Jekyll2025-07-19T17:58:35+00:00https://berndmehnert.github.io/feed.xmlBernds homepageThis is my personal homepage where I share my projects and research.Test latex post.2025-07-19T00:00:00+00:002025-07-19T00:00:00+00:00https://berndmehnert.github.io/jekyll/update/2025/07/19/large-sieveWe let \(\alpha_{1},...,\alpha_{R}\) be \(R\) real numbers that are pairwise distinct modulo one and we let \(a=(a_{n})_{M<n\leq M+N}\) be an arbitrary \(N\)-vector of complex numbers. We let further

\[S(\alpha)=\sum_{M<n\leq M+N}a_{n}e(\alpha n),\]

where as usual \(e(x):=\exp (2\pi ix)\).

Then

\[\sum_{r\leq R}|S(\alpha)|^{2}\leq(N-1+\delta^{-1})\left\Vert a\right\Vert ^{2},\]

where \(\delta\) is the minimum distance between different \(\alpha_{i}\) in \(\mathbb{R}/\mathbb{Z}\). The above is an example of a “Large Sieve”-inequality …

]]>Welcome to my blog!2025-05-30T12:16:04+00:002025-05-30T12:16:04+00:00https://berndmehnert.github.io/jekyll/update/2025/05/30/welcome-to-jekyllGreetings to all out there!

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