small><small><small>[[15/13]]</small></small></small>, <small>[[22/19]]</small>, '''[[29/25]]''', [[7/6]]
 | ^^^d3
+| 27
 |-
 | 10
@@ Line 184: / Line 195: @@
 | <small><small><small>[[27/23]]</small></small></small>, <small>[[20/17]]</small>, '''<u>[[13/11]]'''</u>, '''[[32/27]]''', <small>[[19/16]]</small>, <small><small>[[25/21]]</small></small>, <small><small><small>[[31/26]]</small></small></small>
 | vvm3
+| 30
 |-
 | 11
@@ Line 189: / Line 201: @@
 | '''[[6/5]]''', <small>[[29/24]]</small>, <small><small>[[23/19]]</small></small>
 | ^m3
+| 33
 |-
 | 12
@@ Line 194: / Line 207: @@
 | <small><small><small>[[17/14]]</small></small></small>, <small>[[28/23]]</small>, '''<u>[[11/9]]'''</u>, [[27/22]], <small><small>[[16/13]]</small></small>
 | ~3
+| 36
 |-
 | 13
@@ Line 199: / Line 213: @@
 | <small><small>[[21/17]]</small></small>, <small>[[26/21]]</small>, [[31/25]], <small>[[5/4]]</small>
 | vM3
+| 39
 |-
 | 14
@@ Line 204: / Line 219: @@
 | [[29/23]], '''<u>[[24/19]]'''</u>, '''[[19/15]]''', <small><small>[[14/11]]</small></small>
 | ^^M3
+| 42
 |-
 | 15
@@ Line 209: / Line 225: @@
 | <small><small>[[23/18]]</small></small>, <small>[[32/25]]</small>, '''<u>[[9/7]]'''</u>, <small>[[31/24]]</small>, <small><small>[[22/17]]</small></small>
 | vvvA3
+| 45
 |-
 | 16
@@ Line 214: / Line 231: @@
 | <small><small>[[13/10]]</small></small>, '''[[30/23]]''', '''<u>[[17/13]]'''</u>, [[21/16]], <small><small>[[25/19]]</small></small>, <small><small><small>[[29/22]]</small></small></small>
 | v<sup>4</sup>4
+| 48
 |-
 | 17
@@ Line 219: / Line 237: @@
 | '''[[4/3]]'''
 | v4
+| 51
 |-
 | 18
@@ Line 224: / Line 243: @@
 | [[31/23]], '''[[27/20]]''', '''<u>[[23/17]]'''</u>, [[19/14]], <small><small><small>[[15/11]]</small></small></small>
 | ^^4
+| 54
 |-
 | 19
@@ Line 229: / Line 249: @@
 | <small>[[26/19]]</small>, '''<u>[[11/8]]'''</u>, <small>[[29/21]]</small>, <small><small>[[18/13]]</small></small>
 | vvvA4
+| 57
 |-
 | 20
@@ Line 234: / Line 255: @@
 | <small><small>[[25/18]]</small></small>, <small>[[32/23]]</small>, '''<u>[[7/5]]'''</u>, <small><small><small>[[31/22]]</small></small></small>
 | A4
+| 60
 |-
 | 21
@@ Line 239: / Line 261: @@
 | <small><small><small>[[24/17]]</small></small></small>, [[17/12]], '''<u>[[27/19]]'''</u>, <small>[[10/7]]</small>
 | vd5
+| 63
 |-
 | 22
@@ Line 244: / Line 267: @@
 | <small><small>[[23/16]]</small></small>, '''<u>[[13/9]]'''</u>, '''[[29/20]]''', <small><small>[[16/11]]</small></small>
 | ^^d5
+| 66
 |-
 | 23
@@ Line 249: / Line 273: @@
 | <small><small>[[19/13]]</small></small>, '''[[22/15]]''', '''<u>[[25/17]]'''</u>, '''[[28/19]]''', [[31/21]]
