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SUMMARY:Christian Elsholtz (Graz University of Technology\, Austria)
DTSTART:20200605T150000Z
DTEND:20200605T152500Z
DTSTAMP:20260423T011219Z
UID:CANT2020/61
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2020/61/
 ">Sums of unit fractions</a>\nby Christian Elsholtz (Graz University of Te
 chnology\, Austria) as part of Combinatorial and additive number theory (C
 ANT 2021)\n\n\nAbstract\nLet $f_k(m\,n)$ denote the\nnumber of solutions o
 f $\\frac{m}{n}= \\frac{1}{x_1} + \\cdots +\n\\frac{1}{x_k}$ in positive i
 ntegers $x_i$.\nThe case $k=2$ is essentially a question on a divisor func
 tion\, and \nthe case $k=3$ is closely related to a sum of certain divisor
  functions. \nFor the case $k=3\, m=4$ Erd\\H{o}s and Straus conjectured t
 hat\n\\[\nf_3(4\,n)>0 \\text{ for all } n>1.\n\\] \nThe case $m=n=1$ recei
 ved special attention\, and even has applications in discrete \ngeometry. 
  We give a survey on previous results and report on new results \nover the
  last years. \n\nTheorem 1: There are infinitely many primes $p$ with\n\\[
 \nf_3(m\,p)\\gg\\exp \\left(c_m \\frac{\\log p}{\\log \\log p}\\right).\n\
 \]\n\nTheorem 2: For fixed $m$ and almost all integers $n$ one has: \n\\[\
 nf_3(m\,n)\\gg\n(\\log n)^{\\log 3+o_m(1)}.\n\\]\n\nTheorem 3: $f_3(4\,n)=
 O_{\\varepsilon}\\left(n^{3/5+\\varepsilon}\\right)$\, for\nall $\\varepsi
 lon >0$.\nThere are related but more complicated bounds when $k\\geq 4$.\n
 \nJoint work with T. Browning\, S. Planitzer\, and T. Tao.\n
LOCATION:https://researchseminars.org/talk/CANT2020/61/
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