Multiplicative sequences
Case C=2
Any completely-multiplicative sequence of length [math]\displaystyle{ 247 }[/math] has discrepancy more than [math]\displaystyle{ 2 }[/math].
Data and plots
There are 500 sequences of length [math]\displaystyle{ 246 }[/math] with discrepancy [math]\displaystyle{ 2 }[/math], all of which agree at primes up to and including [math]\displaystyle{ 67 }[/math]. Here is one example:
0 + - - + - + - - + + + - - + + + - - + - + - - + + + - - + - + - - + + + - - + + - - - + - + - - + - + - + + - + - - + + + - - + + + - - + - - - + + - + - - + - + + - + + + - - - + + - - + - + - - + + - - + + - - + - + + + - - + + + - - + - + - + + - + - - + - + - - + + + - - - + + + - + - - - - + + + - - + - + + - - + + - - - + + - - + - + - + + - + - + + - + - - + + + - - + + - - - + - + - - + - + + - + + - - - + + - + + - + + - - - - + - + + + + - - - - + - + + + + - - - + + - - + - -
Here are the values this sequence takes at the first few primes. The primes up to 101 that go to -1 are 2, 3, 5, 7, 13, 17, 23, 37, 41, 43, 47, 67, 71, 83, 89, 97, 101. The primes less than 100 that go to 1 are 11, 19, 29, 31, 53, 59, 61, 73, 79.
The total number of such multiplicative sequences for each length can be generated with Alec's python script.
Here is a plot of this data, and a plot of the log of the data, and here are the precise numbers:
length number
2 2
3 3
4 3
5 4
6 4
7 7
8 7
9 6
10 6
11 10
12 10
13 15
14 15
15 14
16 14
17 21
18 21
19 34
20 34
21 24
22 24
23 38
24 38
25 28
26 28
27 23
28 23
29 34
30 34
31 54
32 54
33 37
34 37
35 28
36 28
37 40
38 40
39 31
40 31
41 48
42 48
43 72
44 72
45 57
46 57
47 89
48 89
49 81
50 81
51 62
52 62
53 92
54 92
55 55
56 55
57 44
58 44
59 68
60 68
61 111
62 111
63 83
64 83
65 71
66 71
67 113
68 113
69 97
70 97
71 157
72 157
73 240
74 240
75 175
76 175
77 125
78 125
79 185
80 185
81 178
82 178
83 286
84 286
85 212
86 212
87 178
88 178
89 276
90 276
91 163
92 163
93 138
94 138
95 119
96 119
97 176
98 176
99 129
100 129
101 198
102 198
103 315
104 315
105 277
106 277
107 426
108 426
109 656
110 656
111 485
112 485
113 846
114 846
115 502
116 502
117 256
118 256
119 198
120 198
121 112
122 112
123 82
124 82
125 82
126 82
127 100
128 100
129 84
130 84
131 134
132 134
133 56
134 56
135 44
136 44
137 61
138 61
139 105
140 105
141 84
142 84
143 72
144 72
145 55
146 55
147 48
148 48
149 72
150 72
151 120
152 120
153 72
154 72
155 72
156 72
157 132
158 132
159 112
160 112
161 112
162 112
163 184
164 184
165 164
166 164
167 246
168 246
169 234
170 234
171 168
172 168
173 246
174 246
175 246
176 246
177 246
178 246
179 408
180 408
181 624
182 624
183 414
184 414
185 384
186 384
187 286
188 286
189 286
190 286
191 304
192 304
193 392
194 392
195 362
196 362
197 468
198 468
199 812
200 812
201 776
202 776
203 626
204 626
205 386
206 386
207 386
208 386
209 386
210 386
211 694
212 694
213 573
214 573
215 471
216 471
217 279
218 279
219 259
220 259
221 259
222 259
223 354
224 354
225 125
226 125
227 125
228 125
229 250
230 250
231 250
232 250
233 375
234 375
235 250
236 250
237 250
238 250
239 500
240 500
241 750
242 750
243 500
244 500
245 500
246 500[math]\displaystyle{ Insert formula here }[/math]
247 0
