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SUMMARY:Wijit Yangjit (University of Michigan)
DTSTART:20220527T193000Z
DTEND:20220527T195500Z
DTSTAMP:20260423T011320Z
UID:CANT2022/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/5/"
 >On the Montgomery–Vaughan weighted generalization of Hilbert's inequali
 ty</a>\nby Wijit Yangjit (University of Michigan) as part of Combinatorial
  and additive number theory (CANT 2022)\n\n\nAbstract\nHilbert's inequalit
 y states that\n$$\n\\left\\vert\\sum_{m=1}^N\\sum_{n=1\\atop n\\neq m}^N\\
 frac{z_m\\overline{z_n}}{m-n}\\right\\vert\\le C_0\\sum_{n=1}^N\\left\\ver
 t z_n\\right\\vert^2\,\n$$\nwhere $C_0$ is an absolute constant. In 1911\,
  Schur showed that the optimal value of $C_0$ is $\\pi$.\n\nIn 1974\, Mont
 gomery and Vaughan proved a weighted generalization of Hilbert's inequalit
 y and used it to estimate mean values of Dirichlet series. This generalize
 d Hilbert inequality is important in the theory of the large sieve. The op
 timal constant $C$ in this inequality is known to satisfy $\\pi\\le C<\\pi
 +1$. It is widely conjectured that $C=\\pi$. In this talk\, I will describ
 e the known approaches to obtain an upper bound for $C$\, which proceed vi
 a a special case of a parametric family of inequalities. We analyze the op
 timal constants in this family of inequalities. A corollary is that the me
 thod in its current form cannot imply an upper bound for $C$ below $3.19$.
 \n
LOCATION:https://researchseminars.org/talk/CANT2022/5/
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