 | vvv5
+| 69
 |-
 | 24
@@ Line 254: / Line 279: @@
 | [[3/2]]
 | P5
+| 72
 |-
 | 25
@@ Line 259: / Line 285: @@
 | '''[[32/21]]''', [[29/19]], <small><small>[[26/17]]</small></small>, <small><small><small>[[23/15]]</small></small></small>
 | ^^^5
+| 75
 |-
 | 26
@@ Line 264: / Line 291: @@
 | <small>[[20/13]]</small>, '''<u>[[17/11]]'''</u>, '''[[31/20]]''', <small><small>[[14/9]]</small></small>
 | vvA5, ^^d6
+| 78
 |-
 | 27
@@ Line 269: / Line 297: @@
 | <small><small>[[25/16]]</small></small>, '''<u>[[11/7]]'''</u>, [[30/19]], <small><small>[[19/12]]</small></small>
 | vvvm6
+| 81
 |-
 | 28
@@ Line 274: / Line 303: @@
 | <small><small>[[27/17]]</small></small>, '''<u>[[8/5]]'''</u>, <small><small><small>[[29/18]]</small></small></small>
 | m6
+| 84
 |-
 | 29
@@ Line 279: / Line 309: @@
 | <small><small>[[21/13]]</small></small>, '''<u>[[13/8]]'''</u>, [[31/19]], <small><small>[[18/11]]</small></small>
 | ^^^m6
+| 87
 |-
 | 30
@@ Line 284: / Line 315: @@
 | <small><small>[[23/14]]</small></small>, [[28/17]], <small><small><small>[[5/3]]</small></small></small>
 | vvM6
+| 90
 |-
 | 31
@@ Line 289: / Line 321: @@
 | '''[[32/19]]''', [[27/16]], <small><small>[[22/13]]</small></small>
 | ^M6
+| 93
 |-
 | 32
@@ Line 294: / Line 327: @@
 | <small><small>[[17/10]]</small></small>, '''[[29/17]]''', '''[[12/7]]''', <small><small><small>[[31/18]]</small></small></small>
 | ^<sup>4</sup>M6
+| 96
 |-
 | 33
@@ Line 299: / Line 333: @@
 | <small><small>[[19/11]]</small></small>, [[26/15]], <small><small>[[7/4]]</small></small>
 | ^^^d7
+| 99
 |-
 | 34
@@ Line 304: / Line 339: @@
 | '''[[30/17]]''', '''<u>[[23/13]]'''</u>, <small>[[16/9]]</small>
 | vvm7
+| 102
 |-
 | 35
@@ Line 309: / Line 345: @@
 | <small><small>[[25/14]]</small></small>, '''<u>[[9/5]]'''</u>, <small><small><small>[[29/16]]</small></small></small>
 | ^m7
+| 105
 |-
 | 36
@@ Line 314: / Line 351: @@
 | <small><small>[[20/11]]</small></small>, '''[[31/17]]''', '''[[11/6]]'''
 | ~7
+| 108
 |-
 | 37
@@ Line 319: / Line 357: @@
 | <small><small><small>[[24/13]]</small></small></small>, '''<u>[[13/7]]'''</u>, [[28/15]], <small><small><small>[[15/8]]</small></small></small>
 | vM7
+| 111
 |-
 | 38
@@ Line 324: / Line 363: @@
 | <small>[[32/17]]</small>, '''<u>[[17/9]]'''</u>, <small>[[19/10]]</small>
 | ^^M7
+| 114
 |-
 | 39
@@ Line 329: / Line 369: @@
 | <small><small><small>[[21/11]]</small></small></small>, [[23/12]], '''<u>[[25/13]]'''</u>, [[27/14]], <small>[[29/15]]</small>, <small><small><small>[[31/16]]</small></small></small>
 | vvvA7
+| 117
 |-
 | 40
@@ Line 334: / Line 375: @@
 |
 | v<sup>4</sup>1 +1oct
+| 120
