用狄立赫利级数所定义的整函数的极大项

Letf(s)=sum from n=1 to ∞ (a_n)/(n~B),be a non-termiaating Dirichlet's series wherea_n0,n=-1,2,….Let the abscissa of convergence of this series be-∞.Then f(s)representsan entire function of order ρ and tgFe τ given by the following formulas:1/ρ+(log log n)/(log log 1/(a_n))=1,τ=((ρ-1)~(ρ-1))/ρ((log n)~ρ)/((log 1/(a_n))~(ρ-1)),Letand (r)be the rank of this term.Thenlogμ(r)=O(1)+integral from 0 to r log(r)dx.Both (r)and μ(r)are monotonely increasing unbounded functions,andIf f(s)is a function of regular growth and 1ρ<∞,thenlog M(r)～logμ(r)andas r→∞.In the case ρ>1,we put (logv(r))/(r~(-1))=then we have/ρ,where is the lower type of f(s).If ρ=1,we put(logM(r))/(r(log r)~ω)= 0<ω<∞then we have=(logn)/(log log 1/(a_n))~ωLetlogv(r)/((logr)~∞)=If f (s)is a function of regular growth,then