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Polynomials in many variables

Hugh L. Montgomery

Séminaire Delange-Pisot-Poitou. Théorie des nombres

Extreme values of the Riemann zeta function.

Hugh L. Montgomery — 1977

Commentarii mathematici Helvetici

A note on rearrangements of Fourier coefficients

Hugh L. Montgomery — 1976

Annales de l'institut Fourier

Let $f (x) \sim Σ a_{n} e^{2 π i n x}, f * (x) \sim \sum_{n = 0}^{\infty} a *_{n} \cos 2 π n x$ , where the $a *_{n}$ are the numbers $| a_{n} |$ rearranged so that $a_{n}^{*} ↘ 0$ . Then for any convex increasing $ψ$ , $∥ ψ (| f |^{2} ∥_{1} \leq ∥ ψ (20 | f * |^{2} ∥_{1}$ . The special case $ψ (t) = t^{q / 2}$ , $q \geq 2$ , gives $∥ f ∥_{q} \leq 5 ∥ f * ∥_{q}$ an equivalent of Littlewood.