{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "# load('ZetaFunctionsNewtonND.py')\n",
    "R.<x, y, z> = PolynomialRing(QQ, order='neglex')\n",
    "S.<u, v> = PolynomialRing(QQ, order='neglex')\n",
    "S.<s> = PolynomialRing(QQ)\n",
    "f = u^5 + v^6\n",
    "F = z^10 + f(u=x, v=y)\n",
    "ZF = ZetaFunctions(F)\n",
    "Zf = ZetaFunctions(f)\n",
    "dd = (1, 2, 3, 4, 5, 6, 8, 10, 15, 30)\n",
    "K = {d: ZF.topological_zeta(local=True, d=d) for d in dd}\n",
    "k = {d: Zf.topological_zeta(local=True, d=d) for d in dd}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{1}{5}\\right) \\cdot (s + \\frac{7}{15})^{-1} \\cdot (s + 1)^{-1} \\cdot (s + \\frac{7}{3})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{1}{5}\\right) \\cdot (s + \\frac{7}{15})^{-1} \\cdot (s + 1)^{-1} \\cdot (s + \\frac{7}{3})$"
      ],
      "text/plain": [
       "(1/5) * (s + 7/15)^-1 * (s + 1)^-1 * (s + 7/3)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "2\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 0\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "3\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{2}{5}\\right) \\cdot (s + \\frac{7}{15})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{2}{5}\\right) \\cdot (s + \\frac{7}{15})^{-1}$"
      ],
      "text/plain": [
       "(2/5) * (s + 7/15)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "4\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 0\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "5\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{1}{30}\\right) \\cdot (s + \\frac{7}{15})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{1}{30}\\right) \\cdot (s + \\frac{7}{15})^{-1}$"
      ],
      "text/plain": [
       "(1/30) * (s + 7/15)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "6\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{2}{5}\\right) \\cdot (s + \\frac{7}{15})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{2}{5}\\right) \\cdot (s + \\frac{7}{15})^{-1}$"
      ],
      "text/plain": [
       "(2/5) * (s + 7/15)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "8\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 0\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "10\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(-\\frac{1}{6}\\right) \\cdot (s + \\frac{7}{15})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(-\\frac{1}{6}\\right) \\cdot (s + \\frac{7}{15})^{-1}$"
      ],
      "text/plain": [
       "(-1/6) * (s + 7/15)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "15\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{7}{30}\\right) \\cdot (s + \\frac{7}{15})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{7}{30}\\right) \\cdot (s + \\frac{7}{15})^{-1}$"
      ],
      "text/plain": [
       "(7/30) * (s + 7/15)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "30\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{7}{30}\\right) \\cdot (s + \\frac{7}{15})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{7}{30}\\right) \\cdot (s + \\frac{7}{15})^{-1}$"
      ],
      "text/plain": [
       "(7/30) * (s + 7/15)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n"
     ]
    }
   ],
   "source": [
    "for d in dd:\n",
    "    if K[d] == 0:\n",
    "        print(d)\n",
    "        show(0)\n",
    "    else:\n",
    "        print(d)\n",
    "        show(K[d].factor())\n",
    "    print(\"========================\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{1}{3}\\right) \\cdot (s + \\frac{11}{30})^{-1} \\cdot (s + 1)^{-1} \\cdot (s + \\frac{11}{10})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{1}{3}\\right) \\cdot (s + \\frac{11}{30})^{-1} \\cdot (s + 1)^{-1} \\cdot (s + \\frac{11}{10})$"
      ],
      "text/plain": [
       "(1/3) * (s + 11/30)^-1 * (s + 1)^-1 * (s + 11/10)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "2\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{2}{15}\\right) \\cdot (s + \\frac{11}{30})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{2}{15}\\right) \\cdot (s + \\frac{11}{30})^{-1}$"
      ],
      "text/plain": [
       "(2/15) * (s + 11/30)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "3\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{2}{15}\\right) \\cdot (s + \\frac{11}{30})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{2}{15}\\right) \\cdot (s + \\frac{11}{30})^{-1}$"
      ],
      "text/plain": [
