On restricted averages of Dedekind sums
We investigate the averages of Dedekind sums over rational numbers in the set $$\mathscr{F}_\alpha(Q):=\{\, {v}/{w}\in \mathbb{Q}: 0
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| creator | Minelli, Paolo Sourmelidis, Athanasios Technau, Marc |
| description | We investigate the averages of Dedekind sums over rational numbers in the set
$$\mathscr{F}_\alpha(Q):=\{\, {v}/{w}\in \mathbb{Q}: 0 |
| doi_str_mv | 10.48550/arxiv.2301.00441 |
| format | Article |
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$$\mathscr{F}_\alpha(Q):=\{\, {v}/{w}\in \mathbb{Q}: 0<w\leq Q\,\}\cap [0,
\alpha)$$
for fixed $\alpha\leq 1/2$.
In previous work, we obtained asymptotics for $\alpha=1/2$, confirming a
conjecture of Ito in a quantitative form.
In the present article we extend our former results, first to all fixed
rational $\alpha$ and then to almost all irrational $\alpha$.
As an intermediate step we obtain a result quantifying the bias occurring in
the second term of the asymptotic for the average running time of the
\textit{by-excess} Euclidean algorithm, which is of independent interest.</description><identifier>DOI: 10.48550/arxiv.2301.00441</identifier><language>eng</language><subject>Mathematics - Number Theory</subject><creationdate>2023-01</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2301.00441$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2301.00441$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Minelli, Paolo</creatorcontrib><creatorcontrib>Sourmelidis, Athanasios</creatorcontrib><creatorcontrib>Technau, Marc</creatorcontrib><title>On restricted averages of Dedekind sums</title><description>We investigate the averages of Dedekind sums over rational numbers in the set
$$\mathscr{F}_\alpha(Q):=\{\, {v}/{w}\in \mathbb{Q}: 0<w\leq Q\,\}\cap [0,
\alpha)$$
for fixed $\alpha\leq 1/2$.
In previous work, we obtained asymptotics for $\alpha=1/2$, confirming a
conjecture of Ito in a quantitative form.
In the present article we extend our former results, first to all fixed
rational $\alpha$ and then to almost all irrational $\alpha$.
As an intermediate step we obtain a result quantifying the bias occurring in
the second term of the asymptotic for the average running time of the
\textit{by-excess} Euclidean algorithm, which is of independent interest.</description><subject>Mathematics - Number Theory</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotzjsLwjAYheEsDqL-ACezObV-aW7NKN6h4KB7SXORoq2SqOi_9zqd5eXwIDQkkLKcc5jo8KjvaUaBpACMkS4ab1scXLyG2lydxfrugj64iM8ez511x7q1ON6a2Ecdr0_RDf7bQ7vlYj9bJ8V2tZlNi0QLSRJBtDFAGDE5rbhRlZGVUEwpKby04t1Qm-WCE6sqcN5TqQQVHDzYjGeW9tDo9_qFlpdQNzo8yw-4_ILpCxpVOXQ</recordid><startdate>20230101</startdate><enddate>20230101</enddate><creator>Minelli, Paolo</creator><creator>Sourmelidis, Athanasios</creator><creator>Technau, Marc</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20230101</creationdate><title>On restricted averages of Dedekind sums</title><author>Minelli, Paolo ; Sourmelidis, Athanasios ; Technau, Marc</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a671-61acc0141c83b5c9bc7b6949976f7d6a673d28651d9b0eff37963650f0d252d3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Mathematics - Number Theory</topic><toplevel>online_resources</toplevel><creatorcontrib>Minelli, Paolo</creatorcontrib><creatorcontrib>Sourmelidis, Athanasios</creatorcontrib><creatorcontrib>Technau, Marc</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Minelli, Paolo</au><au>Sourmelidis, Athanasios</au><au>Technau, Marc</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On restricted averages of Dedekind sums</atitle><date>2023-01-01</date><risdate>2023</risdate><abstract>We investigate the averages of Dedekind sums over rational numbers in the set
$$\mathscr{F}_\alpha(Q):=\{\, {v}/{w}\in \mathbb{Q}: 0<w\leq Q\,\}\cap [0,
\alpha)$$
for fixed $\alpha\leq 1/2$.
In previous work, we obtained asymptotics for $\alpha=1/2$, confirming a
conjecture of Ito in a quantitative form.
In the present article we extend our former results, first to all fixed
rational $\alpha$ and then to almost all irrational $\alpha$.
As an intermediate step we obtain a result quantifying the bias occurring in
the second term of the asymptotic for the average running time of the
\textit{by-excess} Euclidean algorithm, which is of independent interest.</abstract><doi>10.48550/arxiv.2301.00441</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Number Theory |
| title | On restricted averages of Dedekind sums |
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