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SUMMARY:Mikhail Gabdullin (University of Illinois at Urbana-Champaign)
DTSTART:20260713T150000Z
DTEND:20260713T152500Z
DTSTAMP:20260710T111409Z
UID:CANT2026/13
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/13/
 ">Moments of the shifted prime divisor function</a>\nby Mikhail Gabdullin 
 (University of Illinois at Urbana-Champaign) as part of Combinatorial and 
 additive number theory seminar (CANT 2026)\n\nLecture held in Science Cent
 er in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet $\\omega^*(n)
  = \\{d|n: d=p-1\, \\mbox{$p$ is a prime}\\}$ denote the ``shifted prime d
 ivisor'' function. It is easy to see that $\\sum_{n\\leq x}\\omega^*(n)=x\
 \log\\log x+O(x)$\, similar to the average value of $\\omega(n)$\, the num
 ber of prime divisors of $n$. We confirm a recent conjecture of Fan and Po
 merance by proving that\, for each integer $k\\geq2$\, $\n\\qquad \\sum_{n
 \\leq x}\\omega^*(n)^k \\asymp x(\\log x)^{2^k-k-1}\,\n$ \nwhere the impli
 ed constant may depend only on $k$. The proof relies on a combinatorial id
 entity for the least common multiple\, viewed as a multiplicative analogue
  of the inclusion-exclusion principle\, together with the theory of multip
 licative functions.\n
LOCATION:https://researchseminars.org/talk/CANT2026/13/
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