Line data Source code
1 : /* Copyright (C) 2000 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation. It is distributed in the hope that it will be useful, but WITHOUT
8 : ANY WARRANTY WHATSOEVER.
9 :
10 : Check the License for details. You should have received a copy of it, along
11 : with the package; see the file 'COPYING'. If not, write to the Free Software
12 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
13 :
14 : #include "pari.h"
15 : #include "paripriv.h"
16 :
17 : /*****************************************************************/
18 : /* Program to compute L(chi,s) */
19 : /* for Im(s) large, chi primitive Dirichlet character */
20 : /* In the present branch, only Tyagi's method is used */
21 : /*****************************************************************/
22 : /*
23 : In addition, C can also be a polynomial defining an abelian
24 : extension of Q.
25 : */
26 :
27 : /*****************************************************************/
28 : /* Character programs */
29 : /*****************************************************************/
30 : /* A character, always assumed primitive can be given in the following formats:
31 : * - omitted or 0: special to zetaRS,
32 : * - a t_INT: assumed to be a discriminant,
33 : * - a t_INTMOD: a conrey character,
34 : * - a pair [G,chi] or [bnr,chi],
35 : * - [C1,C2,...]~ where the Ci are characters as above with same moduli. */
36 :
37 : /* Given a list of linit/ldata for chars of same conductor F, return
38 : * [Vecan, F, Parities, Gaussums] */
39 : static GEN
40 1085 : mycharinit(GEN C, long bit)
41 : {
42 : GEN L, LVC, LE, LGA;
43 1085 : long F = 0, i, j, lc = lg(C), prec;
44 :
45 1085 : bit += 64; prec = nbits2prec(bit);
46 1085 : L = cgetg(lc, t_VEC);
47 1085 : LE = cgetg(lc, t_VECSMALL);
48 1085 : LGA = cgetg(lc, t_VEC);
49 2205 : for (i = 1; i < lc; i++)
50 : {
51 1120 : GEN gv, ga, gm, ro, ldata = gel(C, i);
52 : long e;
53 1120 : if (is_linit(ldata)) ldata = linit_get_ldata(ldata);
54 1120 : gv = ldata_get_gammavec(ldata); e = itou(gel(gv, 1));
55 1120 : gm = ldata_get_conductor(ldata);
56 1120 : ro = ldata_get_rootno(ldata);
57 1120 : if (isintzero(ro)) ro = lfunrootno(ldata, bit);
58 1120 : ga = gmul(ro, gsqrt(gm, prec)); if (e) ga = mulcxI(ga);
59 1120 : gel(LGA, i) = ga;
60 1120 : LE[i] = e;
61 1120 : if (i == 1) F = itos(gm); /* constant */
62 1120 : gel(L, i) = lfunan(ldata, F, prec);
63 : }
64 1085 : if (lc == 2 && is_vec_t(typ(gmael(L,1,1))))
65 : { /* multichar */
66 7 : LGA = gel(LGA,1); lc = lg(LGA);
67 7 : LVC = gel(L,1);
68 7 : LE = const_vecsmall(lc-1, LE[1]); /* FIXME: can handle mixed values */
69 : }
70 : else
71 : {
72 1078 : LVC = cgetg(F + 1, t_VEC);
73 4459 : for (j = 1; j <= F; j++)
74 : {
75 3381 : GEN v = cgetg(lc, t_VEC);
76 7049 : for (i = 1; i < lc; i++) gel(v, i) = gmael(L, i, j);
77 3381 : gel(LVC, j) = v;
78 : }
79 : }
80 1085 : return mkvec5(LVC, stoi(F), LE, LGA, grootsof1(2*F, prec));
81 : }
82 :
83 : /* n >= 1 and #VC = F, the conductor of the character or multicharacter X.
84 : * VC contains [X(1),X(2),...X(F)] */
85 : static GEN
86 618472 : mycall(GEN VC, long n)
87 : {
88 618472 : long F = lg(VC) - 1;
89 618472 : GEN z = n <= F ? gel(VC, n) : gel(VC, ((n - 1) % F) + 1);
90 618472 : return gequal0(z)? NULL: z;
91 : }
92 :
93 187267 : static GEN get_chivec(GEN VCALL) { return gel(VCALL, 1); }
94 763801 : static long get_modulus(GEN VCALL) { return itos(gel(VCALL, 2)); }
95 185965 : static GEN get_signat(GEN VCALL) { return gel(VCALL, 3); }
96 21 : static GEN get_gauss(GEN VCALL) { return gel(VCALL, 4); }
97 184663 : static GEN get_chiZ(GEN VCALL) { return gel(VCALL, 5); }
98 :
99 : /* (-1)^A[i] * conj(B[i]) */
100 : static GEN
101 7 : gnegconj(GEN A, GEN B)
102 : {
103 7 : long i, l = lg(A);
104 7 : GEN W = cgetg(l, t_VEC);
105 14 : for (i = 1; i < l; i++)
106 7 : { GEN b = gconj(gel(B,i)); gel(W,i) = A[i]? gneg(b): b; }
107 7 : return W;
108 : }
109 : /* g(conj(CHI)) */
110 : static GEN
111 7 : gaussconj(GEN VCALL)
112 7 : { return gnegconj(get_signat(VCALL), get_gauss(VCALL)); }
113 :
114 : static GEN
115 7 : myinitconj(GEN VCALL)
116 : {
117 7 : GEN CONJ = shallowcopy(VCALL);
118 7 : gel(CONJ, 1) = gconj(get_chivec(VCALL));
119 7 : gel(CONJ, 4) = gaussconj(VCALL); return CONJ;
120 : }
121 :
122 : /********************************************************************/
123 : /* Driver Program */
