3 * Riemann zeta function
9 * float x, y, zetacf();
21 * zetac(x) = > k , x > 1,
25 * is related to the Riemann zeta function by
27 * Riemann zeta(x) = zetac(x) + 1.
29 * Extension of the function definition for x < 1 is implemented.
30 * Zero is returned for x > log2(MAXNUM).
32 * An overflow error may occur for large negative x, due to the
33 * gamma function in the reflection formula.
37 * Tabulated values have full machine accuracy.
40 * arithmetic domain # trials peak rms
41 * IEEE 1,50 30000 5.5e-7 7.5e-8
47 Cephes Math Library Release 2.2: July, 1992
48 Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
49 Direct inquiries to 30 Frost Street, Cambridge, MA 02140
55 /* Riemann zeta(x) - 1
56 * for integer arguments between 0 and 30.
58 static float azetacf[] = {
59 -1.50000000000000000000E0,
60 1.70141183460469231730E38, /* infinity. */
61 6.44934066848226436472E-1,
62 2.02056903159594285400E-1,
63 8.23232337111381915160E-2,
64 3.69277551433699263314E-2,
65 1.73430619844491397145E-2,
66 8.34927738192282683980E-3,
67 4.07735619794433937869E-3,
68 2.00839282608221441785E-3,
69 9.94575127818085337146E-4,
70 4.94188604119464558702E-4,
71 2.46086553308048298638E-4,
72 1.22713347578489146752E-4,
73 6.12481350587048292585E-5,
74 3.05882363070204935517E-5,
75 1.52822594086518717326E-5,
76 7.63719763789976227360E-6,
77 3.81729326499983985646E-6,
78 1.90821271655393892566E-6,
79 9.53962033872796113152E-7,
80 4.76932986787806463117E-7,
81 2.38450502727732990004E-7,
82 1.19219925965311073068E-7,
83 5.96081890512594796124E-8,
84 2.98035035146522801861E-8,
85 1.49015548283650412347E-8,
86 7.45071178983542949198E-9,
87 3.72533402478845705482E-9,
88 1.86265972351304900640E-9,
89 9.31327432419668182872E-10
93 /* 2**x (1 - 1/x) (zeta(x) - 1) = P(1/x)/Q(1/x), 1 <= x <= 10 */
95 5.85746514569725319540E11,
96 2.57534127756102572888E11,
97 4.87781159567948256438E10,
98 5.15399538023885770696E9,
99 3.41646073514754094281E8,
100 1.60837006880656492731E7,
101 5.92785467342109522998E5,
102 1.51129169964938823117E4,
103 2.01822444485997955865E2,
105 static float Q[8] = {
106 /* 1.00000000000000000000E0,*/
107 3.90497676373371157516E11,
108 5.22858235368272161797E10,
109 5.64451517271280543351E9,
110 3.39006746015350418834E8,
111 1.79410371500126453702E7,
112 5.66666825131384797029E5,
113 1.60382976810944131506E4,
114 1.96436237223387314144E2,
117 /* log(zeta(x) - 1 - 2**-x), 10 <= x <= 50 */
118 static float A[11] = {
119 8.70728567484590192539E6,
120 1.76506865670346462757E8,
121 2.60889506707483264896E10,
122 5.29806374009894791647E11,
123 2.26888156119238241487E13,
124 3.31884402932705083599E14,
125 5.13778997975868230192E15,
126 -1.98123688133907171455E15,
127 -9.92763810039983572356E16,
128 7.82905376180870586444E16,
129 9.26786275768927717187E16,
131 static float B[10] = {
132 /* 1.00000000000000000000E0,*/
133 -7.92625410563741062861E6,
134 -1.60529969932920229676E8,
135 -2.37669260975543221788E10,
136 -4.80319584350455169857E11,
137 -2.07820961754173320170E13,
138 -2.96075404507272223680E14,
139 -4.86299103694609136686E15,
140 5.34589509675789930199E15,
141 5.71464111092297631292E16,
142 -1.79915597658676556828E16,
145 /* (1-x) (zeta(x) - 1), 0 <= x <= 1 */
147 static float R[6] = {
148 -3.28717474506562731748E-1,
149 1.55162528742623950834E1,
150 -2.48762831680821954401E2,
151 1.01050368053237678329E3,
152 1.26726061410235149405E4,
153 -1.11578094770515181334E5,
155 static float S[5] = {
156 /* 1.00000000000000000000E0,*/
157 1.95107674914060531512E1,
158 3.17710311750646984099E2,
159 3.03835500874445748734E3,
160 2.03665876435770579345E4,
161 7.43853965136767874343E4,
168 * Riemann zeta function, minus one
171 extern float MACHEPF, PIO2F, MAXNUMF, PIF;
174 extern float sinf ( float xx );
175 extern float floorf ( float x );
176 extern float gammaf ( float xx );
177 extern float powf ( float x, float y );
178 extern float expf ( float xx );
179 extern float polevlf ( float xx, float *coef, int N );
180 extern float p1evlf ( float xx, float *coef, int N );
182 float sinf(), floorf(), gammaf(), powf(), expf();
183 float polevlf(), p1evlf();
186 float zetacf(float xx)
196 mtherr( "zetacf", OVERFLOW );
201 b = sinf(PIO2F*x) * powf(2.0*PIF, x) * gammaf(s) * (1.0 + w) / PIF;
206 return(0.0); /* because first term is 2**-x */
208 /* Tabulated values for integer argument */
215 return( azetacf[i] );
223 a = polevlf( x, R, 5 ) / ( w * p1evlf( x, S, 5 ));
229 mtherr( "zetacf", SING );
235 b = powf( 2.0, x ) * (x - 1.0);
237 s = (x * polevlf( w, P, 8 )) / (b * p1evlf( w, Q, 8 ));
244 w = polevlf( x, A, 10 ) / p1evlf( x, B, 10 );
250 /* Basic sum of inverse powers */
261 while( b/s > MACHEPF );