 |-
 | 41
@@ Line 339: / Line 381: @@
 | <small><small>[[2/1]]</small></small>
 | v1 +1oct
+| 123
 |-
 | 42
@@ Line 344: / Line 387: @@
 |
 | ^^1 +1oct
+| 126
 |-
 | 43
@@ Line 349: / Line 393: @@
 | <small>[[31/15]]</small>, <small><small><small>[[29/14]]</small></small></small>
 | vvvA1 +1oct
+| 129
 |-
 | 44
@@ Line 354: / Line 399: @@
 | <small><small>[[27/13]]</small></small>, [[25/12]], '''<u>[[23/11]]'''</u>, <small>[[21/10]]</small>
 | v<sup>4</sup>m2 +1oct
+| 132
 |-
 | 45
@@ Line 359: / Line 405: @@
 | <small><small><small>[[19/9]]</small></small></small>, '''<u>[[17/8]]'''</u>, [[32/15]], <small><small><small>[[15/7]]</small></small></small>
 | vm2 +1oct
+| 135
 |-
 | 46
@@ Line 364: / Line 411: @@
 | [[28/13]], '''[[13/6]]'''
 | ^^m2 +1oct
+| 138
 |-
 | 47
@@ Line 369: / Line 417: @@
 | <small><small><small>[[24/11]]</small></small></small>, '''<u>[[11/5]]'''</u>, <small><small>[[31/14]]</small></small>
 | vvvM2 +1oct
+| 141
 |-
 | 48
@@ Line 374: / Line 423: @@
 | <small><small>[[20/9]]</small></small>, '''[[29/13]]''', <small><small>[[9/4]]</small></small>
 | M2 +1oct
+| 144
 |-
 | 49
@@ Line 379: / Line 429: @@
 | '''<u>[[25/11]]'''</u>, <small>[[16/7]]</small>
 | ^^^M2 +1oct
+| 147
 |-
 | 50
@@ Line 384: / Line 435: @@
 | <small>[[23/10]]</small>, '''[[30/13]]'''
 | vvA2 +1oct, ^^d3 +1oct
+| 150
 |-
 | 51
@@ Line 389: / Line 441: @@
 | <small><small><small>[[7/3]]</small></small></small>, <small>[[26/11]]</small>
 | vvvm3 +1oct
+| 153
 |-
 | 52
@@ Line 394: / Line 447: @@
 | <small><small>[[19/8]]</small></small>, '''[[31/13]]''', [[12/5]]
 | m3 +1oct
+| 156
 |-
 | 53
@@ Line 399: / Line 453: @@
 | <small><small>[[29/12]]</small></small>, '''<u>[[17/7]]'''</u>, <small>[[22/9]]</small>
 | ^^^m3 +1oct
+| 159
 |-
 | 54
@@ Line 404: / Line 459: @@
 | <small><small><small>[[27/11]]</small></small></small>, <small>[[32/13]]</small>
 | vvM3 +1oct
+| 162
 |-
 | 55
@@ Line 409: / Line 465: @@
 | <small><small>[[5/2]]</small></small>
 | ^M3 +1oct
+| 165
 |-
 | 56
@@ Line 414: / Line 471: @@
 | <small>[[28/11]]</small>, '''<u>[[23/9]]'''</u>, <small><small>[[18/7]]</small></small>
 | ^<sup>4</sup>M3 +1oct
+| 168
 |-
 | 57
@@ Line 419: / Line 477: @@
 | <small><small>[[31/12]]</small></small>, '''<u>[[13/5]]'''</u>
 | ^^^d4 +1oct
+| 171
 |-
 | 58
@@ Line 424: / Line 483: @@
 | <small><small><small>[[21/8]]</small></small></small>, [[29/11]]
 | vv4 +1oct
+| 174
 |-
 | 59
@@ Line 429: / Line 489: @@
 | <small><small><small>[[8/3]]</small></small></small>, [[27/10]]
 | ^4 +1oct
+| 177
 |-
 | 60
@@ Line 434: / Line 495: @@
 | <small><small><small>[[19/7]]</small></small></small>, '''[[30/11]]''', <small><small>[[11/4]]</small></small>
