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Motivated by work of Peluse and Soundararajan on Miller's conjecture for character tables of symmetric groups, we study the proportion of entries in the character table of $G_q$ which are not divisible by $\\ell$, in the sense of divisibility in the ring of algebraic integers. We prove that $N_\\ell(q)=\\frac{q^4}{2}+O_\\epsilon(q^{3+\\epsilon})$ for every $\\epsilon>0$, where $N_\\ell(q)$ denotes the number of entries which are not divisible by $\\ell$. We also show that the number of zero entrie"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.28085","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2026-06-26T13:47:45Z","cross_cats_sorted":["math.CO","math.NT"],"title_canon_sha256":"f2e5f8a26ed90312d2a35648f8fe247729c4cfb3289681966b97df9b291c12c5","abstract_canon_sha256":"b1a0e60d2749430004929190f3ed07986df10b40da3011f507c883a81fdc33d2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-29T01:14:57.526409Z","signature_b64":"8SD6/d58X5dr72qrVIFgwENuhqoaX2LlHCjsJF+NZDslMwk1kfnEHCrh46xTOpYrm7t5gZOdp5QkPFuG3VUuCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9283f8e6ad379ad11b0ba0fac7e44f211746cf49c75878d6facd977fb9707a6b","last_reissued_at":"2026-06-29T01:14:57.526016Z","signature_status":"signed_v1","first_computed_at":"2026-06-29T01:14:57.526016Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Average divisibility in character tables of $\\mathrm{GL}_2(\\mathbb{F}_q)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.RT","authors_text":"Anwesh Ray, Mishty Ray","submitted_at":"2026-06-26T13:47:45Z","abstract_excerpt":"Let $q$ range over odd prime powers and let $G_q=\\mathrm{GL}_2(\\mathbb{F}_q)$. Fix a prime number $\\ell$. Motivated by work of Peluse and Soundararajan on Miller's conjecture for character tables of symmetric groups, we study the proportion of entries in the character table of $G_q$ which are not divisible by $\\ell$, in the sense of divisibility in the ring of algebraic integers. We prove that $N_\\ell(q)=\\frac{q^4}{2}+O_\\epsilon(q^{3+\\epsilon})$ for every $\\epsilon>0$, where $N_\\ell(q)$ denotes the number of entries which are not divisible by $\\ell$. 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