{
  "annotation_id": "93045eeb-016a-4711-96de-72acdb3a6880",
  "date_created": "2026-02-17T05:53:16.242000Z",
  "date_modified": "2026-02-17T05:53:16.242000Z",
  "file_hash": "2f6133556cf28ed7cf238cd59ac1bd89f727f0a286ec65438b874e3cf85f0b86",
  "private": false,
  "record": {
    "abstract": "Let $p\u003e3$ be a prime and $b\\ge 2$ an integer such that $p$ does not divide $b$. Then $1/p$ has a periodic digit expansion with respect to the basis $b$. The length $q$ of the period is the (multiplicative) order of $b$ mod $p$. In the case $q=p-1$ a formula for the variance of the digits of a period was given previously. This formula involves a Dedekind sum. We determine the variance in the case $q=(p-1)/2$. If $p\\equiv 3$ mod 4 a Dedekind sum and the class number of $\\mathbb Q(\\sqrt{-p})$ occur in the respective formula. If $p\\equiv 1$ mod 4, the formula may be much more complex since it involves linear combinations of (possibly many) products of two Bernoulli numbers attached to odd characters.",
    "arxiv_id": "2601.08416",
    "authors": [
      "Kurt Girstmair"
    ],
    "categories": [
      "math.NT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "On the variance of the digits of $1/p$",
    "url": "https://arxiv.org/abs/2601.08416",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "e2ef3e95-60af-43a7-b751-42f219051454",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}