 | ~4 +1oct
+| 180
 |-
 | 61
@@ Line 439: / Line 501: @@
 | '''<u>[[25/9]]'''</u>, <small><small>[[14/5]]</small></small>
 | vA4 +1oct
+| 183
 |-
 | 62
@@ Line 444: / Line 507: @@
 | [[31/11]], '''[[17/6]]'''
 | ^^A4 +1oct, vvd5 +1oct
+| 186
 |-
 | 63
@@ Line 449: / Line 513: @@
 | <small><small>[[20/7]]</small></small>, '''<u>[[23/8]]'''</u>, <small>[[26/9]]</small>
 | ^d5 +1oct
+| 189
 |-
 | 64
@@ Line 454: / Line 519: @@
 | <small><small><small>[[29/10]]</small></small></small>, <small>[[32/11]]</small>
 | ~5 +1oct
+| 192
 |-
 | 65
@@ Line 459: / Line 525: @@
 |
 | v5 +1oct
+| 195
 |-
 | 66
@@ Line 464: / Line 531: @@
 | <small><small><small>[[3/1]]</small></small></small>
 | ^^5 +1oct
+| 198
 |-
 | 67
@@ Line 469: / Line 537: @@
 | <small><small><small>[[31/10]]</small></small></small>
 | vvvA5 +1oct
+| 201
 |-
 | 68
@@ Line 474: / Line 543: @@
 | <small>[[28/9]]</small>, '''<u>[[25/8]]'''</u>, <small>[[22/7]]</small>
 | v<sup>4</sup>m6 +1oct
+| 204
 |-
 | 69
@@ Line 479: / Line 549: @@
 | [[19/6]], <small><small>[[16/5]]</small></small>
 | vm6 +1oct
+| 207
 |-
 | 70
@@ Line 484: / Line 555: @@
 | [[29/9]], <small>[[13/4]]</small>
 | ^^m6 +1oct
+| 210
 |-
 | 71
@@ Line 489: / Line 561: @@
 | '''<u>[[23/7]]'''</u>
 | vvvM6 +1oct
+| 213
 |-
 | 72
@@ Line 494: / Line 567: @@
 | [[10/3]]
 | M6 +1oct
+| 216
 |-
 | 73
@@ Line 499: / Line 573: @@
 | <small><small><small>[[27/8]]</small></small></small>, '''<u>[[17/5]]'''</u>, <small><small><small>[[24/7]]</small></small></small>
 | ^^^M6 +1oct
+| 219
 |-
 | 74
@@ Line 504: / Line 579: @@
 | [[31/9]]
 | vvA6 +1oct, ^^d7 +1oct
+| 222
 |-
 | 75
@@ Line 509: / Line 585: @@
 | <small>[[7/2]]</small>
 | vvvm7 +1oct
+| 225
 |-
 | 76
@@ Line 514: / Line 591: @@
 | <small><small>[[32/9]]</small></small>, '''<u>[[25/7]]'''</u>, <small><small>[[18/5]]</small></small>
 | m7 +1oct
+| 228
 |-
 | 77
@@ Line 519: / Line 597: @@
 | [[29/8]], <small><small><small>[[11/3]]</small></small></small>
 | ^^^m7 +1oct
+| 231
 |-
 | 78
@@ Line 524: / Line 603: @@
 | <small>[[26/7]]</small>
 | vvM7 +1oct
+| 234
 |-
 | 79
@@ Line 529: / Line 609: @@
 | '''[[15/4]]'''
 | ^M7 +1oct
+| 237
 |-
 | 80
@@ Line 534: / Line 615: @@
 | <small><small>[[19/5]]</small></small>, '''[[23/6]]'''
 | ^<sup>4</sup>M7 +1oct
+| 240
 |-
 | 81
@@ Line 539: / Line 621: @@
 | <small><small><small>[[27/7]]</small></small></small>, [[31/8]]
 | ^^^d1 +2oct
+| 243
 |-
 | 82
@@ Line 544: / Line 627: @@
 |
 | vv1 +2oct
+| 246
 |-
 | 83
@@ Line 549: / Line 633: @@
 | <small>[[4/1]]</small>
 | ^1 +2oct
+| 249
 |-
 | 84
@@ Line 554: / Line 639: @@
 |
 | ^<sup>4</sup>1 +2oct
+| 252
 |-
 | 85
@@ Line 559: / Line 645: @@
 | [[29/7]], '''[[25/6]]'''
 | ^^^d2 +2oct
+| 255
 |-
 | 86
@@ Line 564: / Line 651: @@
 | <small><small>[[21/5]]</small></small>, <small>[[17/4]]</small>
 | vvm2 +2oct
+| 258
 |-
 | 87
@@ Line 569: / Line 657: @@
 | [[30/7]], <small><small><small>[[13/3]]</small></small></small>
 | ^m2 +2oct
+| 261
 |-
 | 88
@@ Line 574: / Line 663: @@
 | <small><small>[[22/5]]</small></small>
 | ~2 +2oct
+| 264
 |-
 | 89
@@ Line 579: / Line 669: @@
 | [[31/7]]
 | vM2 +2oct
+| 267
 |-
 | 90
@@ Line 584: / Line 675: @@
 | <small>[[9/2]]</small>
 | ^^M2 +2oct
+| 270
 |-
 | 91
@@ Line 589: / Line 681: @@
 | <small><small>[[32/7]]</small></small>, '''<u>[[23/5]]'''</u>
 | vvvA2 +2oct
+| 273
 |-
 | 92
@@ Line 594: / Line 687: @@
 | '''[[14/3]]'''
 | v<sup>4</sup>m3 +2oct
+| 276
 |-
 | 93
@@ Line 599: / Line 693: @@
 | '''<u>[[19/4]]'''</u>
 | vm3 +2oct
+| 279
 |-
 | 94
@@ Line 604: / Line 699: @@
 | <small><small><small>[[24/5]]</small></small></small>, '''<u>[[29/6]]'''</u>
 | ^^m3 +2oct
+| 282
 |-
 | 95
@@ Line 609: / Line 705: @@
 |
 | vvvM3 +2oct
+| 285
 |-
 | 96
@@ Line 614: / Line 711: @@
 | '''<u>[[5/1]]'''</u>
 | M3 +2oct
+| 288
 |-
 | 97
@@ Line 619: / Line 717: @@
 |
 | ^^^M3 +2oct
+| 291
 |-
 | 98
@@ Line 624: / Line 723: @@
 | '''<u>[[31/6]]'''</u>, <small><small>[[26/5]]</small></small>
 | vvA3 +2oct, ^^d4 +2oct
+| 294
 |-
 | 99
@@ Line 629: / Line 729: @@
 | '''[[21/4]]'''
 | vvv4 +2oct
+| 297
 |-
 | 100
@@ Line 634: / Line 735: @@
 | '''[[16/3]]'''
 | P4 +2oct
+| 300
 |-
 | 101
@@ Line 639: / Line 741: @@
 | <small><small>[[27/5]]</small></small>
 | ^^^4 +2oct
+| 303
 |-
 | 102
@@ Line 644: / Line 747: @@
 | <small>[[11/2]]</small>
 | vvA4 +2oct
+| 306
 |-
 | 103
@@ Line 649: / Line 753: @@
 | [[28/5]], <small><small><small>[[17/3]]</small></small></small>
 | ^A4 +2oct
+| 309
 |-
 | 104
@@ Line 654: / Line 759: @@
 | <small><small>[[23/4]]</small></small>
 | d5 +2oct
+| 312
 |-
 | 105
@@ Line 659: / Line 765: @@
 | '''[[29/5]]'''
 | ^^^d5 +2oct
+| 315
 |-
 | 106
@@ Line 664: / Line 771: @@
 |
 | vv5 +2oct
+| 318
 |-
 | 107
@@ Line 669: / Line 777: @@
 | '''[[6/1]]'''
 | ^5 +2oct
+| 321
 |-
 | 108
@@ Line 674: / Line 783: @@
 |
 | ^<sup>4</sup>5 +2oct
+| 324
 |-
 | 109
@@ Line 679: / Line 789: @@
 | [[31/5]], <small>[[25/4]]</small>
 | ^^^d6 +2oct
+| 327
 |-
 | 110
@@ Line 684: / Line 795: @@
 | '''[[19/3]]'''
 | vvm6 +2oct
+| 330
 |-
 | 111
@@ Line 689: / Line 801: @@
 | <small>[[32/5]]</small>
 | ^m6 +2oct
+| 333
 |-
 | 112
@@ Line 694: / Line 807: @@
 | <small><small>[[13/2]]</small></small>
 | ~6 +2oct
+| 336
 |-
 | 113
@@ Line 699: / Line 813: @@
 | '''[[20/3]]'''
 | vM6 +2oct
+| 339
 |-
 | 114
@@ Line 704: / Line 819: @@
 | '''[[27/4]]'''
 | ^^M6 +2oct
+| 342
 |-
 | 115
@@ Line 709: / Line 825: @@
 |
 | vvvA6 +2oct
+| 345
 |-
 | 116
@@ Line 714: / Line 831: @@
 | '''<u>[[7/1]]'''</u>
 | v<sup>4</sup>m7 +2oct
+| 348
 |-
 | 117
@@ Line 719: / Line 837: @@
 |
 | vm7 +2oct
+| 351
 |-
 | 118
@@ Line 724: / Line 843: @@
 | '''[[29/4]]'''
 | ^^m7 +2oct
+| 354
 |-
 | 119
@@ Line 729: / Line 849: @@
 | '''[[22/3]]'''
 | vvvM7 +2oct
+| 357
 |-
 | 120
@@ Line 734: / Line 855: @@
 | [[15/2]]
 | M7 +2oct
+| 360
 |-
 | 121
@@ Line 739: / Line 861: @@
 | <small><small><small>[[23/3]]</small></small></small>
 | ^^^M7 +2oct
+| 363
 |-
 | 122
@@ Line 744: / Line 867: @@
 | '''[[31/4]]'''
 | vvA7 +2oct, ^^d1 +3oct
+| 366
 |-
 | 123
@@ Line 749: / Line 873: @@
 |
 | vvv1 +3oct
+| 369
 |-
 | 124
@@ Line 754: / Line 879: @@
 | '''<u>[[8/1]]'''</u>
 | P1 +3oct
+| 372
 |-
 | 125
@@ Line 759: / Line 885: @@
 |
 | ^^^1 +3oct
+| 375
 |-
 | 126
@@ Line 764: / Line 891: @@
 | <small><small><small>[[25/3]]</small></small></small>
 | vvA1 +3oct, ^^d2 +3oct
+| 378
 |-
 | 127
@@ Line 769: / Line 897: @@
 |
 | vvvm2 +3oct
+| 381
 |-
 | 128
@@ Line 774: / Line 903: @@
 | <small><small>[[17/2]]</small></small>
 | m2 +3oct
+| 384
 |-
 | 129
@@ Line 779: / Line 909: @@
 | [[26/3]]
 | ^^^m2 +3oct
+| 387
 |-
 | 130
@@ Line 784: / Line 915: @@
 |
 | vvM2 +3oct
+| 390
 |-
 | 131
@@ Line 789: / Line 921: @@
 | '''<u>[[9/1]]'''</u>
 | ^M2 +3oct
+| 393
 |-
 | 132
@@ Line 794: / Line 927: @@
 |
 | ^<sup>4</sup>M2 +3oct
+| 396
 |-
 | 133
@@ Line 799: / Line 933: @@
 | [[28/3]]
 | ^^^d3 +3oct
+| 399
 |-
 | 134
@@ Line 804: / Line 939: @@
 | <small>[[19/2]]</small>
 | vvm3 +3oct
+| 402
 |-
 | 135
@@ Line 809: / Line 945: @@
 | <small>[[29/3]]</small>
 | ^m3 +3oct
+| 405
 |-
 | 136
@@ Line 814: / Line 951: @@
 |
 | ~3 +3oct
+| 408
 |-
 | 137
@@ Line 819: / Line 957: @@
 | <small><small>[[10/1]]</small></small>
 | vM3 +3oct
+| 411
 |-
 | 138
@@ Line 824: / Line 963: @@
 |
 | ^^M3 +3oct
+| 414
 |-
 | 139
@@ Line 829: / Line 969: @@
 | <small>[[31/3]]</small>
 | vvvA3 +3oct
+| 417
 |-
 | 140
@@ Line 834: / Line 975: @@
 | <small>[[21/2]]</small>
 | v<sup>4</sup>4 +3oct
+| 420
 |-
 | 141
@@ Line 839: / Line 981: @@
 | [[32/3]]
 | v4 +3oct
+| 423
 |-
 | 142
@@ Line 844: / Line 987: @@
 |
 | ^^4 +3oct
+| 426
 |-
 | 143
@@ Line 849: / Line 993: @@
 | '''<u>[[11/1]]'''</u>
 | vvvA4 +3oct
+| 429
 |-
 | 144
@@ Line 854: / Line 999: @@
 |
 | A4 +3oct
+| 432
 |-
 | 145
@@ Line 859: / Line 1,005: @@
 |
 | vd5 +3oct
+| 435
 |-
 | 146