       "(2/15) * (s + 11/30)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "4\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 0\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "5\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{1}{6}\\right) \\cdot (s + \\frac{11}{30})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{1}{6}\\right) \\cdot (s + \\frac{11}{30})^{-1}$"
      ],
      "text/plain": [
       "(1/6) * (s + 11/30)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "6\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{2}{15}\\right) \\cdot (s + \\frac{11}{30})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{2}{15}\\right) \\cdot (s + \\frac{11}{30})^{-1}$"
      ],
      "text/plain": [
       "(2/15) * (s + 11/30)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "8\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 0\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "10\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(-\\frac{1}{30}\\right) \\cdot (s + \\frac{11}{30})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(-\\frac{1}{30}\\right) \\cdot (s + \\frac{11}{30})^{-1}$"
      ],
      "text/plain": [
       "(-1/30) * (s + 11/30)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "15\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(-\\frac{1}{30}\\right) \\cdot (s + \\frac{11}{30})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(-\\frac{1}{30}\\right) \\cdot (s + \\frac{11}{30})^{-1}$"
      ],
      "text/plain": [
       "(-1/30) * (s + 11/30)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n",
      "30\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(-\\frac{1}{30}\\right) \\cdot (s + \\frac{11}{30})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(-\\frac{1}{30}\\right) \\cdot (s + \\frac{11}{30})^{-1}$"
      ],
      "text/plain": [
       "(-1/30) * (s + 11/30)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "========================\n"
     ]
    }
   ],
   "source": [
    "for d in dd:\n",
    "    if k[d] == 0:\n",
    "        print(d)\n",
    "        show(0)\n",
    "    else:\n",
    "        print(d)\n",
    "        show(k[d].factor())\n",
    "    print(\"========================\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 5]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "divisors(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def corchete(fz, e, N):\n",
    "    res = 0\n",
    "    for l0 in divisors(N / e):\n",
    "        l = l0 * e\n",
    "        res += moebius(l0) * fz[l]\n",
    "    return res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/6/(s + 7/15)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "corchete(k, 2, 10)(s=s + 1 / 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/5/(s + 7/15)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "corchete(k, 5, 10)(s=s + 1 / 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1/30/(s + 7/15)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "corchete(k, 10, 10)(s=s + 1 / 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Ztopmanuel(dic, N):\n",
    "    res =  1 / (N * s + 1) \n",
    "    res += (N - 1) / N * s / (s+1) * ((N * s+ N + 1)/ (N * s + 1)) * dic[1](s=s + 1 / N)\n",
    "    div = [e for e in divisors(N) if e != 1]\n",
    "    for e in div:\n",
    "        res += -s / (s + 1) * (e^2 - 1) / N * corchete(dic, e, N)(s=s + 1 / N)\n",
    "    return res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(Ztopmanuel(k, 10) - K[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4/15/(s^2 + 22/15*s + 7/15),\n",
       " 0,\n",
       " 0,\n",
       " 1/5/(s + 7/15),\n",
       " 1/6/(s + 7/15),\n",
       " -2/5/(s + 7/15),\n",
       " 0,\n",
       " (7/30) * (s + 7/15)^-1)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AK10t30 = K10t30\n",
    "AK10t15 = K10t15 - AK10t30\n",
    "AK10t10 = K10t10  - AK10t30\n",
    "AK10t6 = K10t6 - AK10t30\n",
    "AK10t5 = K10t5 - AK10t10 - AK10t15 - AK10t30\n",
    "AK10t3 = K10t3 - AK10t6  - AK10t30\n",
    "AK10t2 = K10t2  - AK10t6 - AK10t10 - AK10t30\n",
    "AK10 = K10 - AK10t3 - AK10t5 - AK10t6  - AK10t10 - AK10t30\n",
    "\n",
    "AK10, AK10t2, AK10t3, AK10t5, AK10t6, AK10t10, AK10t15, AK10t30"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K10- AK10 - AK10t5 - AK10t6 - AK10t10 - AK10t30"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((1/3*s + 2/5)/(s^2 + 47/30*s + 77/150),\n",
       " 2/15/(s + 7/15),\n",
       " 2/15/(s + 7/15),\n",
       " 0,\n",
       " 1/6/(s + 7/15),\n",
       " 2/15/(s + 7/15),\n",
       " 0,\n",
       " -1/30/(s + 7/15),\n",
       " -1/30/(s + 7/15),\n",