124 : /********************************************************************/
125 :
126 : /* assume |Im(s)| >> 1, in particular s is not a negative integer */
127 : static GEN
128 14 : applyfuneq(GEN gau, GEN s, GEN z, long odd, long q, long bitprec)
129 : {
130 : GEN t, S;
131 : long prec;
132 14 : if (!gequal0(s)) bitprec += maxss(gexpo(s), 0);
133 14 : prec = nbits2prec(bitprec);
134 14 : if (odd) gau = mulcxmI(gau);
135 14 : S = gmul(Pi2n(-1, prec), gsubgs(s, odd));
136 14 : t = ginv(gmul2n(gmul(gcos(S, prec), ggamma(s, prec)), 1));
137 14 : t = gmul(gpow(gdivgs(Pi2n(1, prec), q), s, prec), t);
138 14 : return gmul(gmul(gau, t), z);
139 : }
140 :
141 : static GEN RZchi(GEN VCALL, GEN s, long prec);
142 :
143 : /* VCALL already initialized */
144 : static GEN
145 1085 : lfunlarge_char(GEN VCALL, GEN s, long bitprec)
146 : {
147 1085 : pari_sp av = avma;
148 : GEN sig, tau, z;
149 1085 : long funeq = 0, ts = typ(s), stau, flconj, q;
150 1085 : if (!is_real_t(ts) && ts != t_COMPLEX) pari_err_TYPE("lfunlarge_char", s);
151 1085 : sig = real_i(s); tau = imag_i(s);
152 1085 : if (gexpo(tau) < 1) pari_err_DOMAIN("lfun","im(s)", "<", gen_2, tau);
153 1085 : stau = gsigne(tau);
154 1085 : if (stau < 0) { tau = gneg(tau); VCALL = myinitconj(VCALL); }
155 1085 : if (gcmp(sig, ghalf) < 0) { funeq = 1; sig = gsubsg(1, sig); }
156 1085 : flconj = ((stau > 0 && funeq) || (stau < 0 && !funeq));
157 1085 : q = get_modulus(VCALL); bitprec += gexpo(stoi(q));
158 1085 : z = RZchi(VCALL, mkcomplex(sig, tau), nbits2prec(bitprec));
159 1085 : if (flconj) z = gconj(z);
160 1085 : if (funeq)
161 : {
162 14 : GEN odd = get_signat(VCALL), gau = get_gauss(VCALL), Vz;
163 14 : long lC = lg(gau), j;
164 14 : Vz = cgetg(lC, t_VEC);
165 28 : for (j = 1; j < lC; j++)
166 14 : gel(Vz,j) = applyfuneq(gel(gau,j), s, gel(z,j), odd[j], q, bitprec);
167 14 : z = Vz;
168 : }
169 1085 : return gerepilecopy(av, z);
170 : }
171 :
172 : static GEN
173 14 : lfungetchars(GEN pol)
174 : {
175 14 : GEN L, F, v, bnf = Buchall(pol_x(1), 0, LOWDEFAULTPREC);
176 : GEN w, condall, bnr;
177 : long i, l, lc;
178 14 : condall = rnfconductor(bnf, pol); bnr = gel(condall, 2);
179 14 : L = bnrchar(bnr, gel(condall, 3), NULL); lc = lg(L);
180 14 : F = cgetg(lc, t_VEC);
181 77 : for (i = 1; i < lc; i++)
182 : {
183 63 : GEN chi = gel(L, i), cond = bnrconductor_raw(bnr, chi);
184 63 : gel(F, i) = gcoeff(gel(cond,1), 1, 1);
185 : }
186 14 : w = vec_equiv(F); l = lg(w); v = cgetg(l, t_COL);
187 42 : for (i = 1; i < l; i++)
188 : {
189 28 : GEN wi = gel(w, i), vi;
190 28 : long j, li = lg(wi);
191 28 : gel(v,i) = vi = cgetg(li, t_VEC);
192 28 : if (li == 2 && equali1(gel(F, wi[1]))) /* common conductor is 1 */
193 14 : gel(vi,1) = lfunmisc_to_ldata_shallow(gen_1);
194 : else
195 : {
196 63 : for (j = 1; j < li; j++)
197 49 : gel(vi,j) = lfunmisc_to_ldata_shallow(mkvec2(bnr, gel(L, wi[j])));
198 : }
199 : }
200 14 : return v;
201 : }
202 :
203 : /********************************************************************/
204 : /* NEW RS IMPLEMENTATION FROM SANDEEP TYAGI */
205 : /********************************************************************/
206 : /* See arXiv:2203.02509v2 */
207 :
208 5523 : static long m_n0(GEN sel) { return itos(gel(sel, 1)); }
209 576072 : static GEN m_r0(GEN sel) { return gel(sel, 2); }
210 1085 : static GEN m_al(GEN sel) { return gel(sel, 3); }
211 576072 : static GEN m_aleps(GEN sel) { return gel(sel, 4); }
212 4396 : static GEN m_h(GEN sel) { return gel(sel, 5); }
213 1113 : static GEN m_lin(GEN sel) { return gel(sel, 6); }
214 1113 : static long m_np(GEN sel) { return itou(gel(sel, 7)); }
215 :
216 : static GEN
217 3297 : phi_hat(GEN x, long prec)
218 : {
219 : GEN y;
220 3297 : if (gsigne(imag_i(x)) > 0)
221 1078 : y = gneg(gexpm1(gneg(gmul(PiI2(prec), x)), prec));
222 : else
223 2219 : y = gexpm1(gmul(PiI2(prec), x), prec);
224 3297 : return ginv(y);
225 : }
226 :
227 : static GEN
228 3241 : phi_hat_h0(GEN sel, long k, long prec)
229 : {
230 3241 : GEN t = gdiv(gsubsg(m_n0(sel) + k, m_r0(sel)), m_aleps(sel));
231 3241 : return phi_hat(gdiv(gasinh(t, prec), m_h(sel)), prec);
232 : }
233 :
234 : /* v[i] = A[i] * (a + (-1)^E[i] b) */
235 : static GEN
236 612774 : mul_addsub(GEN A, GEN a, GEN b, GEN E)
237 : {
238 612774 : long i, l = lg(E);
239 612774 : GEN v = cgetg(l, t_VEC);
240 1308996 : for (i = 1; i < l; i++)
241 696222 : gel(v,i) = gmul(gel(A,i), E[i]? gsub(a, b): gadd(a, b));
242 612774 : return v;
