Entropy, Statistical, and Dynamical Analysis of σ-Iterates: Heavy Tails, Fractal Geometry, and Empirical Evidence on the Schinzel Conjecture for k ≥ 3

Entropy, Statistical, and Dynamical Analysis of σ-Iterates: Heavy Tails, Fractal Geometry, and Empirical Evidence on the Schinzel Conjecture for k ≥ 3 | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Entropy, Statistical, and Dynamical Analysis of σ-Iterates: Heavy Tails, Fractal Geometry, and Empirical Evidence on the Schinzel Conjecture for k ≥ 3 Zeraoulia Rafik This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7377221/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We study the statistical and dynamical properties of iterates of the sum-of-divisors function $\sigma(n)$ via the normalized ratio \[ R_q(n) = \frac{\sigma^{(q)}(n)}{n}, \quad q \ge 3, \] using large-scale computation ($n \le 10^6$), regression modeling, extreme-value analysis, and a finite-difference analogue of Lyapunov diagnostics. Empirically, $R_q(n)$ is strongly right-skewed and heavy-tailed, with rare large spikes linked to highly composite integers; Lyapunov analysis shows a contraction-dominated local sensitivity consistent with boundedness. Regression on arithmetic predictors (log-scale, divisor count, prime factor indicators) explains much central variation but leaves structured extreme residuals, motivating peaks-over-threshold analysis. We introduce an \emph{entropy-based lower-tail criterion} linking bounded empirical Shannon entropy to exponential bounds on upper-tail mass and proving that bounded entropy with a vanishing-tail condition forces infinitely many $n$ with $R_q(n) \le T$. Combined with a fractal-geometry analysis (box--counting dimension $D_{\mathrm{box}} \approx 0.9925$) of the integer-dynamic attractor, this yields measurable constraints supporting the Schinzel Conjecture for $q \ge 3$. Our entropy–fractal framework, supported by reproducible computations, offers a statistically grounded and computationally verified pathway toward resolving this conjecture Applied Statistics sum-of-divisors function Schinzel Conjecture entropy bounds fractal geometry extreme value theory Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7377221","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":500759997,"identity":"66a4d041-1e8d-4fc1-8d3e-c41caf115119","order_by":0,"name":"Zeraoulia Rafik","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA60lEQVRIie3QvQ4BQRDA8blccpoR7YiPe4WTS3wkEq+ymqtoRak6jetJPMRqiI5IKBxaoqFRKegkJOyJ0h46xf6zxU7xyyQDoFL9b3mESAOAAWgcxYyfCSHg6EciHnv+PhOz6e1PZ5eSsc4xu9u7kBrgnMHxJidaa2bHOy4hbSs5q+yCPfQ8rnVD1ujkQCIqCGwrWWL+vcxXUa4jyYlhHvRrQMyNHxB4EUtOkAzjucVaoyB1QRaeIExOCB2j0F4SZnynFhB72FrwcXckJ2Zzqq9PtWIpPZv04xcruFi1twu72PtCdqhUKpXqmx5Kh0n71Vo69wAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-5436-3320","institution":"khemiss Miliana university","correspondingAuthor":true,"prefix":"","firstName":"Zeraoulia","middleName":"","lastName":"Rafik","suffix":""}],"badges":[],"createdAt":"2025-08-14 23:39:34","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7377221/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7377221/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":89277642,"identity":"68d10f85-b719-433f-8d56-8bb179b3d3f9","added_by":"auto","created_at":"2025-08-18 09:50:46","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1175047,"visible":true,"origin":"","legend":"","description":"","filename":"formdpstat.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7377221/v1_covered_e4bf37fe-ca70-42f5-80ac-1d12134ed7cf.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eEntropy, Statistical, and Dynamical Analysis of σ-Iterates: Heavy Tails, Fractal Geometry, and Empirical Evidence on the Schinzel Conjecture for \u003cem\u003ek ≥ 3\u003c/em\u003e\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"sum-of-divisors function, Schinzel Conjecture, entropy bounds, fractal geometry, extreme value theory","lastPublishedDoi":"10.21203/rs.3.rs-7377221/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7377221/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe study the statistical and dynamical properties of iterates of the sum-of-divisors function $\\sigma(n)$ via the normalized ratio\u003c/p\u003e\n\u003cp\u003e\\[\u003c/p\u003e\n\u003cp\u003eR_q(n) = \\frac{\\sigma^{(q)}(n)}{n}, \\quad q \\ge 3,\u003c/p\u003e\n\u003cp\u003e\\]\u003c/p\u003e\n\u003cp\u003eusing large-scale computation ($n \\le 10^6$), regression modeling, extreme-value analysis, and a finite-difference analogue of Lyapunov diagnostics. Empirically, $R_q(n)$ is strongly right-skewed and heavy-tailed, with rare large spikes linked to highly composite integers; Lyapunov analysis shows a contraction-dominated local sensitivity consistent with boundedness. Regression on arithmetic predictors (log-scale, divisor count, prime factor indicators) explains much central variation but leaves structured extreme residuals, motivating peaks-over-threshold analysis.\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eWe introduce an \\emph{entropy-based lower-tail criterion} linking bounded empirical Shannon entropy to exponential bounds on upper-tail mass and proving that bounded entropy with a vanishing-tail condition forces infinitely many $n$ with $R_q(n) \\le T$. Combined with a fractal-geometry analysis (box--counting dimension $D_{\\mathrm{box}} \\approx 0.9925$) of the integer-dynamic attractor, this yields measurable constraints supporting the Schinzel Conjecture for $q \\ge 3$. Our entropy–fractal framework, supported by reproducible computations, offers a statistically grounded and computationally verified pathway toward resolving this conjecture\u003c/p\u003e","manuscriptTitle":"Entropy, Statistical, and Dynamical Analysis of σ-Iterates: Heavy Tails, Fractal Geometry, and Empirical Evidence on the Schinzel Conjecture for k ≥ 3","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-18 09:42:38","doi":"10.21203/rs.3.rs-7377221/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"d2a09209-3b5b-43d7-9df2-c4b1f4e104df","owner":[],"postedDate":"August 18th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":53301495,"name":"Applied Statistics"}],"tags":[],"updatedAt":"2025-08-18T09:42:38+00:00","versionOfRecord":[],"versionCreatedAt":"2025-08-18 09:42:38","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7377221","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7377221","identity":"rs-7377221","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}