@@ Line 864: / Line 1,011: @@
 | <small>[[23/2]]</small>
 | ^^d5 +3oct
+| 438
 |-
 | 147
@@ Line 869: / Line 1,017: @@
 |
 | vvv5 +3oct
+| 441
 |-
 | 148
@@ Line 874: / Line 1,023: @@
 | [[12/1]]
 | P5 +3oct
+| 444
 |-
 | 149
@@ Line 879: / Line 1,029: @@
 |
 | ^^^5 +3oct
+| 447
 |-
 | 150
@@ Line 884: / Line 1,035: @@
 |
 | vvA5 +3oct, ^^d6 +3oct
+| 450
 |-
 | 151
@@ Line 889: / Line 1,041: @@
 | <small><small>[[25/2]]</small></small>
 | vvvm6 +3oct
+| 453
 |-
 | 152
@@ Line 894: / Line 1,047: @@
 |
 | m6 +3oct
+| 456
 |-
 | 153
@@ Line 899: / Line 1,053: @@
 | '''<u>[[13/1]]'''</u>
 | ^^^m6 +3oct
+| 459
 |-
 | 154
@@ Line 904: / Line 1,059: @@
 |
 | vvM6 +3oct
+| 462
 |-
 | 155
@@ Line 909: / Line 1,065: @@
 | [[27/2]]
 | ^M6 +3oct
+| 465
 |-
 | 156
@@ Line 914: / Line 1,071: @@
 |
 | ^<sup>4</sup>M6 +3oct
+| 468
 |-
 | 157
@@ Line 919: / Line 1,077: @@
 | <small><small>[[14/1]]</small></small>
 | ^^^d7 +3oct
+| 471
 |-
 | 158
@@ Line 924: / Line 1,083: @@
 |
 | vvm7 +3oct
+| 474
 |-
 | 159
@@ Line 929: / Line 1,089: @@
 |
 | ^m7 +3oct
+| 477
 |-
 | 160
@@ Line 934: / Line 1,095: @@
 | <small><small><small>[[29/2]]</small></small></small>
 | ~7 +3oct
+| 480
 |-
 | 161
@@ Line 939: / Line 1,101: @@
 |
 | vM7 +3oct
+| 483
 |-
 | 162
@@ Line 944: / Line 1,107: @@
 | <small><small><small>[[15/1]]</small></small></small>
 | ^^M7 +3oct
+| 486
 |-
 | 163
@@ Line 949: / Line 1,113: @@
 | <small><small><small>[[31/2]]</small></small></small>
 | vvvA7 +3oct
+| 489
 |-
 | 164
@@ Line 954: / Line 1,119: @@
 |
 | v<sup>4</sup>1 +4oct
+| 492
 |-
 | 165
@@ Line 959: / Line 1,125: @@
 | <small><small>[[16/1]]</small></small>
 | v1 +4oct
+| 495
 |-
 | 166
@@ Line 964: / Line 1,131: @@
 |
 | ^^1 +4oct
+| 498
 |-
 | 167
@@ Line 969: / Line 1,137: @@
 |
 | vvvA1 +4oct
+| 501
 |-
 | 168
@@ Line 974: / Line 1,143: @@
 |
 | v<sup>4</sup>m2 +4oct
+| 504
 |-
 | 169
@@ Line 979: / Line 1,149: @@
 | '''<u>[[17/1]]'''</u>
 | vm2 +4oct
+| 507
 |-
 | 170
@@ Line 984: / Line 1,155: @@
 |
 | ^^m2 +4oct
+| 510
 |-
 | 171
@@ Line 989: / Line 1,161: @@
 |
 | vvvM2 +4oct
+| 513
 |-
 | 172
@@ Line 994: / Line 1,167: @@
 | <small><small>[[18/1]]</small></small>
 | M2 +4oct
+| 516
 |-
 | 173
@@ Line 999: / Line 1,173: @@
 |
 | ^^^M2 +4oct
+| 519
 |-
 | 174
@@ Line 1,004: / Line 1,179: @@
 |
 | vvA2 +4oct, ^^d3 +4oct
+| 522
 |-
 | 175
@@ Line 1,009: / Line 1,185: @@
 |
 | vvvm3 +4oct
+| 525
 |-
 | 176
@@ Line 1,014: / Line 1,191: @@
 | <small><small>[[19/1]]</small></small>
 | m3 +4oct
+| 528
 |-
 | 177
@@ Line 1,019: / Line 1,197: @@
 |
 | ^^^m3 +4oct
+| 531
 |-
 | 178
@@ Line 1,024: / Line 1,203: @@
 |
 | vvM3 +4oct
+| 534
 |-