And the same data in Mathematica format: {{1,1},{2,2},{3,3},{4,3},{5,4},{6,4},{7,7},{8,7},{9,6},{10,6},{11,10},{12,10},{13,15},{14,15},{15,14},{16,14},{17,21},{18,21},{19,34},{20,34},{21,24},{22,24},{23,38},{24,38},{25,28},{26,28},{27,23},{28,23},{29,34},{30,34},{31,54},{32,54},{33,37},{34,37},{35,28},{36,28},{37,40},{38,40},{39,31},{40,31},{41,48},{42,48},{43,72},{44,72},{45,57},{46,57},{47,89},{48,89},{49,81},{50,81},{51,62},{52,62},{53,92},{54,92},{55,55},{56,55},{57,44},{58,44},{59,68},{60,68},{61,111},{62,111},{63,83},{64,83},{65,71},{66,71},{67,113},{68,113},{69,97},{70,97},{71,157},{72,157},{73,240},{74,240},{75,175},{76,175},{77,125},{78,125},{79,185},{80,185},{81,178},{82,178},{83,286},{84,286},{85,212},{86,212},{87,178},{88,178},{89,276},{90,276},{91,163},{92,163},{93,138},{94,138},{95,119},{96,119},{97,176},{98,176},{99,129},{100,129},{101,198},{102,198},{103,315},{104,315},{105,277},{106,277},{107,426},{108,426},{109,656},{110,656},{111,485},{112,485},{113,846},{114,846},{115,502},{116,502},{117,256},{118,256},{119,198},{120,198},{121,112},{122,112},{123,82},{124,82},{125,82},{126,82},{127,100},{128,100},{129,84},{130,84},{131,134},{132,134},{133,56},{134,56},{135,44},{136,44},{137,61},{138,61},{139,105},{140,105},{141,84},{142,84},{143,72},{144,72},{145,55},{146,55},{147,48},{148,48},{149,72},{150,72},{151,120},{152,120},{153,72},{154,72},{155,72},{156,72},{157,132},{158,132},{159,112},{160,112},{161,112},{162,112},{163,184},{164,184},{165,164},{166,164},{167,246},{168,246},{169,234},{170,234},{171,168},{172,168},{173,246},{174,246},{175,246},{176,246},{177,246},{178,246},{179,408},{180,408},{181,624},{182,624},{183,414},{184,414},{185,384},{186,384},{187,286},{188,286},{189,286},{190,286},{191,304},{192,304},{193,392},{194,392},{195,362},{196,362},{197,468},{198,468},{199,812},{200,812},{201,776},{202,776},{203,626},{204,626},{205,386},{206,386},{207,386},{208,386},{209,386},{210,386},{211,694},{212,694},{213,573},{214,573},{215,471},{216,471},{217,279},{218,279},{219,259},{220,259},{221,259},{222,259},{223,354},{224,354},{225,125},{226,125},{227,125},{228,125},{229,250},{230,250},{231,250},{232,250},{233,375},{234,375},{235,250},{236,250},{237,250},{238,250},{239,500},{240,500},{241,750},{242,750},{243,500},{244,500},{245,500},{246,500},{247,0}}
Case C=3
The maximum length for [math]\displaystyle{ C=3 }[/math] is at least [math]\displaystyle{ 819 }[/math]. Here is an example of that length:
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+ +-++--+-++-----+-++-++--+-+++++--+-++--+--+-++--+--+-++-++-- +-++-----+-++-++--+-++-+++-+-+---+--+-++--+--+-++-+++-+-+-+- +--+-++--++-+-++-+---+-++--+--+-++-++-++-+--+++---++--+--+-+ +--+--+-++-+++-+-++--+--+--++----+-++-+++-+-++--+----++-++-- ++++-++--+-++--+--+-++-++--+-+--++--+-++--+---+++--+--+++--+ +--+++---+--+-++-+--+-+++-++--+-++---+-+-++--+--+-++-++--+++ ---+--+-+++-+--+--+-+++--+++--+-+--+-++++---++-++--+-++--+-- +-++-+---+-++-++--+-+++-+---++---+--+--++++--+-++--+-++----+ ++-+-++-++--+-++--+----++-++--+--+-++--+-+++-+--+-++--+-++-+ +-+---+-++--+--+++--+++-+-++-+---++++--++---++--+--+-++--++- -+++--+-++-+---+--+--++++-++-+-+----+-+++++--+-+--+---+++++- +--+-++-+----+++-++--+-++--+--+-+++--+-+--+++---+-++--+--+-+ +-++----++-+++---++-+++-+--+-+--++-++-+