243 : }
244 :
245 : static GEN
246 184663 : wd(GEN VCALL, GEN pmd, GEN x, long prec)
247 : {
248 184663 : GEN VC = get_chivec(VCALL), E = get_signat(VCALL), Z = get_chiZ(VCALL);
249 184663 : GEN ex, emx, xpmd = gmul(x, pmd), y = NULL;
250 184663 : long md = get_modulus(VCALL), N = 2*md, k;
251 184663 : ex = gexp(mulcxI(xpmd), prec); emx = ginv(ex);
252 797437 : for (k = 1; k <= (md-1) / 2; k++)
253 : {
254 612774 : GEN xc = mycall(VC, k);
255 612774 : if (xc)
256 : {
257 612774 : GEN p3, p2, p1 = gmul(xc, gel(Z, Fl_neg(Fl_sqr(k,N), N) + 1));
258 612774 : GEN a = gmul(ex, gel(Z, N - k + 1)), b = gmul(emx, gel(Z, k + 1));
259 612774 : GEN c = gmul(ex, gel(Z, k + 1)), d = gmul(emx, gel(Z, N - k + 1));
260 612774 : if (odd(md))
261 : {
262 594831 : p2 = ginv(mulcxmI(gmul2n(gsub(a,b), -1))); /* 1 / sin(xpmd - kpmd) */
263 594831 : p3 = ginv(mulcxmI(gmul2n(gsub(c,d), -1))); /* 1 / sin(xpmd + kpmd) */
264 : }
265 : else
266 : {
267 17943 : p2 = mulcxI(gdiv(gadd(a,b), gsub(a,b))); /* cotan(xpmd - kpmd) */
268 17943 : p3 = mulcxI(gdiv(gadd(c,d), gsub(c,d))); /* cotan(xpmd + kpmd) */
269 : }
270 612774 : p1 = mul_addsub(p1, p2, p3, E);
271 612774 : y = y ? gadd(y, p1) : p1;
272 : }
273 : }
274 184663 : return mulcxmI(gdivgs(y, N));
275 : }
276 :
277 : static GEN
278 1085 : series_h0(long n0, GEN s, GEN VCALL, long fl, long prec)
279 : {
280 1085 : GEN C = get_modulus(VCALL) == 1? NULL: get_chivec(VCALL);
281 1085 : GEN R = pardirpowerssumfun(C, n0, gneg(s), fl, prec);
282 1085 : if (C) return R;
283 686 : if (fl) return mkvec2(mkvec(gel(R,1)), mkvec(gel(R,2)));
284 658 : return mkvec(R);
285 : }
286 :
287 : static GEN
288 1113 : series_residues_h0(GEN sel, GEN s, GEN VCALL, long prec)
289 : {
290 1113 : long n0 = m_n0(sel), np = m_np(sel), k;
291 1113 : GEN val = gen_0, VC = get_chivec(VCALL);
292 4424 : for (k = maxss(1 - np, 1 - n0); k <= 1 + np; k++)
293 : {
294 3311 : GEN nk = mycall(VC, n0 + k); /* n0 + k > 0 */
295 3311 : if (nk) val = gadd(val, gmul(gmul(phi_hat_h0(sel, k, prec), nk),
296 : gpow(stoi(n0 + k), gneg(s), prec)));
297 : }
298 1113 : return val;
299 : }
300 :
301 : static GEN
302 572831 : integrand_h0(GEN sel, GEN s, GEN VCALL, GEN x, long prec)
303 : {
304 572831 : pari_sp av = avma;
305 572831 : long md = get_modulus(VCALL);
306 572831 : GEN r0 = m_r0(sel), aleps = m_aleps(sel), zn, p1;
307 572831 : GEN pmd = divru(mppi(prec), md), ix = ginv(x);
308 572831 : zn = gadd(r0, gdivgs(gmul(aleps, gsub(x, ix)), 2));
309 572831 : p1 = gmul(expIxy(pmd, gsqr(zn), prec),
310 : gmul(gpow(zn, gneg(s), prec), gmul(aleps, gadd(x, ix))));
311 572831 : if (md == 1)
312 388168 : p1 = gdiv(mkvec(mulcxI(p1)), gmul2n(gsin(gmul(pmd, zn), prec), 2));
313 : else
314 184663 : p1 = gdivgs(gmul(p1, wd(VCALL, pmd, zn, prec)), -2);
315 572831 : return gerepileupto(av, p1);
316 : }
317 :
318 : static GEN
319 1113 : integral_h0(GEN sel, GEN s, GEN VCALL, long prec)
320 : {
321 1113 : GEN lin_grid = m_lin(sel), S = gen_0;
322 1113 : pari_sp av = avma;
323 1113 : long j, l = lg(lin_grid);
324 569604 : for (j = 1; j < l; j++)
325 : {
326 568491 : S = gadd(S, integrand_h0(sel, s, VCALL, gel(lin_grid, j), prec));
327 568491 : if ((j & 0xff) == 0) S = gerepileupto(av, S);
328 : }
329 1113 : return gerepileupto(av, gmul(m_h(sel), S));
330 : }
331 :
332 : /* a + log |x|, a t_REAL, low accuracy */
333 : static GEN
334 4508 : addrlogabs(GEN a, GEN x)
335 : {
336 4508 : long prec = DEFAULTPREC;
337 4508 : if (gequal0(x)) return real_m2n(64, prec); /* -oo */
338 4438 : switch(typ(x))
339 : {
340 4438 : case t_COMPLEX: return gmul2n(glog(cxnorm(x), prec), -1);
341 0 : case t_REAL: break;
342 0 : default: x = gtofp(x, prec);
343 : }
344 0 : return addrr(a, logr_abs(x));
345 : }
346 :
347 : struct fun_q_t { GEN sel, s, VCALL, B; };
348 : static GEN
349 4340 : fun_q(void *E, GEN x)
350 : {
351 4340 : struct fun_q_t *S = (struct fun_q_t *)E;
352 4340 : GEN z = integrand_h0(S->sel,S->s,S->VCALL, gexp(x,DEFAULTPREC), DEFAULTPREC);
353 4340 : if (typ(z) == t_VEC) z = vecsum(z);
354 4340 : return addrlogabs(S->B, z);
355 : }
356 : static GEN
357 2254 : brent_q(void *E, GEN (*f)(void*,GEN), GEN q_low, GEN q_hi)
358 : {
359 2254 : if (gsigne(f(E, q_low)) * gsigne(f(E, q_hi)) >= 0) return NULL;
360 0 : return zbrent(E, f, q_low, q_hi, LOWDEFAULTPREC);
361 : }
362 : static GEN
363 1127 : findq(void *E, GEN (*f)(void*,GEN), GEN lq, long B)
364 : {
