{
  "annotation_id": "aa4c8453-94cf-4ef9-af28-f227f07b29e9",
  "date_created": "2026-02-17T05:53:24.099000Z",
  "date_modified": "2026-02-17T05:53:24.099000Z",
  "file_hash": "6bab12aac2a001debd50ff638c526bc46cd778e06dcd4b64edd0279d56475c3e",
  "private": false,
  "record": {
    "abstract": "We establish various upper bounds on Type-I and Type-II shifted bilinear sums with Sali\\\u0027e sums modulo a large prime $q$. We use these bounds to study, for fixed integers $a,b\\not \\equiv 0 \\bmod q$, the distribution ofsolutions to the congruence $x^2 \\equiv ap+b \\bmod q$, over primes $p\\le P$. This is similar to the recently studied case of $b = 0$, however the case $b\\not \\equiv 0 \\bmod q$ exhibits some new difficulties.",
    "arxiv_id": "2601.10113",
    "authors": [
      "Igor E. Shparlinski",
      "Yixiu Xiao"
    ],
    "categories": [
      "math.NT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "Shifted bilinear sums of Sali\\\u0027e sums and the distribution of modular square roots of shifted primes",
    "url": "https://arxiv.org/abs/2601.10113",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "95ac91b9-2bc3-4d7d-ac40-ef4ccd4e13c4",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}