Comparative growth of an entire function and the integrated counting function of its zeros - Andrusyak, I.V.; Filevych, P.V. (Carpathian Mathematical Publications, 2024)
Let (ζ_n) be a sequence of complex numbers such that ζ_n→∞ as n→∞, N(r) be the integrated counting function of this sequence, and let α be a positive continuous and increasing to +∞ function on R for which α(r)=o((N(r)/r)) as r→+∞. It is proved that for any set E(1,+∞) satisfying _Er^α(r)dr=+∞, there exists an entire function f whose zeros are precisely the ζ_n, with multiplicities taken into account, such that the relation _rE, r→+∞M(r)r(N(r)/r)=0 holds, where M(r) is the maximum modulus of the function f. It is also shown that this relation is best possible in a certain sense.