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Fedor Petrov
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Is the following true (and if yes, where the best proof is written?)?

For any $c>0$ for large enough positive integers $N$ we have $\sum_{k=0}^{N-1} \cos(k^2t)\geqslant -cN$ for all real $t$?

Hm, if true, it should be hard: it allows to get signs of certain Gauss type sums, for example.

Is the following true (and if yes, where the best proof is written?)?

For any $c>0$ for large enough positive integers $N$ we have $\sum_{k=0}^{N-1} \cos(k^2t)\geqslant -cN$ for all real $t$?

Is the following true (and if yes, where the best proof is written?)?

For any $c>0$ for large enough positive integers $N$ we have $\sum_{k=0}^{N-1} \cos(k^2t)\geqslant -cN$ for all real $t$?

Hm, if true, it should be hard: it allows to get signs of certain Gauss type sums, for example.

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Fedor Petrov
  • 108.9k
  • 9
  • 264
  • 459

uniform one-sided van der Corput inequality

Is the following true (and if yes, where the best proof is written?)?

For any $c>0$ for large enough positive integers $N$ we have $\sum_{k=0}^{N-1} \cos(k^2t)\geqslant -cN$ for all real $t$?