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The Dirichlet-Bohr radius

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Carando D., Defant A., Garcí D., Maestre M., Sevilla-Peris P.

Acta Arithmetica

2015

tom 171

nr 1

23-37

Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial $∑_{n ≤ x} a_n n^{-s}$ we have $∑_{n ≤ x} |a_n| r^{Ω(n)} ≤ sup_{t∈ ℝ} |∑_{n ≤ x} a_n n^{-it}|$. We prove that the asymptotically correct order of L(x) is $(log x)^{1/4} x^{-1/8}$. Following Bohr's vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows translating various results on Bohr radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa.

Ograniczanie wyników

1 Acta Arithmetica

1 Carando D.

1 Defant A.

1 Garcí D.

1 Maestre M.

1 Sevilla-Peris P.

1 2015

Wyniki wyszukiwania

The Dirichlet-Bohr radius