Length 627:
+--+-++-++--+-++--+--+-++--+--+-++-++--+-++--+--+-++-++--+-+ +-++--+-++-----+-++-++--+-+++++--+-++--+--+-++--+--+-++-++-- +-++-----+-++-++--+-++-+++-+-+---+--+-++--+--+-++-+++-+-+-+- +--+-++--++-+-++-+---+-++--+--+-++-++-++-+--+++---++--+--+-+ +--+--+-++-+++-+-++--+--+--++----+-++-+++-+-++--+----++-++-- ++++-++--+-+---+--+-++-++--+-++-++--+-++--+---+++--+--+++--+ +--+++---+--+-++-+--+-+++-++--+-++---+-+-++--+--+-++-++--+++ ---+--+-+++-+--+--+-+++--+++--+-+--+-++++---++-++--+-++--+-- +-++-+---+-++-++--+-+++-+---++---+--+--++++--+-++--+-++----+ ++-+-++-++--+-++--+----++-++--+--+-++--+-+++-+--+-++--+-++-+ +-+---+-++--+--+++--+++-+++
General Case
Here is some data on how the discrepancy of completely-multiplicative sequences grows as a function of length, depending on how missing values at prime indices are chosen.
Without loss of generality, one may always set [math]\displaystyle{ x_1=+1 }[/math]
Uniform choice
Here is some data when the values of [math]\displaystyle{ x_n }[/math] are: a given +/-1 sequence when [math]\displaystyle{ n\gt 1 }[/math] is one of the first [math]\displaystyle{ N }[/math] primes, only the value [math]\displaystyle{ -1 }[/math] for [math]\displaystyle{ n }[/math] any other prime, and a multiplicatively computed value when [math]\displaystyle{ n }[/math] is composite.
- N=1
- [math]\displaystyle{ x_1=+1 }[/math], [math]\displaystyle{ x_2=+1 }[/math]: D(100)=21, D(1000)=107, D(10000)=407
- ...
- N=2
- [math]\displaystyle{ x_1=+1 }[/math], [math]\displaystyle{ x_2=+1 }[/math], [math]\displaystyle{ x_3=+1 }[/math]: D(100)=34, D(1000)=262, D(10000)=1190
- ...
- N=3
- [math]\displaystyle{ x_1=+1 }[/math], [math]\displaystyle{ x_2=+1 }[/math], [math]\displaystyle{ x_3=+1 }[/math], [math]\displaystyle{ x_5=+1 }[/math]: D(100)=25, D(1000)=413, D(10000)=2332
- ...
Minimizing D up to the next prime
Here is a plot of D(n) as well as the partial sums of the first few HAP when one starts with [math]\displaystyle{ x_1=1 }[/math] and ask that the value at prime [math]\displaystyle{ p }[/math] be either +1 or -1 depending on which allows to minimize [math]\displaystyle{ D(q) }[/math], where [math]\displaystyle{ q }[/math] is the next prime.
Here is a plot doing the same thing but instead starting with the first 1124 sequence as a seed.
The two plots show that the partial sums do grow at least logarithmically.
Sum of partial sums
A method to choose a value at an undertermined prime [math]\displaystyle{ p }[/math] is to choose to impose [math]\displaystyle{ x_p=+1 }[/math] or [math]\displaystyle{ x_p=-1 }[/math] depending on which gave the smallest quantity [math]\displaystyle{ \ell_s(q) }[/math], where [math]\displaystyle{ q }[/math] is the next prime and [math]\displaystyle{ \ell_s(q):=\sum_{d=1}^q s_d(q) }[/math] with [math]\displaystyle{ s_d(q) }[/math] itself the partial sum of the d-HAP up to [math]\displaystyle{ q }[/math].
Here is a plot obtained when setting only [math]\displaystyle{ x_1=+1 }[/math]. On it is shown the function [math]\displaystyle{ f(x):=\log (x) }[/math] (the very flat curve), the partial sums of the sequence and its first few HAPs, and both [math]\displaystyle{ D(n) }[/math] and [math]\displaystyle{ -D(n) }[/math].
