{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,12,2]],"date-time":"2023-12-02T00:34:37Z","timestamp":1701477277890},"reference-count":28,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","funder":[{"name":"NSF","award":["DMS-1905641"],"award-info":[{"award-number":["DMS-1905641"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2023,2]]},"abstract":"<jats:p>Motivated by results about \u201cuntangling\u201d closed curves on hyperbolic surfaces, Gupta and Kapovich introduced the primitivity and simplicity index functions for finitely generated free groups, [Formula: see text] and [Formula: see text], where [Formula: see text], and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence [Formula: see text] as [Formula: see text]. Answering a question from [17], we prove that this sequence is unbounded and that for [Formula: see text], we have [Formula: see text]. By contrast, we show that for all [Formula: see text], one has [Formula: see text]. In addition to topological and group-theoretic arguments, number-theoretic considerations, particularly the use of asymptotic properties of the second Chebyshev function, turn out to play a key role in the proofs.<\/jats:p>","DOI":"10.1142\/s021819672350008x","type":"journal-article","created":{"date-parts":[[2022,11,22]],"date-time":"2022-11-22T09:30:55Z","timestamp":1669109455000},"page":"105-122","source":"Crossref","is-referenced-by-count":0,"title":["Primitivity index bounds in free groups, and the second Chebyshev function"],"prefix":"10.1142","volume":"33","author":[{"given":"Ilya","family":"Kapovich","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Hunter College of CUNY, 695 Park Ave, New York, NY 10065, USA"}]},{"given":"Zachary","family":"Simon","sequence":"additional","affiliation":[{"name":"Advanced Technology Center, Lockheed Martin Space, 1111 Lockheed Martin Way, Sunnyvale, CA 94089, USA"}]}],"member":"219","published-online":{"date-parts":[[2022,11,21]]},"reference":[{"issue":"12","key":"S021819672350008XBIB001","doi-asserted-by":"crossref","first-page":"8343","DOI":"10.1090\/tran\/6865","volume":"369","author":"Biringer I.","year":"2017","journal-title":"Trans. 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