365 1127 : GEN q_low, q_hi, q_right, q_left, q_est = gasinh(lq, LOWDEFAULTPREC);
366 1127 : q_low = gdivgs(gmulsg(4, q_est), 5);
367 1127 : q_hi = gdivgs(gmulsg(3, q_est), 2);
368 1127 : q_right = brent_q(E, f, q_low, q_hi); if (!q_right) q_right = q_est;
369 1127 : q_left = brent_q(E, f, gneg(q_low), gneg(q_hi)); if (!q_left) q_left = q_est;
370 1127 : return bitprecision0(gmax(q_right, q_left), B);
371 : }
372 :
373 : static GEN
374 1085 : set_q_value(GEN sel, GEN s, GEN VCALL, long prec)
375 : {
376 : struct fun_q_t E;
377 1085 : GEN al = m_al(sel), lq;
378 1085 : long md = get_modulus(VCALL), LD = DEFAULTPREC;
379 1085 : E.sel = sel; E.s = s; E.VCALL = VCALL, E.B = mulur(prec, mplog2(LD));
380 1085 : lq = gdiv(gsqrt(gdiv(gmulsg(md, E.B), Pi2n(1, LD)), LD), al);
381 1085 : return findq((void*)&E, &fun_q, lq, prec);
382 : }
383 :
384 : static GEN
385 1099 : setlin_grid_exp(GEN h, long m, long prec)
386 : {
387 1099 : GEN w, vex = gpowers(gexp(h, prec), (m - 1)/2);
388 : long i;
389 1099 : w = cgetg(m+1, t_VEC); gel(w, (m + 1)/2) = gen_1;
390 283108 : for (i = (m + 3)/2; i <= m; i++)
391 : {
392 282009 : GEN t1 = gel(vex, i - ((m - 1)/2));
393 282009 : gel(w, i) = t1; gel(w, (m + 1) - i) = ginv(t1);
394 : }
395 1099 : return w;
396 : }
397 :
398 : static long
399 1099 : get_m(GEN q, long prec)
400 : {
401 1099 : GEN t = divrr(mulur(4 * prec2nbits(prec), mplog2(prec)), sqrr(mppi(prec)));
402 1099 : return 2*itos(gfloor(mulrr(q, t))) + 1;
403 : }
404 :
405 : static GEN
406 1085 : RZinit(GEN s, GEN VCALL, GEN numpoles, long prec)
407 : {
408 : GEN sel, al, aleps, n0, r0, q, h;
409 1085 : long md = get_modulus(VCALL), m;
410 1085 : al = gcmpgs(gabs(imag_i(s), prec), 100) < 0 ? ginv(stoi(4)) : gen_1;
411 1085 : r0 = gsqrt(gdiv(gmulgs(s, md), PiI2(prec)), prec);
412 1085 : n0 = gfloor(gsub(real_i(r0), imag_i(r0)));
413 1085 : aleps = gmul(al, gexp(PiI2n(-2, prec), prec));
414 1085 : sel = mkvecn(7, n0, r0, al, aleps, NULL, NULL, numpoles);
415 1085 : q = set_q_value(sel, s, VCALL, prec);
416 1085 : m = get_m(q, prec);
417 1085 : gel(sel,5) = h = divru(q, (m - 1) >> 1);
418 1085 : gel(sel,6) = setlin_grid_exp(h, m, prec);
419 1085 : return sel;
420 : }
421 :
422 : static GEN
423 441 : xpquo_one(GEN s, GEN cs, GEN ga, long odd, long md, long prec)
424 : {
425 441 : GEN rho, a = odd? gen_1: gen_0, z = divsr(md, mppi(prec));
426 441 : rho = gmul(gdiv(gpow(gen_I(), gdivgs(gneg(a), 2), prec), gsqrt(ga, prec)),
427 : gpow(stoi(md), ginv(stoi(4)), prec));
428 441 : return gmul(gdiv(gconj(gmul(rho, gpow(z, gdivgs(cs, 2), prec))),
429 : gmul(rho, gpow(z, gdivgs(s, 2), prec))),
430 : gexp(gsub(gconj(glngamma(gdivgs(gadd(cs, a), 2), prec)),
431 : glngamma(gdivgs(gadd(s, a), 2), prec)), prec));
432 : }
433 :
434 : static GEN
435 1085 : xpquo(GEN s, GEN VCALL, long prec)
436 : {
437 1085 : long md = get_modulus(VCALL), n, lve, i;
438 1085 : GEN cd = NULL, z, pz, cs, VC = get_chivec(VCALL), ve, R;
439 1085 : if (!gequal0(s)) prec = nbits2prec(prec2nbits(prec) + maxss(gexpo(s), 0));
440 1085 : z = gexp(gdivgs(PiI2(prec), -md), prec);
441 1085 : if (md == 1)
442 686 : return gmul(gpow(mppi(prec), gsub(s, ghalf), prec),
443 : gexp(gsub(glngamma(gdivgs(gsubsg(1, s), 2), prec),
444 : glngamma(gdivgs(s, 2), prec)), prec));
445 399 : pz = gpowers(z, md - 1);
446 2786 : for (n = 1; n < md; n++)
447 : {
448 2387 : GEN xn = mycall(VC, n);
449 2387 : if (xn)
450 : {
451 2352 : GEN tmp = gmul(xn, gel(pz, n + 1));
452 2352 : cd = cd ? gadd(cd, tmp) : tmp;
453 : }
454 : }
455 399 : cs = gsubsg(1, gconj(s));
456 399 : ve = get_signat(VCALL); lve = lg(ve); R = cgetg(lve, t_VEC);
457 840 : for (i = 1; i < lve; i++)
458 441 : gel(R, i) = xpquo_one(s, cs, gel(cd, i), ve[i], md, prec);
459 399 : if (lve == 2) R = gel(R, 1);
460 399 : return R;
461 : }
462 :
463 : static GEN
464 1113 : total_value(GEN serh0, GEN sel, GEN s, GEN VCALL, long prec)
465 : {
466 1113 : return gadd(integral_h0(sel, s, VCALL, prec),
467 : gsub(serh0, series_residues_h0(sel, s, VCALL, prec)));
468 : }
469 : static GEN
470 1085 : dirichlet_ours(GEN s, GEN VCALL, long prec)
471 : {
472 1085 : int fl = !gequal(real_i(s), ghalf);
473 1085 : GEN sel = RZinit(s, VCALL, gen_1, prec);
474 1085 : GEN S1, S2, serh0 = series_h0(m_n0(sel), s, VCALL, fl, prec);
475 1085 : if (!fl)
476 1057 : S2 = S1 = total_value(serh0, sel, s, VCALL, prec);
477 : else
478 : {
479 28 : S1 = total_value(gel(serh0,1), sel, s, VCALL, prec);
