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SUMMARY:Daodao Yang (Graz University of Technology\, Austria)
DTSTART:20220527T170000Z
DTEND:20220527T172500Z
DTSTAMP:20260423T011437Z
UID:CANT2022/50
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/50/
 ">Extreme values of derivatives of the Riemann zeta function\, log-type GC
 D sums\, and spectral norms</a>\nby Daodao Yang (Graz University of Techno
 logy\, Austria) as part of Combinatorial and additive number theory (CANT 
 2022)\n\n\nAbstract\nFirst I will recall the research on greatest common d
 ivisor (GCD) sums and   extreme values of  the Riemann zeta function. The 
 motivation for the study and the connection between the two problems will 
 be discussed.\n Then I will explain how to establish lower bounds for maxi
 mums of $|\\zeta^{(\\ell)}\\left(\\sigma+it\\right)|$ when $\\sigma \\in [
 \\frac{1}{2}\, 1]$\, $\\ell \\in \n$.  One of my results states that   as 
 $T \\to \\infty$\,  uniformly for all positive integers $\\ell \\leqslant 
 (\\log_3 T) / (\\log_4 T)$\, we have\n$  \n\\max_{T\\leqslant t\\leqslant 
 2T}\\left|\\zeta^{(\\ell)}\\left(1+it\\right)\\right| \\geqslant \\left(\\
 mathbf Y_{\\ell}+ o\\left(1\\right)\\right)\\left(\\log_2 T \\right)^{\\el
 l+1} $\, where $\\mathbf Y_{\\ell} = \\int_0^{\\infty} u^{\\ell} \\rho (u)
  du$\, and $\\rho(u)$ denotes the Dickman function. This  generalizes resu
 lts of Bohr-Landau and Littlewood on $\\left|\\zeta\\left(1+it\\right)\\ri
 ght|$ in 1910s.  The tools are Soundararajan's resonance methods and ingre
 dients are certain combinatorial optimization problems. On the other hand\
 , assuming the Riemann hypothesis\, we have $|\\zeta^{(\\ell)}\\left(1+it\
 \right)| \\ll_{\\ell}\\left(\\log \\log t\\right)^{\\ell+1}$.\nThen I will
  talk on the log-type GCD sums $\\Gamma^{(\\ell)}_{\\sigma}(N)$\, which I 
 define it as $\\Gamma_{\\sigma}^{(\\ell)}(N):\\\,= \\sup_{|\\mathcal{M}| =
  N} \\frac{1}{N}\\sum_{m\, n\\in \\mathcal{M}} \\frac{(m\,n)^{\\sigma}}{[m
 \,n]^{\\sigma}}\\log^{\\ell} \\left(\\frac{m}{(m\,n)}\\right)\\log^{\\ell}
 \\left(\\frac{n}{(m\,n)}\\right)\,$\nwhere the supremum is taken over all 
 subsets   $\\mathcal{M} \\subset \\mathbb N$ with size $N$.\nI will explai
 n how $\\Gamma^{(\\ell)}_{\\sigma}(N)$ can be related to $|\\zeta^{(\\ell)
 }(1+it)|$ and how to prove that $\\left(\\log\\log N\\right)^{2+2\\ell} \\
 ll _{\\ell}\\Gamma^{(\\ell)}_1(N)\\ll_{\\ell} \\left(\\log \\log N\\right)
 ^{2+2\\ell}$\, which generalizes Gál's theorem (corresponding to the case
  $\\ell = 0$). The lower bounds could be used to produce large values of $
 |\\zeta^{(\\ell)}\\left(1+it\\right)|$.   Using a random model for the zet
 a function via methods  of Lewko-Radziwiłł\,   upper bounds for spectral
  norms on $\\alpha$-line are established\,  when $\\alpha \\to 1^{-}$ with
  certain fast rates. As a corollary\,  upper bounds of correct order of   
 the log-type GCD sums are established.\n
LOCATION:https://researchseminars.org/talk/CANT2022/50/
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