The Dirichlet-Bohr radius

DANIEL CARANDO; ANDREAS DEFANT; DOMINGO GARCÍA; MANUEL MAESTRE; PABLO SEVILLA PERIS

ACTA ARITHMETICA

POLISH ACAD SCIENCES INST MATHEMATICS

Lugar: VARSOVIA; Año: 2015 vol. 171 p. 23 - 37

0065-1036

Denote by $Omega(n)$ the number of prime divisors of $n in mathbb{N}$(counted with multiplicities). For $xin mathbb{N}$ define the Dirichlet-Bohr radius $L(x)$ to be the best $r>0$ such that for every finite Dirichlet polynomial $sum_{n leq x} a_n n^{-s}$ we have$$sum_{n leq x} |a_n| r^{Omega(n)} leq sup_{tin mathbb{R}} ig|sum_{n leq x} a_n n^{-it}ig|,.$$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 to translate various results on Bohr radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa.