def plot_c_q_m():
q_range = 500
m_range = 20
s = [m_range, q_range]
c_q_m_2d = np.zeros(s, dtype = np.complex64)
for m in range(1, m_range + 1):
for q in range(1, q_range + 1):
c_q_m = rama_sum_use_prim_unit_root(q, m)
c_q_m_2d[m - 1][q - 1] = c_q_m
plt.figure()
t_range_list = range(1, q_range + 1)
title_str = 'c_q_m, q_range %d'%(q_range)
plot_one_list(np.asarray(c_q_m_2d[0]).real, title_str, t_range_list, 'b-*')
# plot_one_list(np.asarray(c_q_m_2d[1]).real, title_str, t_range_list, 'c-*')
# plot_one_list(np.asarray(c_q_m_2d[2]).real, title_str, t_range_list, 'y-*')
# plot_one_list(np.asarray(c_q_m_2d[3]).real, title_str, t_range_list, 'k-*')
plot_one_list(np.asarray(c_q_m_2d[19]).real, title_str, t_range_list, 'r-*')
xlabel_str = 'blue: m=1, cyran: m=2, yellow: m=3, black, m=4, red: m=19'
plt.xlabel(xlabel_str)
def gaus_sum_for_principle_char_upto_n(N):
gaus_sum_arr = [1]*N
# start from 1, because dirichi_char_for_n() is not defined for 1
for n in range(2, N + 1):
# print 'n = ', n
# get dirich_char_mat, which is a 2D matrix
dirich_char_mat, phi_n, coprime_list, prim_char_stat_all = dirichi_char_for_n(n)
# char_index = 0 will give principle char
G_chi = gaus_sum_for_dirich_char_by_index(0, n, dirich_char_mat, phi_n)
gaus_sum_arr[n - 1] = G_chi
plt.figure()
t_range_list = range(1, N+1)
title_str = 'gaus_sum, for principle character N %d'%(N)
plot_one_list(np.absolute(gaus_sum_arr), title_str, t_range_list, 'b-*')
plot_one_list(np.asarray(gaus_sum_arr).real, title_str, t_range_list, 'c-*')
plot_one_list(np.asarray(gaus_sum_arr).imag, title_str, t_range_list, 'y-*')
xlabel_str = 'blue: abs, cyran: real, yellow: imag'
plt.xlabel(xlabel_str)
# need the round to int first, then astype(int)
gaus_sum_principle_char = np.round(np.asarray(gaus_sum_arr).real).astype(int)
print gaus_sum_principle_char
return gaus_sum_principle_char
def gaus_sum_for_N(N, more_verify = False):
# get dirich_char_mat, which is a 2D matrix
dirich_char_mat, phi_n, coprime_list, prim_char_stat_all = dirichi_char_for_n(N)
gaus_sum_arr = [0]*phi_n
# this is the theoretical result, absolute(G(1, chi)) = sqrt(N)
gaus_sum_for_prim_char = np.sqrt(N)
ALMOST_ZERO = 0.001
gaus_sum_prim_char_cnt = 0
# print np.asarray(dirich_char_mat)
if(more_verify == True):
verify_all_propertiers(dirich_char_mat, N, phi_n, coprime_list)
# Note, in the following plot, 1st row skipped, so its gaus_sum is not correct.
verify_poly_vinograd_gaus_sum(dirich_char_mat, N, phi_n, prim_char_stat_all)
# number of row. Note: 1st row is principle
for i in range(0, phi_n):
G_chi = gaus_sum_for_dirich_char_by_index(i, N, dirich_char_mat, phi_n)
gaus_sum_arr[i] = G_chi
# For prim_char, its gaus_sum == sqrt(N), let's verify this
diff = np.absolute(G_chi) - gaus_sum_for_prim_char
if(np.absolute(diff) < ALMOST_ZERO):
gaus_sum_prim_char_cnt += 1
plt.figure()
t_range_list = range(phi_n)
title_str = 'gaus_sum, N %d, phi_N %d, gaus_sum_prim_char_cnt %d'%(N, phi_n, gaus_sum_prim_char_cnt)
# for prim_char, gaus_sum shall be equal to sqrt(N)
plot_one_list(np.absolute(gaus_sum_arr), title_str, t_range_list, 'b-*')
plot_one_list(np.asarray(gaus_sum_arr).real, title_str, t_range_list, 'c-*')
plot_one_list(np.asarray(gaus_sum_arr).imag, title_str, t_range_list, 'y-*')
xlabel_str = 'sqrt(N): %f, blue: abs, cyran: real, yellow: imag'%(np.sqrt(N))
plt.xlabel(xlabel_str)