480 28 : S2 = total_value(gel(serh0,2), sel, gsubsg(1, gconj(s)), VCALL, prec);
481 : }
482 1085 : return gadd(S1, vecmul(xpquo(s, VCALL, prec), gconj(S2)));
483 : }
484 :
485 : /* assume |Im(s)| > 2^-bitprec */
486 : static GEN
487 1085 : RZchi(GEN VCALL, GEN s, long prec)
488 : {
489 1085 : long prec2 = prec + EXTRAPREC64;
490 1085 : return gprec_wtrunc(dirichlet_ours(gprec_w(s, prec2), VCALL, prec2), prec);
491 : }
492 :
493 : /********************************************************************/
494 : /* Utility Functions */
495 : /********************************************************************/
496 : /* lam = 0, return L(s); else Lambda(s) */
497 : static GEN
498 1085 : lfuncharall(GEN VCALL, GEN s, long lam, long bitprec)
499 : {
500 1085 : GEN ve, P, Q, R, z = lfunlarge_char(VCALL, s, bitprec);
501 : long l, i, q, prec;
502 1085 : if (!lam) return z;
503 882 : ve = get_signat(VCALL); l = lg(ve);
504 882 : q = get_modulus(VCALL); prec = nbits2prec(bitprec);
505 882 : R = cgetg(l, t_VEC);
506 882 : Q = divur(q, mppi(prec));
507 882 : P = (q == 1 || zv_equal0(ve))? NULL: gsqrt(utoipos(q), prec);
508 1764 : for (i = 1; i < l; i++)
509 : {
510 882 : GEN se = gmul2n(gaddgs(s, ve[i]), -1), r;
511 882 : if (lam == 1)
512 : {
513 0 : r = gmul(gpow(Q, se, prec), ggamma(se, prec));
514 0 : if (P && ve[i]) r = gdiv(r, P);
515 : }
516 : else
517 : {
518 882 : r = gadd(gmul(se, glog(Q, prec)), glngamma(se, prec));
519 882 : if (P && ve[i]) r = gsub(r, glog(P, prec));
520 : }
521 882 : gel(R, i) = r;
522 : }
523 882 : return lam == 1 ? vecmul(R, z) : gadd(R, glog(z, prec));
524 : }
525 :
526 : static GEN
527 28 : lfunlargeall_from_chars(GEN v, GEN s, long lam, long bit)
528 : {
529 28 : long i, l = lg(v);
530 84 : for (i = 1; i < l; i++)
531 : {
532 56 : GEN w = mycharinit(gel(v, i), bit), L = lfuncharall(w, s, lam, bit);
533 56 : gel(v, i) = lam==-1 ? vecsum(L): vecprod(L);
534 : }
535 28 : return lam==-1 ? vecsum(v): vecprod(v);
536 : }
537 : static GEN
538 1057 : lfunlargeall(GEN ldata, GEN s, long lam, long bit)
539 : {
540 : GEN w, an;
541 1057 : if (lg(ldata) == 2)
542 : { /* HACK: ldata[1] a t_DESC_PRODUCT from lfunabelianrelinit / Q */
543 14 : GEN v = lfunprod_get_fact(linit_get_tech(gel(ldata,1)));
544 : long i, l;
545 14 : v = shallowcopy(gel(v,1)); l = lg(v);
546 42 : for (i = 1; i < l; i++) gel(v,i) = mkvec(gel(v,i));
547 14 : return lfunlargeall_from_chars(v, s, lam, bit);
548 : }
549 1043 : an = gel(ldata_get_an(ldata), 2);
550 1043 : switch(ldata_get_type(ldata))
551 : {
552 14 : case t_LFUN_NF:
553 : {
554 14 : GEN v = lfungetchars(nf_get_pol(an));
555 14 : return lfunlargeall_from_chars(v, s, lam, bit);
556 : }
557 7 : case t_LFUN_CHIGEN:
558 : {
559 7 : GEN chi = gmael(an, 2, 2);
560 7 : if (lg(chi) > 1 && is_vec_t(typ(gel(chi,1))))
561 : { /* multi char */
562 7 : w = mycharinit(mkcol(ldata), bit);
563 7 : return lfuncharall(w, s, lam, bit);
564 : }
565 : }
566 : default: /* single char */
567 1022 : w = mycharinit(mkcol(ldata), bit);
568 1022 : return gel(lfuncharall(w, s, lam, bit), 1);
569 : }
570 : }
571 :
572 : GEN
573 189 : lfunlarge(GEN CHI, GEN s, long bit)
574 189 : { return lfunlargeall(CHI, s, 0, bit); }
575 :
576 : GEN
577 0 : lfunlambdalarge(GEN CHI, GEN s, long bit)
578 0 : { return lfunlargeall(CHI, s, 1, bit); }
579 :
580 : GEN
581 868 : lfunloglambdalarge(GEN CHI, GEN s, long bit)
582 868 : { return lfunlargeall(CHI, s, -1, bit); }
583 :
584 : /********************************************************************/
585 : /* LERCH RS IMPLEMENTATION FROM SANDEEP TYAGI */
586 : /********************************************************************/
587 :
588 : static GEN
589 56 : double_exp_residue_pos_h(GEN selsm, long k, long ind, long prec)
590 : {
591 56 : long nk = itos(gel(selsm, 1)) + k;
592 56 : GEN r = gel(selsm, 2), ale = gel(selsm, 3), aor = gel(selsm, 4);
593 56 : GEN h = gel(selsm, 5), t = gen_0;
594 56 : switch(ind)
595 : {
596 28 : case 0: t = gaddsg(nk, aor); break;
597 0 : case 1: t = gneg(gaddsg(nk, aor)); break;
598 28 : case 2: t = gsubsg(nk, aor); break;
599 : }
600 56 : return gdiv(gasinh(gdiv(gsub(t, r), ale), prec), h);
601 : }
602 :
603 : static GEN
604 56 : phi_hat_h(GEN selsm, long m, long ind, long prec)
605 56 : { return phi_hat(double_exp_residue_pos_h(selsm, m, ind, prec), prec); }
606 :
607 : static long
608 56 : myex(GEN is)
609 56 : { return gequal0(is) ? 0 : maxss(0, 2 + gexpo(is)); }
