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%md










︠%auto

a = 5




a
5 
def fig_riemann_spectrum_gaps():
    t = stats.TimeSeries(zeta_zeros())
    g = t.diffs().plot_histogram(bins=500, normalize=False)
    g.show()

fig_riemann_spectrum_gaps()

def fig_mini_phihat_even():
    G = plot_symbolic_phihat(5, 1, 100, 10000, zeros=True)
    G.show()


def fig_phihat_even():
    for bound in [5, 10, 20]:
        G = plot_symbolic_phihat(bound, 2, 100,
                            plot_points=10^5)
        G.show()

def fig_phihat_even_all():
    p = [plot_symbolic_phihat(n, 2,100) for n in [5,20,50,500]]
    [a.ymin(0) for a in p]
    for a in p:
        a.show()
    g = graphics_array([[a] for a in p],4,1)
    g.show()

def symbolic_phihat(bound):
    t = var('t')
    f = SR(0)#what is this - all expressions learned in calculus
    for pn in prime_powers(bound+1):
        if pn == 1: continue
        p, e = factor(pn)[0]
        f += - log(p)/sqrt(pn) * cos(t*log(pn))
    return f

#omit 2 and - signs for p = 3 (mod 4)
#do Fourier trans on sum in phi
def plot_symbolic_phihat(bound, xmin, xmax, plot_points=1000, zeros=True):
    f = symbolic_phihat(bound)
    P = plot(f, (t,xmin, xmax), plot_points=plot_points)
    if not zeros:
        return P
    ym = P.ymax()
    Z = []
    for y in zeta_zeros():
        if y > xmax: break
        Z.append(y)
    zeros = sum([arrow((x,ym),(x,0),rgbcolor='red',width=0.5,arrowsize=2)
                 for i, x in enumerate(Z)])
    return P + zeros
fig_mini_phihat_even()
#fig_phihat_even()
Error in lines 1-4 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "sage/structure/sage_object.pyx", line 1097, in sage.structure.sage_object.save (/projects/sage/sage-6.10/src/build/cythonized/sage/structure/sage_object.c:12013) if not os.path.splitext(filename)[1] or not hasattr(obj, "save"): File "/projects/sage/sage-6.10/local/lib/python/posixpath.py", line 98, in splitext return genericpath._splitext(p, sep, altsep, extsep) File "/projects/sage/sage-6.10/local/lib/python/genericpath.py", line 99, in _splitext sepIndex = p.rfind(sep) AttributeError: 'NoneType' object has no attribute 'rfind' 
@auto
def gauss_primepowers(bound):
    """
    Computes all powers of Gaussian primes less than a given norm.
    Returns them as a list
    """
    def pgen(I):
        I.is_principal()
        assert len(I._reduced_generators) == 1
        return I._reduced_generators[0]

    def exponentiate(base, power):
        res = base
        while power > 1:
            res = res * base
            power = power - 1
        return res

    K.<i> = QQ[sqrt(-1)]
    eles = K.primes_of_bounded_norm(bound-1)
    print "eles: ",
    print eles
    primes = [pgen(I) for I in eles]
    print "primes: ",
    print primes
    primepowers = []
    for p in primes:
        primepowers.append(p)
        n = 2
        p = exponentiate(p,n)
        while p.norm() < bound:
            primepowers.append(p)
            n = n + 1
            p = exponentiate(p,n)
    return primepowers

def zeros(y):
    s = os.popen('lcalc --zeros-interval -x 1 -y %s --stepsize .01 --twist-quadratic -s -4 -f -4'%y).readlines()
    return [float(z.split()[1]) for z in s]

