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SUMMARY:Satoru Oshima and Takafumi Tsurumaki (University of Tsukuba)
DTSTART:20261117T130000Z
DTEND:20261117T140000Z
DTSTAMP:20261109T050833Z
UID:OWNS/174
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OWNS/174/">A
  new criterion for algebraic independence of special values of power serie
 s at Pisot or Salem numbers</a>\nby Satoru Oshima and Takafumi Tsurumaki (
 University of Tsukuba) as part of One World Numeration seminar\n\n\nAbstra
 ct\nIn this talk\, we introduce a new criterion for the algebraic independ
 ence of special values of power series at Pisot or Salem numbers. \nMore p
 recisely\,  we derive the algebraic independence of the following three va
 lues: \n\\[\n\\sum_{n\\geq0} t_1(n)\\beta^{-w_1(n)}\,\\quad \\sum_{n\\geq0
 } t_2(n)\\beta^{-w_1(n)}\,\\quad \\sum_{n\\geq0} t_3(n)\\beta^{-w_2(n)}\,\
 n\\]\nwhere $(t_1(n))_{n\\geq0}\, (t_2(n))_{n\\geq0}\, (t_3(n))_{n\\geq0}$
  are sequences of positive integers\, $(w_1(n))_{n\\geq0}\, (w_2(n))_{n\\g
 eq0}$ are  eventually strictly increasing sequences of nonnegative integer
 s\, and $\\beta$ is a Pisot or Salem number.\n\nFor instance\, we prove th
 at the following three values are algebraically independent: \n\\[\n\\sum_
 {n=3}^{\\infty}\\lfloor n^{y}\\rfloor\\beta^{-\\lfloor n^{\\log \\log n}\\
 rfloor}\,\\quad\n\\sum_{n=3}^{\\infty}\\beta^{-\\lfloor n^{\\log \\log n}\
 \rfloor}\,\\quad \n\\sum_{n=1}^{\\infty}\\beta^{-\\lfloor n^{\\log n}\\rfl
 oor}\,\n\\]\nwhere $y$ is an arbitrary positive real number. \nOur criteri
 on is flexible because it can also be used to prove the algebraic independ
 ence without using functional equations.\nThis is joint work with Hajime K
 aneko.\n
LOCATION:https://researchseminars.org/talk/OWNS/174/
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