def dc_calc_kloosterman(modulus, number):
arg = request.args.get("val", [])
if not arg:
return flask.abort(404)
try:
a, b = map(int, arg.split(','))
if modulus == 1:
# there is a bug in sage for modulus = 1
return r"""
\( \displaystyle K(%s,%s,\chi_{1}(1,·))
= \sum_{r \in \mathbb{Z}/\mathbb{Z}}
\chi_{1}(1,r) 1^{%s r + %s r^{-1}}
= 1 \)
""" % (a, b, a, b)
from dirichlet_conrey import DirichletGroup_conrey
chi = DirichletGroup_conrey(modulus)[number]
chi = chi.sage_character()
k = chi.kloosterman_sum_numerical(100, a, b)
real = round(k.real(), 5)
imag = round(k.imag(), 5)
if imag == 0:
k = str(real)
elif real == 0:
k = str(imag) + "i"
else:
k = latex(k)
return r"""
\( \displaystyle K(%s,%s,\chi_{%s}(%s,·))
= \sum_{r \in \mathbb{Z}/%s\mathbb{Z}}
\chi_{%s}(%s,r) e\left(\frac{%s r + %s r^{-1}}{%s}\right)
= %s. \)""" % (a, b, modulus, number, modulus, modulus, number, a, b, modulus, k)
except Exception, e:
return "<span style='color:red;'>ERROR: %s</span>" % e
def dc_calc_jacobi(modulus, number):
arg = request.args.get("val", [])
if not arg:
return flask.abort(404)
arg = map(int, arg.split('.'))
try:
num = arg[0]
from dirichlet_conrey import DirichletGroup_conrey
chi = DirichletGroup_conrey(modulus)[number]
psi = DirichletGroup_conrey(modulus)[num]
chi = chi.sage_character()
psi = psi.sage_character()
jacobi_sum = chi.jacobi_sum(psi)
return r"\( \displaystyle J(\chi_{%s}(%s,·),\chi_{%s}(%s,·)) = \sum_{r\in \mathbb{Z}/%s\mathbb{Z}} \chi_{%s}(%s,r) \chi_{%s}(%s,1-r) = %s.\)" % (modulus, number, modulus, num, modulus, modulus, number, modulus, num, latex(jacobi_sum))
except Exception, e:
return "<span style='color:red;'>ERROR: %s</span>" % e
def dc_calc_gauss(modulus, number):
arg = request.args.get("val", [])
if not arg:
return flask.abort(404)
try:
if modulus == 1:
# there is a bug in sage for modulus = 1
return r"""\(\displaystyle \tau_{%s}(\chi_{1}(1,·)) =
\sum_{r\in \mathbb{Z}/\mathbb{Z}} \chi_{1}(1,r) 1^{%s}= 1. \)""" % (int(arg), int(arg))
from dirichlet_conrey import DirichletGroup_conrey
chi = DirichletGroup_conrey(modulus)[number]
chi = chi.sage_character()
g = chi.gauss_sum_numerical(100, int(arg))
real = round(g.real(), 10)
imag = round(g.imag(), 10)
if imag == 0.:
g = str(real)
elif real == 0.:
g = str(imag) + "i"
else:
g = latex(g)
from sage.rings.rational import Rational
x = Rational('%s/%s' % (int(arg), modulus))
n = x.numerator()
n = str(n) + "r" if not n == 1 else "r"
d = x.denominator()
return r"\(\displaystyle \tau_{%s}(\chi_{%s}(%s,·)) = \sum_{r\in \mathbb{Z}/%s\mathbb{Z}} \chi_{%s}(%s,r) e\left(\frac{%s}{%s}\right) = %s. \)" % (int(arg), modulus, number, modulus, modulus, number, n, d, g)
except Exception, e:
return "<span style='color:red;'>ERROR: %s</span>" % e
def dc_calc_kloosterman(modulus, number):
arg = request.args.get("val", [])
if not arg:
return flask.abort(404)
try:
a, b = map(int, arg.split(','))
if modulus == 1:
# there is a bug in sage for modulus = 1
return r"""
\( \displaystyle K(%s,%s,\chi_{1}(1,·))
= \sum_{r \in \mathbb{Z}/\mathbb{Z}}
\chi_{1}(1,r) 1^{%s r + %s r^{-1}}
= 1 \)
""" % (a, b, a, b)
from dirichlet_conrey import DirichletGroup_conrey
chi = DirichletGroup_conrey(modulus)[number]
chi = chi.sage_character()
k = chi.kloosterman_sum_numerical(100, a, b)
real = round(k.real(), 5)
imag = round(k.imag(), 5)
if imag == 0:
k = str(real)
elif real == 0:
k = str(imag) + "i"
else:
k = latex(k)
return r"""
\( \displaystyle K(%s,%s,\chi_{%s}(%s,·))
= \sum_{r \in \mathbb{Z}/%s\mathbb{Z}}
\chi_{%s}(%s,r) e\left(\frac{%s r + %s r^{-1}}{%s}\right)
= %s. \)""" % (a, b, modulus, number, modulus, modulus, number, a, b,
modulus, k)
except Exception, e:
return "<span style='color:red;'>ERROR: %s</span>" % e
def dc_calc_gauss(modulus, number):
arg = request.args.get("val", [])
if not arg:
return flask.abort(404)
try:
if modulus == 1:
# there is a bug in sage for modulus = 1
return r"""\(\displaystyle \tau_{%s}(\chi_{1}(1,·)) =
\sum_{r\in \mathbb{Z}/\mathbb{Z}} \chi_{1}(1,r) 1^{%s}= 1. \)""" % (
int(arg), int(arg))
from dirichlet_conrey import DirichletGroup_conrey
chi = DirichletGroup_conrey(modulus)[number]
chi = chi.sage_character()
g = chi.gauss_sum_numerical(100, int(arg))
real = round(g.real(), 10)
imag = round(g.imag(), 10)
if imag == 0.:
g = str(real)
elif real == 0.:
g = str(imag) + "i"
else:
g = latex(g)
from sage.rings.rational import Rational
x = Rational('%s/%s' % (int(arg), modulus))
n = x.numerator()
n = str(n) + "r" if not n == 1 else "r"
d = x.denominator()
return r"\(\displaystyle \tau_{%s}(\chi_{%s}(%s,·)) = \sum_{r\in \mathbb{Z}/%s\mathbb{Z}} \chi_{%s}(%s,r) e\left(\frac{%s}{%s}\right) = %s. \)" % (
int(arg), modulus, number, modulus, modulus, number, n, d, g)
except Exception, e:
return "<span style='color:red;'>ERROR: %s</span>" % e
def dc_calc_jacobi(modulus, number):
arg = request.args.get("val", [])
if not arg:
return flask.abort(404)
arg = map(int, arg.split('.'))
try:
num = arg[0]
from dirichlet_conrey import DirichletGroup_conrey
chi = DirichletGroup_conrey(modulus)[number]
psi = DirichletGroup_conrey(modulus)[num]
chi = chi.sage_character()
psi = psi.sage_character()
jacobi_sum = chi.jacobi_sum(psi)
return r"\( \displaystyle J(\chi_{%s}(%s,·),\chi_{%s}(%s,·)) = \sum_{r\in \mathbb{Z}/%s\mathbb{Z}} \chi_{%s}(%s,r) \chi_{%s}(%s,1-r) = %s.\)" % (
modulus, number, modulus, num, modulus, modulus, number, modulus,
num, latex(jacobi_sum))
except Exception, e:
return "<span style='color:red;'>ERROR: %s</span>" % e