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DTSTART;TZID=America/Toronto:20250611T093000
SEQUENCE:0
TRANSP:TRANSPARENT
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URL:https://uwaterloo.ca/pure-mathematics/events/phd-thesis-defense-11
SUMMARY:PhD thesis defense
CLASS:PUBLIC
DESCRIPTION:SOURABHASHIS DAS\, UNIVERSITY OF WATERLOO\n\nOn the distributio
 ns of prime divisor counting function\n\nIn 1917\, Hardy and Ramanujan est
 ablished that $\\omega(n)$\, the number\nof distinct prime factors of a na
 tural number $n$\, and $\\Omega(n)$\,\nthe total number of prime factors o
 f $n$ have normal order $\\log \\log\nn$. In 1940\, Erdős and Kac refined
  this understanding by proving that\n$\\omega(n)$ follows a Gaussian distr
 ibution over the natural numbers. \n\nIn this talk\, we extend these class
 ical results to the subsets of\n$h$-free and $h$-full numbers. We show tha
 t $\\omega_1(n)$\, the number\nof distinct prime factors of $n$ with multi
 plicity exactly $1$\, has\nnormal order $\\log \\log n$ over $h$-free numb
 ers. Similarly\,\n$\\omega_h(n)$\, the number of distinct prime factors wi
 th multiplicity\nexactly $h$\, has normal order $\\log \\log n$ over $h$-f
 ull numbers.\nHowever\, for $1 < k < h$\, we prove that $\\omega_k(n)$ doe
 s not have a\nnormal order over $h$-free numbers\, and for $k > h$\, $\\om
 ega_k(n)$\ndoes not have a normal order over $h$-full numbers. \n\nFurther
 more\, we establish that $\\omega_1(n)$ satisfies the Erdős-Kac\ntheorem 
 over $h$-free numbers\, while $\\omega_h(n)$ does so over\n$h$-full number
 s. These results provide a deeper insight into the\ndistribution of prime 
 factors within structured subsets of natural\nnumbers\, revealing intrigui
 ng asymptotic behavior in these settings.\n\nMC 5417
DTSTAMP:20260502T042125Z
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