#omit 2 and - signs for p = 3 (mod 4)
#do Fourier trans on sum in phi
def gauss_symbolic_phihat(bound):
    t = var('t')
    f = SR(0)#what is this - all expressions learned in calculus
    for pn in prime_powers(bound+1):
        if pn == 1: continue
        if pn % 2 != 0:
            p, e = factor(pn)[0]
            if p % 4 == 3:
                f += log(p)/sqrt(pn) * cos(t*log(pn))
            else:
                f += -log(p)/sqrt(pn) * cos(t*log(pn))
    return f

def gauss_plot_symbolic_phihat(bound, xmin, xmax, plot_points=1000, show_zeros=True):
    f = gauss_symbolic_phihat(bound)
    P = plot(f, (t,xmin, xmax), plot_points=plot_points)
    if not show_zeros:
        return P
    ym = P.ymax()
    Z = []
    for y in zeros(xmax):
        Z.append(y)
    zeros_arrows = sum([arrow((x,ym),(x,0),rgbcolor='red',width=0.5,arrowsize=2)
                 for i, x in enumerate(Z)])
    return P + zeros_arrows

def gauss_fig_mini_phihat_even():
    G = gauss_plot_symbolic_phihat(5, 1, 100, 10000)
    G.show()

plot_symbolic_phihat(5, 2,100)

%time
frames = []
for n in [6,7,8]:
    print n
    p = gauss_plot_symbolic_phihat(n, 2,100)
    p.xmin(0); p.xmax(100); p.ymin(-2); p.ymax(2)
    frames.append(p)
6 7 8 CPU time: 12.98 s, Wall time: 13.63 s 
animate(frames).save('a.gif', delay=50)


for n in [5, 20, 50]:
    g = gauss_plot_symbolic_phihat(n, 2,100)
    g.show()





︠

zeros(50)
[6.0209489046976, 10.243770304167, 12.988098012312, 16.342607104587, 18.291993196124, 21.450611343983, 23.27837652046, 25.728756425089, 28.359634343025, 29.656384014593, 32.592186527117, 34.199957509213, 36.142880458303, 38.511923141719, 40.322674066691, 41.807084620005, 44.617891058662, 45.599584396792, 47.741562280939, 49.723129323783] 
def lbl(name, f):
    open("thesis/graphics/%s.tex"%name, 'w').write("$$f(t)=%s$$"%latex(f))
f = gauss_symbolic_phihat(3)
lbl("intro1", f)
p = f.plot(0,20)
save(p, 'thesis/graphics/intro1.pdf')



f = gauss_symbolic_phihat(5); show(f)
lbl('intro2', f)
p = f.plot(0, 20)
p
save(p, 'thesis/graphics/intro2.pdf')
−15 5cos⁡(tlog⁡(5))log⁡(5)+13 3cos⁡(tlog⁡(3))log⁡(3)\displaystyle -\frac{1}{5} \, \sqrt{5} \cos\left(t \log\left(5\right)\right) \log\left(5\right) + \frac{1}{3} \, \sqrt{3} \cos\left(t \log\left(3\right)\right) \log\left(3\right)
f = gauss_symbolic_phihat(7)
lbl("intro3", f)
p = f.plot(0, 20)
save(p, 'thesis/graphics/intro3.pdf')

f = gauss_symbolic_phihat(11); show(f)
f.plot(0, 20)
111 11cos⁡(tlog⁡(11))log⁡(11)+17 7cos⁡(tlog⁡(7))log⁡(7)−15 5cos⁡(tlog⁡(5))log⁡(5)+13 3cos⁡(tlog⁡(3))log⁡(3)+13 cos⁡(tlog⁡(9))log⁡(3)\displaystyle \frac{1}{11} \, \sqrt{11} \cos\left(t \log\left(11\right)\right) \log\left(11\right) + \frac{1}{7} \, \sqrt{7} \cos\left(t \log\left(7\right)\right) \log\left(7\right) - \frac{1}{5} \, \sqrt{5} \cos\left(t \log\left(5\right)\right) \log\left(5\right) + \frac{1}{3} \, \sqrt{3} \cos\left(t \log\left(3\right)\right) \log\left(3\right) + \frac{1}{3} \, \cos\left(t \log\left(9\right)\right) \log\left(3\right)
f = gauss_symbolic_phihat(13); show(f)
f.plot(0, 20)
−113 13cos⁡(tlog⁡(13))log⁡(13)+111 11cos⁡(tlog⁡(11))log⁡(11)+17 7cos⁡(tlog⁡(7))log⁡(7)−15 5cos⁡(tlog⁡(5))log⁡(5)+13 3cos⁡(tlog⁡(3))log⁡(3)+13 cos⁡(tlog⁡(9))log⁡(3)\displaystyle -\frac{1}{13} \, \sqrt{13} \cos\left(t \log\left(13\right)\right) \log\left(13\right) + \frac{1}{11} \, \sqrt{11} \cos\left(t \log\left(11\right)\right) \log\left(11\right) + \frac{1}{7} \, \sqrt{7} \cos\left(t \log\left(7\right)\right) \log\left(7\right) - \frac{1}{5} \, \sqrt{5} \cos\left(t \log\left(5\right)\right) \log\left(5\right) + \frac{1}{3} \, \sqrt{3} \cos\left(t \log\left(3\right)\right) \log\left(3\right) + \frac{1}{3} \, \cos\left(t \log\left(9\right)\right) \log\left(3\right)

