On the equivalence of the sum and the maximal term of the Dirichlet series absolutely convergent in the half-plane
For absolutely convergent in the half-plane ${zcolon {mRe,}z<0}$ Dirichlet series $F(z)=sumlimits_{n=0}^{+infty}a_ne^{zlambda_n},$ where $0leqlambda_nuparrow +infty (0leq nuparrow+infty),$ we establish conditions on the coefficients of itsNewton majorant, sufficient for the relation $F(x+iy)=(1+o...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Vasyl Stefanyk Precarpathian National University
2009-06-01
|
| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Online Access: | http://journals.pu.if.ua/index.php/cmp/article/view/18/15 |
| Summary: | For absolutely convergent in the half-plane ${zcolon {mRe,}z<0}$ Dirichlet series $F(z)=sumlimits_{n=0}^{+infty}a_ne^{zlambda_n},$ where $0leqlambda_nuparrow +infty (0leq nuparrow+infty),$ we establish conditions on the coefficients of itsNewton majorant, sufficient for the relation $F(x+iy)=(1+o(1))a_{u(x)}e^{(x+iy)lambda_{u(x)}}$ to hold as$xo -0$ outside some set $E$ of zero logarithmic density in thepoint $0,$ uniformly by $yin{mathbb R}$. |
|---|---|
| ISSN: | 2075-9827 |