files / problems / chowla-cosine / compute / certificate.json

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{
  "lemma72": {
    "psi_max": [
      "4",
      "1/2"
    ],
    "phi_B": [
      "4",
      "1"
    ],
    "phi_max": [
      "4",
      "2"
    ],
    "gap": [
      "0",
      "1/2"
    ],
    "verified_by": "verify_lemma72.py"
  },
  "lemma71": {
    "C_Q1": 3,
    "C_Q2": 14,
    "note": "|Q1| <= 3(hat1_A+K), ||Q2||_2 <= 14 K^2"
  },
  "C_r": 32,
  "quadratic": {
    "A_alg": [
      "140",
      "224"
    ],
    "B_alg": [
      "980",
      "1856"
    ],
    "A_up": "2284/5",
    "B_up": "126172/35",
    "disc_up": "39040032/175",
    "sqrt_disc_up": "29519981126591/62500000000",
    "z_up": "58069981126591/125000000000",
    "Cstar_up": "3372122708042634945567281281/15625000000000000000000",
    "Cstar_pub": "3372138333042634945567281281/15625000000000000000000",
    "Cstar_int": 215816,
    "sqrt2_up": "99/70",
    "sqrt2_dn": "7/5"
  },
  "symmetric": {
    "N_le": "6906112 K^7 on the energy branch",
    "C7": 6906112,
    "C7_frac": "3372138333042634945567281281/488281250000000000000",
    "K_sym_ge_min_of": [
      "sqrt(N/2)",
      "(N/6906112)^(1/7)"
    ]
  },
  "chowla_positive": {
    "K_n_ge_min_of": [
      "n^(1/2)/2",
      "1/18 n^(1/7)"
    ],
    "c_1_7": "1/18",
    "c_1_7_float_preview": 0.05820240297604766,
    "seventh_root_upper_of_64_C7": 18,
    "statement": "For every n>=1 and every n-element set A of positive integers, min_x sum_{a in A} cos(a x) <= - min(sqrt(n)/2, (1/18) n^{1/7})."
  },
  "polylog_rows": [
    {
      "M": 5,
      "pi_M": 3,
      "N_exp_num": 1,
      "N_exp_den": 3,
      "denom": 21581600
    },
    {
      "M": 7,
      "pi_M": 4,
      "N_exp_num": 2,
      "N_exp_den": 4,
      "denom": 42299936
    },
    {
      "M": 11,
      "pi_M": 5,
      "N_exp_num": 3,
      "N_exp_den": 5,
      "denom": 104454944
    },
    {
      "M": 13,
      "pi_M": 6,
      "N_exp_num": 4,
      "N_exp_den": 6,
      "denom": 145891616
    },
    {
      "M": 19,
      "pi_M": 8,
      "N_exp_num": 6,
      "N_exp_den": 8,
      "denom": 311638304
    },
    {
      "M": 23,
      "pi_M": 9,
      "N_exp_num": 7,
      "N_exp_den": 9,
      "denom": 456666656
    },
    {
      "M": 31,
      "pi_M": 11,
      "N_exp_num": 9,
      "N_exp_den": 11,
      "denom": 829596704
    },
    {
      "M": 43,
      "pi_M": 14,
      "N_exp_num": 12,
      "N_exp_den": 14,
      "denom": 1596175136
    },
    {
      "M": 61,
      "pi_M": 18,
      "N_exp_num": 16,
      "N_exp_den": 18,
      "denom": 3212205344
    },
    {
      "M": 79,
      "pi_M": 22,
      "N_exp_num": 20,
      "N_exp_den": 22,
      "denom": 5387630624
    },
    {
      "M": 97,
      "pi_M": 25,
      "N_exp_num": 23,
      "N_exp_den": 25,
      "denom": 8122450976
    }
  ]
}