       " -1/30/(s + 7/15))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g=x^5 + y^6\n",
    "Zg = ZetaFunctions(g)\n",
    "G=Zg.topological_zeta(local=True).value()\n",
    "GS=G(s=s+1/10)\n",
    "G2=Zg.topological_zeta(local=True, d=2).value()\n",
    "GS2=G2(s=s+1/10)\n",
    "G3=Zg.topological_zeta(local=True, d=3).value()\n",
    "GS3=G3(s=s+1/10)\n",
    "G4=Zg.topological_zeta(local=True, d=4)\n",
    "GS4=G4(s=s+1/10)\n",
    "G5=Zg.topological_zeta(local=True, d=5).value()\n",
    "GS5=G5(s=s+1/10)\n",
    "G6=Zg.topological_zeta(local=True, d=6).value()\n",
    "GS6=G6(s=s+1/10)\n",
    "G8=Zg.topological_zeta(local=True, d=8)\n",
    "GS8=G8(s=s+1/10)\n",
    "G10=Zg.topological_zeta(local=True, d=10).value()\n",
    "GS10=G10(s=s+1/10)\n",
    "G15=Zg.topological_zeta(local=True, d=15).value()\n",
    "GS15=G15(s=s+1/10)\n",
    "G30=Zg.topological_zeta(local=True, d=30).value()\n",
    "GS30=G30(s=s+1/10)\n",
    "GS, GS2, GS3, GS4, GS5, GS6, GS8, GS10, GS15, GS30"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/30/(s^2 + 47/30*s + 77/150),\n",
       " 0,\n",
       " 0,\n",
       " 1/5/(s + 7/15),\n",
       " 1/6/(s + 7/15),\n",
       " 0,\n",
       " 0,\n",
       " -1/30/(s + 7/15))"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AGS30 = GS30\n",
    "AGS15 = GS15 - AGS30\n",
    "AGS10 = GS10  - AGS30\n",
    "AGS6 = GS6 - AGS30\n",
    "AGS5 = GS5 - AGS10 - AGS15 - AGS30\n",
    "AGS3 = GS3 - AGS6  - AGS30\n",
    "AGS2 = GS2  - AGS6 - AGS10 - AGS30\n",
    "AGS = GS - AGS2 - AGS3 - AGS5 - AGS6  - AGS10 - AGS15 - AGS30\n",
    "\n",
    "AGS, AGS2, AGS3, AGS5, AGS6, AGS10, AGS15, AGS30"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GS - AGS - AGS5 - AGS6 - AGS30 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "aux = K10 - 1/((10*s+1)) - (9/10)*(s/(s+1))*((10*s+11)/(10*s+1))*GS + (3/10)*(s/(s+1))*AGS6 + (24/10)*(s/(s+1))*AGS5 + (99/10)*(s/(s+1))*AGS30\n",
    "aux"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[0;31mSignature:\u001b[0m      \u001b[0mmoebius\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
       "\u001b[0;31mType:\u001b[0m           Moebius\n",
       "\u001b[0;31mString form:\u001b[0m    The Moebius function\n",
       "\u001b[0;31mFile:\u001b[0m           /usr/local/sagedev/src/sage/arith/misc.py\n",
       "\u001b[0;31mDocstring:\u001b[0m     \n",
       "   Return the value of the Möbius function of abs(n), where n is an\n",
       "   integer.\n",
       "\n",
       "   DEFINITION: \\mu(n) is 0 if n is not square free, and otherwise\n",
       "   equals (-1)^r, where n has r distinct prime factors.\n",
       "\n",
       "   For simplicity, if n=0 we define \\mu(n) = 0.\n",
       "\n",
       "   IMPLEMENTATION: Factors or - for integers - uses the PARI C\n",
       "   library.\n",
       "\n",
       "   INPUT:\n",
       "\n",
       "   * \"n\" - anything that can be factored.\n",
       "\n",
       "   OUTPUT: 0, 1, or -1\n",
       "\n",
       "   EXAMPLES:\n",
       "\n",
       "      sage: moebius(-5)\n",
       "      -1\n",
       "      sage: moebius(9)\n",
       "      0\n",
       "      sage: moebius(12)\n",
       "      0\n",
       "      sage: moebius(-35)\n",
       "      1\n",
       "      sage: moebius(-1)\n",
       "      1\n",
       "      sage: moebius(7)\n",
       "      -1\n",
       "\n",
       "      sage: moebius(0)   # potentially nonstandard!\n",
       "      0\n",
       "\n",
       "   The moebius function even makes sense for non-integer inputs.\n",
       "\n",
       "      sage: x = GF(7)['x'].0\n",
       "      sage: moebius(x + 2)\n",
       "      -1\n",
       "\n",
       "   Tests with numpy and gmpy2 numbers:\n",
       "\n",
       "      sage: from numpy import int8\n",
       "      sage: moebius(int8(-5))\n",
       "      -1\n",
       "      sage: from gmpy2 import mpz\n",
       "      sage: moebius(mpz(-5))\n",
       "      -1\n",
       "\u001b[0;31mInit docstring:\u001b[0m Initialize self.  See help(type(self)) for accurate signature.\n",
       "\u001b[0;31mCall docstring:\u001b[0m\n",
       "   EXAMPLES:\n",
       "\n",
       "      sage: from sage.arith.misc import Moebius\n",
       "      sage: Moebius().__call__(7)\n",
       "      -1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "moebius?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1, -1)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "moebius(5), moebius(-5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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