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SUMMARY:Alisa Sedunova (Purdue University)
DTSTART:20250521T133000Z
DTEND:20250521T135500Z
DTSTAMP:20260423T010133Z
UID:CANT2025/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/4/"
 >The multiplication table constant and sums of two squares</a>\nby Alisa S
 edunova (Purdue University) as part of Combinatorial and additive number t
 heory (CANT 2025)\n\nLecture held in CUNY Graduate Center - Science Center
  (4th floor).\n\nAbstract\nLet $r_1(n)$ be the number of representations o
 f $n$ as the sum of a square and a square of a prime. We discuss the errat
 ic behavior of $r_1$\, which is similar to the one of the divisor function
 . \nWe will show that the number of integers up to $x$ that have at least 
 one such representation \nis asymptotic to $(\\pi/2) x \\log x$ minus a se
 condary term of size $x/(\\log x)^{1+d+o(1)}$\, \nwhere $d$ is the multipl
 ication table constant. Detailed heuristics suggest very precise asymptoti
 c \nfor the secondary term as well. In particular\, our proofs imply that 
 the main contribution to the mean \nvalue of $r_1(n)$ comes from integers 
 with “unusual” number of prime factors\, i.e. those with\n $\\omega(n)
  \\sim 2 \\log \\log x$ (for which $r_1(n) \\sim (\\log x)^{\\log 4-1}$)\,
  where $\\omega(n)$ \n is the number of district prime factors of $n$.   \
 \\\\nIn the talk we will review the results of several works that include 
 a recent joint preprint with Andrew Granville and Cihan Sabuncu and my pap
 er from 2022 as well as some work in progress.\n
LOCATION:https://researchseminars.org/talk/CANT2025/4/
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