{
  "as_of": "2026-09-07",
  "references": [
    {
      "title": "254A, Notes 1: Elementary multiplicative number theory",
      "author": "Terence Tao",
      "year": 2014,
      "role": "Fixed-k summatory formula sum Lambda_k = x R_k(log x)+O_k(x), degree k-1 leading coefficient k"
    },
    {
      "title": "On the vanishing of the generalized von Mangoldt function Lambda_k",
      "venue": "MathOverflow",
      "role": "Definition, support on at most k distinct prime factors, and recursion"
    },
    {
      "title": "Elementary methods in the study of the distribution of prime numbers",
      "author": "Harold G. Diamond",
      "role": "Historical higher-weight Selberg/Bombieri/Wirsing amplification; arbitrary log-power and stretched-exponential elementary PNT errors"
    },
    {
      "title": "An elementary proof of the prime number theorem with a remainder term",
      "authors": [
        "Harold G. Diamond",
        "John Steinig"
      ],
      "venue": "Inventiones Mathematicae 11 (1970), 199-258",
      "role": "Higher-weight elementary PNT subpower error calibration"
    },
    {
      "title": "Parity problem and asymptotic sieve discussion",
      "author": "Terence Tao",
      "role": "Higher generalized von Mangoldt information can leave the prime/parity component undetermined"
    }
  ]
}