610 :
611 : static GEN
612 56 : gaminus(GEN s, long prec)
613 : {
614 56 : GEN is = imag_i(s), tmp;
615 56 : long B = prec2nbits(prec);
616 56 : if (gcmpgs(is, -5*B) < 0) return gen_0;
617 56 : prec = nbits2prec(B + myex(is));
618 56 : tmp = gexp(gsub(glngamma(s, prec), gmul(PiI2n(-1, prec), s)), prec);
619 56 : return bitprecision0(tmp, B);
620 : }
621 :
622 : static GEN
623 28 : gaplus(GEN s, long prec)
624 : {
625 28 : GEN is = imag_i(s), tmp;
626 28 : long B = prec2nbits(prec);
627 28 : if (gcmpgs(is, 5*B) > 0) return gen_0;
628 0 : prec = nbits2prec(B + myex(is));
629 0 : tmp = gexp(gadd(glngamma(s, prec), gmul(PiI2n(-1, prec), s)), prec);
630 0 : return bitprecision0(tmp, B);
631 : }
632 :
633 : GEN
634 3514 : serh_worker(GEN k, GEN z, GEN a, GEN ns, GEN gprec)
635 : {
636 3514 : long prec = itou(gprec);
637 3514 : return gmul(gpow(z, k, prec), gpow(gadd(a, k), ns, prec));
638 : }
639 :
640 : static void
641 28 : set_arg(GEN worker, GEN z, GEN a, GEN ns, long prec)
642 28 : { gel(worker, 7) = mkvec4(z, a, ns, stoi(prec)); }
643 :
644 :
645 : static GEN
646 14 : series_h0l(GEN worker, long n0, GEN s, GEN a, GEN lam, long prec)
647 : {
648 14 : GEN z = typ(lam) == t_INT ? gen_1 : gexp(gmul(PiI2(prec), lam), prec);
649 14 : set_arg(worker, z, a, gneg(s), prec);
650 14 : return parsum(gen_0, utoi(n0), worker);
651 : }
652 :
653 : static GEN
654 14 : series_h1(GEN worker, long n1, GEN s, GEN a, GEN lam, long prec)
655 : {
656 14 : GEN mP, pre_factor, z, sn = gsubgs(s, 1);
657 14 : GEN ini = gequal0(lam) ? gen_1 : gen_0;
658 14 : pre_factor = gaplus(gneg(sn), prec);
659 14 : if (gequal0(pre_factor)) return gen_0;
660 0 : mP = gneg(PiI2(prec));
661 0 : pre_factor = gmul(gmul(pre_factor, gexp(gmul(mP, gmul(a, lam)), prec)),
662 : gpow(Pi2n(1, prec), sn, prec));
663 0 : z = typ(a) == t_INT ? gen_1 : gexp(gmul(mP, a), prec);
664 0 : set_arg(worker, z, lam, sn, prec);
665 0 : return gmul(pre_factor, parsum(ini, stoi(n1 - 1), worker));
666 : }
667 :
668 : static GEN
669 14 : series_h2(GEN worker, long n2, GEN s, GEN a, GEN lam, long prec)
670 : {
671 14 : GEN P, pre_factor, z, sn = gsubgs(s, 1);
672 14 : pre_factor = gaminus(gneg(sn), prec);
673 14 : if (gequal0(pre_factor)) return gen_0;
674 14 : P = PiI2(prec);
675 14 : pre_factor = gmul(gmul(pre_factor, gexp(gmul(gneg(P), gmul(a, lam)), prec)),
676 : gpow(Pi2n(1, prec), sn, prec));
677 14 : z = typ(a) == t_INT ? gen_1 : gexp(gmul(P, a), prec);
678 14 : set_arg(worker, z, gneg(lam), sn, prec);
679 14 : return gmul(pre_factor, parsum(gen_1, stoi(n2), worker));
680 : }
681 :
682 : static GEN
683 14 : series_residues_h0l(long numpoles, GEN selsm0, GEN s, GEN a, GEN lam, long prec)
684 : {
685 14 : GEN P, val = gen_0, ra = real_i(a);
686 14 : long n0 = m_n0(selsm0), k;
687 14 : P = PiI2(prec);
688 42 : for (k = -numpoles + 1; k <= numpoles; k++)
689 28 : if (gsigne(gaddsg(n0 + k, ra)) > 0)
690 28 : val = gadd(val, gmul(gmul(phi_hat_h(selsm0, k, 0, prec),
691 : gexp(gmul(P, gmulgs(lam, n0 + k)), prec)),
692 : gpow(gaddsg(n0 + k, a), gneg(s), prec)));
693 14 : return val;
694 : }
695 :
696 : static GEN
697 14 : series_residues_h1(long numpoles, GEN selsm1, GEN s, GEN a, GEN lam, long prec)
698 : {
699 14 : GEN mP, val = gen_0, rlam = real_i(lam), pre_factor, sn = gsubgs(s, 1);
700 14 : long n1 = m_n0(selsm1), k;
701 14 : pre_factor = gaplus(gneg(sn), prec);
702 14 : if (gequal0(pre_factor)) return gen_0;
703 0 : mP = gneg(PiI2(prec));
704 0 : pre_factor = gmul(gmul(pre_factor, gexp(gmul(mP, gmul(a, lam)), prec)),
705 : gpow(Pi2n(1, prec), sn, prec));
706 0 : for (k = -numpoles; k <= numpoles - 1; k++)
707 0 : if (gsigne(gaddsg(n1 + k, rlam)) > 0)
708 0 : val = gadd(val, gmul(gmul(phi_hat_h(selsm1, k, 1, prec),
709 : gexp(gmul(mP, gmulgs(a, n1 + k)), prec)),
710 : gpow(gaddsg(n1 + k, lam), sn, prec)));
711 0 : return gmul(pre_factor, val);
712 : }
713 :
714 : static GEN
715 14 : series_residues_h2(long numpoles, GEN selsm2, GEN s, GEN a, GEN lam, long prec)
716 : {
717 14 : GEN P, val = gen_0, rlam = real_i(lam), pre_factor, sn = gsubgs(s, 1);
718 14 : long n2 = m_n0(selsm2), k;
719 14 : pre_factor = gaminus(gneg(sn), prec);
720 14 : if (gequal0(pre_factor)) return gen_0;
721 14 : P = PiI2(prec);
722 14 : pre_factor = gmul(gmul(pre_factor, gexp(gmul(gneg(P), gmul(a, lam)), prec)),
723 : gpow(Pi2n(1, prec), sn, prec));
724 42 : for (k = -numpoles + 1; k <= numpoles; k++)
725 28 : if (gsigne(gsubsg(n2 + k, rlam)) > 0)
