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URL:https://uwaterloo.ca/pure-mathematics/events/number-theory-seminar-143
SUMMARY:Number Theory Seminar
CLASS:PUBLIC
DESCRIPTION:SOURABHASHIS DAS\, UNIVERSITY OF WATERLOO \n\nOn the distributi
 ons of divisor counting functions: From\nHardy-Ramanujan to Erdős-Kac\n\n
 In 1917\, Hardy and Ramanujan established that w(n)\, the number of\ndisti
 nct prime factors of a natural number n\, and Omega(n)\, the total\nnumber
  of prime factors of n have normal order log log n. In 1940\,\nErdős and 
 Kac refined this understanding by proving that w(n) follows\na Gaussian di
 stribution over the natural numbers. \n\nIn this talk\, we extend these cl
 assical results to the subsets of\nh-free and h-full numbers. We show that
  w_1(n)\, the number of distinct\nprime factors of n with multiplicity exa
 ctly 1\, has normal order log\nlog n over h-free numbers. Similarly\, w_h(
 n)\, the number of distinct\nprime factors with multiplicity exactly h\, h
 as normal order log log n\nover h-full numbers. However\, for 1 < k < h\, 
 we prove that w_k(n) does\nnot have a normal order over h-free numbers\, a
 nd for k > h\, w_k(n)\ndoes not have a normal order over h-full numbers. \
 n\nFurthermore\, we establish that w_1(n) satisfies the Erdős-Kac theorem
 \nover h-free numbers\, while w_h(n) does so over h-full numbers. These\nr
 esults provide a deeper insight into the distribution of prime\nfactors wi
 thin structured subsets of natural numbers\, revealing\nintriguing asympto
 tic behavior in these settings.\n\nMC 5479
DTSTAMP:20260407T211616Z
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