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SUMMARY:Cathy Swaenepol (Institut de Mathématiques de Jussieu-Paris Rive 
 Gauche)
DTSTART:20230621T150000Z
DTEND:20230621T160000Z
DTSTAMP:20260418T070222Z
UID:LNTS/104
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/104/">P
 rimes and squares with preassigned digits</a>\nby Cathy Swaenepol (Institu
 t de Mathématiques de Jussieu-Paris Rive Gauche) as part of London number
  theory seminar\n\nLecture held in KCL\, Strand Building\, Room S3.30.\n\n
 Abstract\nBourgain (2015) estimated the number of prime numbers with a pos
 itive\nproportion of preassigned digits in base 2.  We first present a\nge
 neralization of this result to any base $g\\geq 2$.  We then discuss\na mo
 re recent result for the set of squares\, which may be seen as one\nof the
  most interesting sets after primes.  More precisely\, for any\nbase $g\\g
 eq 2$\, we obtain an asymptotic formula for the number of\nsquares with a 
 proportion $c>0$ of preassigned digits. Moreover we\nprovide explicit admi
 ssible values for $c$ depending on $g$.  Our\nproof mainly follows the str
 ategy developed by Bourgain for primes in\nbase 2\, with new difficulties 
 for squares. It is based on the circle\nmethod and combines techniques fro
 m harmonic analysis together with\narithmetic properties of squares and bo
 unds for quadratic Weyl sums.\n
LOCATION:https://researchseminars.org/talk/LNTS/104/
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