726 28 : val = gsub(val, gmul(gmul(phi_hat_h(selsm2, k, 2, prec),
727 : gexp(gmul(P, gmulgs(a, n2 + k)), prec)),
728 : gpow(gsubsg(n2 + k, lam), sn, prec)));
729 14 : return gmul(pre_factor, val);
730 : }
731 :
732 : static GEN
733 3430 : integrand_h0l(GEN selsm0, GEN s, GEN alam1, GEN x, long prec)
734 : {
735 3430 : GEN r0 = gel(selsm0, 2), ale = gel(selsm0, 3), a = gel(selsm0, 4);
736 3430 : GEN ix = ginv(x), zn = gadd(r0, gmul2n(gmul(ale, gsub(x, ix)), -1));
737 3430 : GEN P = PiI2n(0, prec), den, num;
738 3430 : den = gexpm1(gmul(P, gmul2n(gsub(zn,a), 1)), prec);
739 3430 : num = gexp(gmul(gmul(P, zn), gsub(alam1, zn)), prec);
740 3430 : num = gmul(gmul(gmul(num, ale), gmul2n(gadd(x, ix), -1)),
741 : gpow(zn, gneg(s), prec));
742 3430 : return gdiv(num, den);
743 : }
744 :
745 : static GEN
746 6860 : integrand_h12(GEN selsm1, GEN s, GEN alam1, GEN x, long prec)
747 : {
748 6860 : GEN r1 = gel(selsm1, 2), ale = gel(selsm1, 3), lam = gel(selsm1, 4);
749 6860 : GEN ix = ginv(x), zn = gadd(r1, gmul2n(gmul(ale, gsub(x, ix)), -1));
750 6860 : GEN P = PiI2n(0, prec), den, num, y;
751 6860 : den = gexpm1(gmul(P, gmul2n(gadd(zn,lam), 1)), prec);
752 6860 : num = gexp(gmul(gmul(P, zn), gadd(alam1, zn)), prec);
753 6860 : num = gmul(gmul(gmul(num, ale), gmul2n(gadd(x, ix), -1)),
754 : gpow(zn, gsubgs(s, 1), prec));
755 6860 : y = gdiv(num, den);
756 6860 : if (gcmp(garg(zn, prec), Pi2n(-2, prec)) > 0)
757 1722 : y = gmul(y, gexp(gmul(PiI2(prec), gsubsg(1, s)), prec));
758 6860 : return y;
759 : }
760 :
761 : static GEN
762 14 : integral_h0l(GEN lin_grid, GEN selsm0, GEN s, GEN a, GEN lam, long prec)
763 : {
764 14 : GEN A = gaddgs(gmul2n(gadd(a, lam),1), 1), S = gen_0;
765 14 : pari_sp av = avma;
766 14 : long j, l = lg(lin_grid);
767 :
768 3388 : for (j = 1; j < l; j++)
769 : {
770 3374 : S = gadd(S, integrand_h0l(selsm0, s, A, gel(lin_grid, j), prec));
771 3374 : if ((j & 0xff) == 0) S = gerepileupto(av, S);
772 : }
773 14 : S = gmul(m_h(selsm0), S);
774 14 : A = gmul(a, gaddsg(1, gadd(a, gmul2n(lam, 1))));
775 14 : return gmul(S, gexp(gneg(gmul(PiI2n(0, prec), A)), prec));
776 : }
777 :
778 : /* do not forget a minus sign for index 2 */
779 : static GEN
780 28 : integral_h12(GEN lin_grid, GEN selsm1, GEN s, GEN a, GEN lam, long prec)
781 : {
782 28 : GEN A, E, S = gen_0, ga = gaminus(gsubsg(1, s), prec);
783 28 : pari_sp av = avma;
784 28 : long j, l = lg(lin_grid);
785 :
786 28 : if (gequal0(ga)) return S;
787 28 : A = gaddgs(gmul2n(gadd(a,lam), 1), 1);
788 6776 : for (j = 1; j < l; j++)
789 : {
790 6748 : S = gadd(S, integrand_h12(selsm1, s, A, gel(lin_grid, j), prec));
791 6748 : if ((j & 0xff) == 0) S = gerepileupto(av, S);
792 : }
793 28 : if (gequal0(S)) return gen_0;
794 28 : S = gmul(m_h(selsm1), S);
795 28 : E = gexp(gmul(PiI2n(0, prec), gmul(lam, gaddgs(lam, 1))), prec);
796 28 : return gmul(gmul(gmul(S, ga), E), gpow(Pi2n(1, prec), gsubgs(s, 1), prec));
797 : }
798 :
799 : struct _fun_q0_t { GEN sel, s, alam1, B; };
800 : static GEN
801 56 : _fun_q0(void *E, GEN x)
802 : {
803 56 : struct _fun_q0_t *S = (struct _fun_q0_t*)E;
804 56 : GEN z = integrand_h0l(S->sel, S->s, S->alam1, x, DEFAULTPREC);
805 56 : return addrlogabs(S->B, z);
806 : }
807 : static GEN
808 112 : _fun_q12(void *E, GEN x)
809 : {
810 112 : struct _fun_q0_t *S = (struct _fun_q0_t*)E;
811 112 : GEN z = integrand_h12(S->sel, S->s, S->alam1, x, DEFAULTPREC);
812 112 : return addrlogabs(S->B, z);
813 : }
814 :
815 : static GEN
816 14 : RZLERinit(GEN s, GEN a, GEN lam, GEN al, GEN numpoles, long prec)
817 : {
818 : GEN eps, r0, r1, r2, h, lin_grid, q, q0, q1, q2, sel0, sel1, sel2, lq;
819 14 : GEN pinv = ginv(PiI2(prec)), c = gmul2n(gadd(a, lam), -1), n0, n1, n2, c2;
820 : long m;
821 : struct _fun_q0_t E;
822 :
823 14 : if (!al || gequal0(al))
824 0 : al = gcmpgs(gabs(imag_i(s), prec), 100) < 0 ? ginv(stoi(4)) : gen_1;
825 14 : c2 = gsub(gsqr(c), gmul(s, pinv));
826 14 : r0 = gadd(c, gsqrt(c2, prec));
827 14 : r1 = gsqrt(gadd(c2, pinv), prec);
828 14 : r2 = gsub(r1, c);
829 14 : r1 = gneg(gadd(r1, c));
830 14 : n0 = gfloor(gsub(gadd(real_i(r0), imag_i(r0)), a));
831 14 : n1 = gneg(gfloor(gadd(gsub(real_i(r1), imag_i(r1)), real_i(lam))));
832 14 : n2 = gfloor(gadd(gsub(real_i(r2), imag_i(r2)), real_i(lam)));
833 :
834 14 : E.s = s; E.alam1 = gaddgs(gmul2n(gadd(a, lam), 1), 1);
835 14 : E.B = mulur(prec, mplog2(prec));
