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SUMMARY:Carlo Francisco E. Adajar (University of Georgia)
DTSTART:20250523T140000Z
DTEND:20250523T142500Z
DTSTAMP:20260423T024828Z
UID:CANT2025/60
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/60/
 ">On the distribution of $v_p(\\sigma(n))$</a>\nby Carlo Francisco E. Adaj
 ar (University of Georgia) as part of Combinatorial and additive number th
 eory (CANT 2025)\n\nLecture held in CUNY Graduate Center - Science Center 
 (4th floor).\n\nAbstract\nFor a positive integer $m$ and a prime $p$\, we 
 write $\\sigma(m) := \\sum_{d \\mid m} d$ for the sum of the divisors of $
 m$\, and $v_p(m) := \\max\\{ k \\in \\mathbf{Z}_{\\ge 0} : p^k \\mid m \\}
 $ for the $p$-adic valuation of $m$\, i.e.\, the exponent of $p$ in the pr
 ime factorization of $m$. For each prime $p$\, we give an asymptotic expre
 ssion for the count\n$$ \\#\\{ n \\le x : v_p(\\sigma(n)) = k \\} $$\nas $
 x\\to\\infty$\, uniformly for $k \\ll \\log\\log{x}$. We then deduce an as
 ymptotic for the count of $n \\le x$ such that $v_p(\\sigma(n)) < v_p(n)$ 
 as $x \\to \\infty$. \\\\\nThis talk is based on ongoing work with Paul Po
 llack.\n
LOCATION:https://researchseminars.org/talk/CANT2025/60/
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