{"paper":{"title":"Expected number of real zeros of random Taylor Series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hendrik Flasche, Zakhar Kabluchko","submitted_at":"2017-09-09T09:54:53Z","abstract_excerpt":"Let $\\xi_0,\\xi_1,\\ldots$ be i.i.d. random variables with zero mean and unit variance. Consider a random Taylor series of the form $f(z)=\\sum_{k=0}^\\infty \\xi_k c_k z^k$, where $c_0,c_1,\\ldots$ is a real sequence such that $c_n^2$ is regularly varying with index $\\gamma-1$, where $\\gamma>0$. We prove that $\\mathbb{E} N[0,1-\\epsilon] \\sim \\frac{\\sqrt{\\gamma}}{2\\pi} |\\log \\epsilon|$ as $\\epsilon \\downarrow 0$, where $N[0,r]$ denotes the number of real zeroes of $f$ in the interval $[0,r]$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.02937","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}