f = gauss_symbolic_phihat(20); show(f)
f.plot(0, 20)
119 19cos⁡(tlog⁡(19))log⁡(19)−117 17cos⁡(tlog⁡(17))log⁡(17)−113 13cos⁡(tlog⁡(13))log⁡(13)+111 11cos⁡(tlog⁡(11))log⁡(11)+17 7cos⁡(tlog⁡(7))log⁡(7)−15 5cos⁡(tlog⁡(5))log⁡(5)+13 3cos⁡(tlog⁡(3))log⁡(3)+13 cos⁡(tlog⁡(9))log⁡(3)\displaystyle \frac{1}{19} \, \sqrt{19} \cos\left(t \log\left(19\right)\right) \log\left(19\right) - \frac{1}{17} \, \sqrt{17} \cos\left(t \log\left(17\right)\right) \log\left(17\right) - \frac{1}{13} \, \sqrt{13} \cos\left(t \log\left(13\right)\right) \log\left(13\right) + \frac{1}{11} \, \sqrt{11} \cos\left(t \log\left(11\right)\right) \log\left(11\right) + \frac{1}{7} \, \sqrt{7} \cos\left(t \log\left(7\right)\right) \log\left(7\right) - \frac{1}{5} \, \sqrt{5} \cos\left(t \log\left(5\right)\right) \log\left(5\right) + \frac{1}{3} \, \sqrt{3} \cos\left(t \log\left(3\right)\right) \log\left(3\right) + \frac{1}{3} \, \cos\left(t \log\left(9\right)\right) \log\left(3\right)
%time
f = gauss_symbolic_phihat(200); show(f)
f.plot(0, 20, plot_points=100)
1199 199cos⁡(tlog⁡(199))log⁡(199)−1197 197cos⁡(tlog⁡(197))log⁡(197)−1193 193cos⁡(tlog⁡(193))log⁡(193)+1191 191cos⁡(tlog⁡(191))log⁡(191)−1181 181cos⁡(tlog⁡(181))log⁡(181)+1179 179cos⁡(tlog⁡(179))log⁡(179)−1173 173cos⁡(tlog⁡(173))log⁡(173)+1167 167cos⁡(tlog⁡(167))log⁡(167)+1163 163cos⁡(tlog⁡(163))log⁡(163)−1157 157cos⁡(tlog⁡(157))log⁡(157)+1151 151cos⁡(tlog⁡(151))log⁡(151)−1149 149cos⁡(tlog⁡(149))log⁡(149)+1139 139cos⁡(tlog⁡(139))log⁡(139)−1137 137cos⁡(tlog⁡(137))log⁡(137)+1131 131cos⁡(tlog⁡(131))log⁡(131)+1127 127cos⁡(tlog⁡(127))log⁡(127)−1113 113cos⁡(tlog⁡(113))log⁡(113)−1109 109cos⁡(tlog⁡(109))log⁡(109)+1107 107cos⁡(tlog⁡(107))log⁡(107)+1103 103cos⁡(tlog⁡(103))log⁡(103)−1101 101cos⁡(tlog⁡(101))log⁡(101)−197 97cos⁡(tlog⁡(97))log⁡(97)−189 89cos⁡(tlog⁡(89))log⁡(89)+183 83cos⁡(tlog⁡(83))log⁡(83)+179 79cos⁡(tlog⁡(79))log⁡(79)−173 73cos⁡(tlog⁡(73))log⁡(73)+171 71cos⁡(tlog⁡(71))log⁡(71)+167 67cos⁡(tlog⁡(67))log⁡(67)−161 61cos⁡(tlog⁡(61))log⁡(61)+159 59cos⁡(tlog⁡(59))log⁡(59)−153 53cos⁡(tlog⁡(53))log⁡(53)+147 47cos⁡(tlog⁡(47))log⁡(47)+143 43cos⁡(tlog⁡(43))log⁡(43)−141 41cos⁡(tlog⁡(41))log⁡(41)−137 37cos⁡(tlog⁡(37))log⁡(37)+131 31cos⁡(tlog⁡(31))log⁡(31)−129 29cos⁡(tlog⁡(29))log⁡(29)+123 23cos⁡(tlog⁡(23))log⁡(23)+119 19cos⁡(tlog⁡(19))log⁡(19)−117 17cos⁡(tlog⁡(17))log⁡(17)−113 13cos⁡(tlog⁡(13))log⁡(13)+111 11cos⁡(tlog⁡(11))log⁡(11)+17 7cos⁡(tlog⁡(7))log⁡(7)−125 5cos⁡(tlog⁡(125))log⁡(5)−15 5cos⁡(tlog⁡(5))log⁡(5)+19 3cos⁡(tlog⁡(27))log⁡(3)+13 3cos⁡(tlog⁡(3))log⁡(3)−113 cos⁡(tlog⁡(169))log⁡(13)+111 cos⁡(tlog⁡(121))log⁡(11)+17 cos⁡(tlog⁡(49))log⁡(7)−15 cos⁡(tlog⁡(25))log⁡(5)+19 cos⁡(tlog⁡(81))log⁡(3)+13 cos⁡(tlog⁡(9))log⁡(3)\displaystyle \frac{1}{199} \, \sqrt{199} \cos\left(t \log\left(199\right)\right) \log\left(199\right) - \frac{1}{197} \, \sqrt{197} \cos\left(t \log\left(197\right)\right) \log\left(197\right) - \frac{1}{193} \, \sqrt{193} \cos\left(t \log\left(193\right)\right) \log\left(193\right) + \frac{1}{191} \, \sqrt{191} \cos\left(t \log\left(191\right)\right) \log\left(191\right) - \frac{1}{181} \, \sqrt{181} \cos\left(t \log\left(181\right)\right) \log\left(181\right) + \frac{1}{179} \, \sqrt{179} \cos\left(t \log\left(179\right)\right) \log\left(179\right) - \frac{1}{173} \, \sqrt{173} \cos\left(t \log\left(173\right)\right) \log\left(173\right) + \frac{1}{167} \, \sqrt{167} \cos\left(t \log\left(167\right)\right) \log\left(167\right) + \frac{1}{163} \, \sqrt{163} \cos\left(t \log\left(163\right)\right) \log\left(163\right) - \frac{1}{157} \, \sqrt{157} \cos\left(t \log\left(157\right)\right) \log\left(157\right) + \frac{1}{151} \, \sqrt{151} \cos\left(t \log\left(151\right)\right) \log\left(151\right) - \frac{1}{149} \, \sqrt{149} \cos\left(t \log\left(149\right)\right) \log\left(149\right) + \frac{1}{139} \, \sqrt{139} \cos\left(t \log\left(139\right)\right) \log\left(139\right) - \frac{1}{137} \, \sqrt{137} \cos\left(t \log\left(137\right)\right) \log\left(137\right) + \frac{1}{131} \, \sqrt{131} \cos\left(t \log\left(131\right)\right) \log\left(131\right) + \frac{1}{127} \, \sqrt{127} \cos\left(t \log\left(127\right)\right) \log\left(127\right) - \frac{1}{113} \, \sqrt{113} \cos\left(t \log\left(113\right)\right) \log\left(113\right) - \frac{1}{109} \, \sqrt{109} \cos\left(t \log\left(109\right)\right) \log\left(109\right) + \frac{1}{107} \, \sqrt{107} \cos\left(t \log\left(107\right)\right) \log\left(107\right) + \frac{1}{103} \, \sqrt{103} \cos\left(t \log\left(103\right)\right) \log\left(103\right) - \frac{1}{101} \, \sqrt{101} \cos\left(t \log\left(101\right)\right) \log\left(101\right) - \frac{1}{97} \, \sqrt{97} \cos\left(t \log\left(97\right)\right) \log\left(97\right) - \frac{1}{89} \, \sqrt{89} \cos\left(t \log\left(89\right)\right) \log\left(89\right) + \frac{1}{83} \, \sqrt{83} \cos\left(t \log\left(83\right)\right) \log\left(83\right) + \frac{1}{79} \, \sqrt{79} \cos\left(t \log\left(79\right)\right) \log\left(79\right) - \frac{1}{73} \, \sqrt{73} \cos\left(t \log\left(73\right)\right) \log\left(73\right) + \frac{1}{71} \, \sqrt{71} \cos\left(t \log\left(71\right)\right) \log\left(71\right) + \frac{1}{67} \, \sqrt{67} \cos\left(t \log\left(67\right)\right) \log\left(67\right) - \frac{1}{61} \, \sqrt{61} \cos\left(t \log\left(61\right)\right) \log\left(61\right) + \frac{1}{59} \, \sqrt{59} \cos\left(t \log\left(59\right)\right) \log\left(59\right) - \frac{1}{53} \, \sqrt{53} \cos\left(t \log\left(53\right)\right) \log\left(53\right) + \frac{1}{47} \, \sqrt{47} \cos\left(t \log\left(47\right)\right) \log\left(47\right) + \frac{1}{43} \, \sqrt{43} \cos\left(t \log\left(43\right)\right) \log\left(43\right) - \frac{1}{41} \, \sqrt{41} \cos\left(t \log\left(41\right)\right) \log\left(41\right) - \frac{1}{37} \, \sqrt{37} \cos\left(t \log\left(37\right)\right) \log\left(37\right) + \frac{1}{31} \, \sqrt{31} \cos\left(t \log\left(31\right)\right) \log\left(31\right) - \frac{1}{29} \, \sqrt{29} \cos\left(t \log\left(29\right)\right) \log\left(29\right) + \frac{1}{23} \, \sqrt{23} \cos\left(t \log\left(23\right)\right) \log\left(23\right) + \frac{1}{19} \, \sqrt{19} \cos\left(t \log\left(19\right)\right) \log\left(19\right) - \frac{1}{17} \, \sqrt{17} \cos\left(t \log\left(17\right)\right) \log\left(17\right) - \frac{1}{13} \, \sqrt{13} \cos\left(t \log\left(13\right)\right) \log\left(13\right) + \frac{1}{11} \, \sqrt{11} \cos\left(t \log\left(11\right)\right) \log\left(11\right) + \frac{1}{7} \, \sqrt{7} \cos\left(t \log\left(7\right)\right) \log\left(7\right) - \frac{1}{25} \, \sqrt{5} \cos\left(t \log\left(125\right)\right) \log\left(5\right) - \frac{1}{5} \, \sqrt{5} \cos\left(t \log\left(5\right)\right) \log\left(5\right) + \frac{1}{9} \, \sqrt{3} \cos\left(t \log\left(27\right)\right) \log\left(3\right) + \frac{1}{3} \, \sqrt{3} \cos\left(t \log\left(3\right)\right) \log\left(3\right) - \frac{1}{13} \, \cos\left(t \log\left(169\right)\right) \log\left(13\right) + \frac{1}{11} \, \cos\left(t \log\left(121\right)\right) \log\left(11\right) + \frac{1}{7} \, \cos\left(t \log\left(49\right)\right) \log\left(7\right) - \frac{1}{5} \, \cos\left(t \log\left(25\right)\right) \log\left(5\right) + \frac{1}{9} \, \cos\left(t \log\left(81\right)\right) \log\left(3\right) + \frac{1}{3} \, \cos\left(t \log\left(9\right)\right) \log\left(3\right)
CPU time: 33.98 s, Wall time: 34.76 s 
zeros(100)
[6.0209489046976, 10.243770304167, 12.988098012312, 16.342607104587, 18.291993196124, 21.450611343983, 23.27837652046, 25.728756425089, 28.359634343025, 29.656384014593, 32.592186527117, 34.199957509213, 36.142880458303, 38.511923141719, 40.322674066691, 41.807084620005, 44.617891058662, 45.599584396792, 47.741562280939, 49.723129323783, 51.686093452871, 52.768820767805, 55.267543584699, 56.934374055202, 58.116707110674, 60.421713949008, 62.008632285768, 63.714641118785, 64.976170573096, 67.636920863546, 68.365884503834, 70.185879908802, 72.155484974382, 73.767635521486, 75.143121647433, 76.69630320343, 78.809998314321, 80.210131238367, 81.213951626883, 83.666656014471, 84.731740363782, 86.57766016839, 87.629718119588, 89.801131616696, 91.349703814698, 92.237499910454, 94.16661958596, 96.136011161781, 96.961741579417, 98.755300415755] 
g = f.derivative()