 | 179
@@ Line 1,029: / Line 1,209: @@
 | <small>[[20/1]]</small>
 | ^M3 +4oct
+| 537
 |-
 | 180
@@ Line 1,034: / Line 1,215: @@
 |
 | ^<sup>4</sup>M3 +4oct
+| 540
 |-
 | 181
@@ Line 1,039: / Line 1,221: @@
 |
 | ^^^d4 +4oct
+| 543
 |-
 | 182
@@ Line 1,044: / Line 1,227: @@
 | <small><small>[[21/1]]</small></small>
 | vv4 +4oct
+| 546
 |-
 | 183
@@ Line 1,049: / Line 1,233: @@
 |
 | ^4 +4oct
+| 549
 |-
 | 184
@@ Line 1,054: / Line 1,239: @@
 | <small><small>[[22/1]]</small></small>
 | ~4 +4oct
+| 552
 |-
 | 185
@@ Line 1,059: / Line 1,245: @@
 |
 | vA4 +4oct
+| 555
 |-
 | 186
@@ Line 1,064: / Line 1,251: @@
 |
 | ^^A4 +4oct, vvd5 +4oct
+| 558
 |-
 | 187
@@ Line 1,069: / Line 1,257: @@
 | '''<u>[[23/1]]'''</u>
 | ^d5 +4oct
+| 561
 |-
 | 188
@@ Line 1,074: / Line 1,263: @@
 |
 | ~5 +4oct
+| 564
 |-
 | 189
@@ Line 1,079: / Line 1,269: @@
 |
 | v5 +4oct
+| 567
 |-
 | 190
@@ Line 1,084: / Line 1,275: @@
 | <small><small><small>[[24/1]]</small></small></small>
 | ^^5 +4oct
+| 570
 |-
 | 191
@@ Line 1,089: / Line 1,281: @@
 |
 | vvvA5 +4oct
+| 573
 |-
 | 192
@@ Line 1,094: / Line 1,287: @@
 | '''<u>[[25/1]]'''</u>
 | v<sup>4</sup>m6 +4oct
+| 576
 |-
 | 193
@@ Line 1,099: / Line 1,293: @@
 |
 | vm6 +4oct
+| 579
 |-
 | 194
@@ Line 1,104: / Line 1,299: @@
 | <small><small>[[26/1]]</small></small>
 | ^^m6 +4oct
+| 582
 |-
 | 195
@@ Line 1,109: / Line 1,305: @@
 |
 | vvvM6 +4oct
+| 585
 |-
 | 196
@@ Line 1,114: / Line 1,311: @@
 |
 | M6 +4oct
+| 588
 |-
 | 197
@@ Line 1,119: / Line 1,317: @@
 | <small><small>[[27/1]]</small></small>
 | ^^^M6 +4oct
+| 591
 |-
 | 198
@@ Line 1,124: / Line 1,323: @@
 |
 | vvA6 +4oct, ^^d7 +4oct
+| 594
 |-
 | 199
@@ Line 1,129: / Line 1,329: @@
 | [[28/1]]
 | vvvm7 +4oct
+| 597
 |-
 | 200
@@ Line 1,134: / Line 1,335: @@
 |
 | m7 +4oct
+| 600
 |-
 | 201
@@ Line 1,139: / Line 1,341: @@
 | '''[[29/1]]'''
 | ^^^m7 +4oct
+| 603
 |-
 | 202
@@ Line 1,144: / Line 1,347: @@
 |
 | vvM7 +4oct
+| 606
 |-
 | 203
@@ Line 1,149: / Line 1,353: @@
 | '''[[30/1]]'''
 | ^M7 +4oct
+| 609
 |-
 | 204
@@ Line 1,154: / Line 1,359: @@
 |
 | ^<sup>4</sup>M7 +4oct
+| 612
 |-
 | 205
@@ Line 1,159: / Line 1,365: @@
 | [[31/1]]
 | ^^^d1 +5oct
+| 615
 |-
 | 206
@@ Line 1,164: / Line 1,371: @@
 |
 | vv1 +5oct
+| 618
 |-
 | 207
@@ Line 1,169: / Line 1,377: @@
 | <small>[[32/1]]</small>
 | ^1 +5oct
+| 621
 |}

186zpi: Difference between revisions

Revision as of 14:56, 10 August 2024

Theory

Record on the Riemann zeta function with primes 2 and 3 removed

Harmonic series

Approximation of EDONOIs

Intervals and notation

Approximation to JI

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