836 14 : lq = gmul(al, sqrtr_abs(mulrr(divsr(prec, Pi2n(1, DEFAULTPREC)),
837 : mplog2(DEFAULTPREC))));
838 14 : eps = gexp(PiI2n(-2, prec), prec);
839 14 : E.sel = sel0 = mkvec5(n0, r0, gdiv(al, eps), a, gen_0);
840 14 : q0 = findq(&E, &_fun_q0, lq, prec);
841 :
842 14 : if (!gequal1(al)) lq = gdiv(lq, gsqr(al));
843 14 : E.sel = sel1 = mkvec5(n1, r1, gmul(al, eps), lam, gen_0);
844 14 : q1 = findq(&E, &_fun_q12, lq, prec);
845 14 : E.sel = sel2 = mkvec5(n2, r2, gmul(al, eps), lam, gen_0);
846 14 : q2 = findq(&E, &_fun_q12, lq, prec);
847 14 : q = vecmax(mkvec3(q0, q1, q2)); m = get_m(q, prec);
848 14 : gel(sel0, 5) = gel(sel1, 5) = gel(sel2, 5) = h = divru(q, (m-1) >> 1);
849 14 : lin_grid = setlin_grid_exp(h, m, prec);
850 14 : if (!numpoles) numpoles = gen_1;
851 14 : return mkvec5(sel0, sel1, sel2, lin_grid, numpoles);
852 : }
853 :
854 42 : static GEN add3(GEN x, GEN y, GEN z) { return gadd(x, gadd(y,z)); }
855 14 : static GEN addsub(GEN x, GEN y, GEN z) { return gadd(x, gsub(y,z)); }
856 :
857 : static GEN
858 14 : lerch_ours(GEN sel, GEN s, GEN a, GEN lam, long prec)
859 : {
860 14 : GEN selsm0 = gel(sel, 1), selsm1 = gel(sel, 2), selsm2 = gel(sel, 3);
861 14 : GEN lin_grid = gel(sel, 4), v0, v1, v2;
862 14 : long numpoles = itos(gel(sel, 5));
863 14 : GEN worker = snm_closure(is_entry("_serh_worker"),
864 : mkvec4(NULL, NULL, NULL, NULL));
865 14 : v0 = add3(series_h0l(worker, m_n0(selsm0), s, a, lam, prec),
866 : series_residues_h0l(numpoles, selsm0, s, a, lam, prec),
867 : integral_h0l(lin_grid, selsm0, s, a, lam, prec));
868 14 : v1 = add3(series_h1(worker, m_n0(selsm1), s, a, lam, prec),
869 : series_residues_h1(numpoles, selsm1, s, a, lam, prec),
870 : integral_h12(lin_grid, selsm1, s, a, lam, prec));
871 14 : v2 = addsub(series_h2(worker, m_n0(selsm2), s, a, lam, prec),
872 : series_residues_h2(numpoles, selsm2, s, a, lam, prec),
873 : integral_h12(lin_grid, selsm2, s, a, lam, prec));
874 14 : return add3(v0, v1, v2);
875 : }
876 :
877 : static GEN
878 0 : RZlerch_easy(GEN s, GEN a, GEN lam, long prec)
879 : {
880 0 : pari_sp av = avma;
881 : GEN z, y, N;
882 0 : long B = prec2nbits(prec), LD = LOWDEFAULTPREC;
883 0 : N = gdiv(gmulsg(B + 5, mplog2(LD)), gmul(Pi2n(1, LD), imag_i(lam)));
884 0 : if (gexpo(N) > 40) pari_err_IMPL("precision too large in lerchzeta");
885 0 : N = gceil(N); prec += EXTRAPREC64;
886 0 : z = typ(lam) == t_INT ? gen_1 : gexp(gmul(PiI2(prec), lam), prec);
887 0 : y = parsum(gen_0, N, snm_closure(is_entry("_serh_worker"),
888 : mkvec4(z, a, gneg(s), stoi(prec))));
889 0 : return gerepilecopy(av, gprec_wtrunc(y, prec));
890 : }
891 :
892 : static GEN
893 14 : mygfrac(GEN z)
894 0 : { return typ(z) == t_COMPLEX ? mkcomplex(gfrac(real_i(z)), imag_i(z))
895 14 : : gfrac(z); }
896 :
897 : static GEN
898 42 : lerchlarge(GEN s, GEN a, GEN lam, GEN al, GEN numpoles, long prec)
899 : {
900 42 : pari_sp av = avma;
901 42 : GEN val, sel, imlam = imag_i(lam);
902 : long prec2;
903 42 : switch(gsigne(imlam))
904 : {
905 0 : case -1: pari_err_IMPL("imag(lam) < 0");
906 0 : case 1: if (gexpo(imlam) >= -16) return RZlerch_easy(s, a, lam, prec);
907 : }
908 42 : if (gcmpgs(real_i(a), 1) < 0)
909 : {
910 14 : GEN P = gexp(gmul(PiI2(prec), lam), prec);
911 14 : GEN L = lerchlarge(s, gaddgs(a, 1), lam, al, numpoles, prec);
912 14 : return gerepileupto(av, gadd(gpow(a, gneg(s), prec), gmul(P, L)));
913 : }
914 28 : if (gcmpgs(real_i(a), 2) >= 0)
915 : {
916 0 : GEN L, P = gexp(gneg(gmul(PiI2(prec), lam)), prec);
917 0 : a = gsubgs(a, 1); L = lerchlarge(s, a, lam, al, numpoles, prec);
918 0 : return gerepileupto(av, gmul(P, gsub(L, gpow(a, gneg(s), prec))));
919 : }
920 28 : if (gsigne(imag_i(s)) > 0)
921 : {
922 : GEN L;
923 14 : lam = mygfrac(gneg(gconj(lam)));
924 14 : L = lerchlarge(gconj(s), a, lam, al, numpoles, prec);
925 14 : return gerepileupto(av, gconj(L));
926 : }
927 14 : prec2 = prec + EXTRAPREC64;
928 14 : a = gprec_w(a, prec2);
929 14 : s = gprec_w(s, prec2);
930 14 : lam = gprec_w(lam, prec2);
931 14 : sel = RZLERinit(s, a, lam, al, numpoles, prec2);
932 14 : val = lerch_ours(sel, s, a, lam, prec2);
933 14 : return gerepilecopy(av, gprec_wtrunc(val, prec));
934 : }
935 :
936 : GEN
937 7 : zetahurwitzlarge(GEN s, GEN a, long prec)
938 7 : { return lerchlarge(s, a, gen_0, gen_1, gen_1, prec); }
939 :
940 : GEN
941 7 : lerchzetalarge(GEN s, GEN a, GEN lam, long prec)
942 7 : { return lerchlarge(s, a, lam, gen_1, gen_1, prec); }
|