f.find_local_maximum(5.5,6.5)
(4.625184318694296, 6.0932418136308746) 
%time f = gauss_symbolic_phihat(20000)
f.find_local_maximum(5.5,6.5)
CPU time: 1.40 s, Wall time: 1.42 s (9.609716603100983, 6.0190156444824714) 
%time f = gauss_symbolic_phihat(100000)
f.find_local_maximum(5.5,6.5)
CPU time: 9.16 s, Wall time: 9.49 s (10.306059947208393, 6.0248752427482559) 
g = f._fast_float_()
/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py:1387: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...) See http://trac.sagemath.org/5930 for details. return FastFloatConverter(ex, *vars)() 
%time plot(g, 5, 100)
CPU time: 8.78 s, Wall time: 8.88 s 
p = 1e9
N(log(p)/sqrt(p))
0.000655327206019062 

@cached_function
def zeros(y, ring):
    s = os.popen('lcalc --zeros-interval -x 1 -y %s --stepsize .01 --twist-quadratic -s %s -f %s'%(y, ring.discriminant(),
                                                                                                  ring.discriminant())).readlines()
    return [float(z.split()[1]) for z in s]

#omit 2 and - signs for p = 3 (mod 4)
#do Fourier trans on sum in phi
def arb_symbolic_phihat(bound,delta):
    t = var('t')
    f = SR(0)#what is this - all expressions learned in calculus
    for pn in prime_powers(bound+1):
        if pn == 1: continue
        if pn % 2 != 0:
            p, e = factor(pn)[0]
            #print "Kronecker for prime %s of delta %s is %s" % (p, delta, kronecker(delta, p))
            f += -kronecker(delta,p) * log(p)/sqrt(pn) * cos(t*log(pn))
    return f

def arb_plot_symbolic_phihat(bound, xmin, xmax, delta = -1, plot_points=1000, show_zeros=True):
    f = arb_symbolic_phihat(bound, delta)
    K.<i> = QQ[sqrt(delta)]
    P = plot(f, (t,xmin, xmax), plot_points=plot_points)
    if not show_zeros:
        return P
    ym = P.ymax()
    Z = []
    for y in zeros(xmax, K):
        Z.append(y)
    zeros_arrows = sum([arrow((x,ym),(x,0),rgbcolor='red',width=0.5,arrowsize=2)
                 for i, x in enumerate(Z)])
    return P + zeros_arrows

p = arb_plot_symbolic_phihat(100, 2, 30, -5)
p.show()

g = gauss_plot_symbolic_phihat(100, 2, 30)
g.show()

K.<i> = QQ[sqrt(-1)]
K.discriminant??