The class $e$ of continuous complex-valued functions, of non-negative real variable forms, a commutative algebre without zero divisors where the product is defined as the finite convolution and the sums and scalar products are defined in the usual way. The quotient field of this algebra is the operator field $K$ of Mikusinski. In this operator field the limit, differentiation and integration are defined. The UT-limit of convergent series operators is defined in this paper. The series: \[ um_{n=0}^{ıfty}a_ne^{-sb_n} \] where: $s$ is the differential operator in the operator field $K$ $a_n$ is the sequence of complex numbers $0\leq b_0<b_1<\ldots<b_n<\ldots$ $_n\to+\infty$, $n\to\infty$ is always convergent. For the definition of the UT-limit, I shall use two linear, continuous transformations $T^{-z}$ and $U_k$ of the operator field $K$. The following theorems are proved: Theorem 1: If a Dirichlet series: \[ ǎrphi(z)=um_{n=0}^{ıfty}a_ne^{-zb_n} \] converges, in the classical sense, for $Rez>\sigma\geq0$, then for each $z$, in the operator field $K$, the limit UT of convergent series operators $\sum_{n=0}^{\infty}a_ne^{-sb_n}$ exists, in the following form: \[ UT-im_{koıfty}um_{n=0}^{ıfty}a_ne^{(\frac sk+z)^{b_n}}=im_{koıfty}U_kT^{-z}\Big(um_{n=0}^{ıfty}a_ne^{-sb_n} \Big) \] and the following holds: \[ im_{koıfty}U_k\bigg[t^{-z}\Big(um_{n=0}^{ıfty}a_ne^{sb_n} \Big)\bigg]=um_{n=0}^{ıfty}a_ne{-b_nz}=\bigg\{zıt_{0}^{ıfty}e^{zt}F(t)dt\bigg\} \] Theorem 2: If a Dirichlet series is $(R,b,k)$, $k>0$, Riesz summable in $z=0$, then for each $z\in S_z(0,\Psi<\frac\pi2)$ in the operator field $K$, the UT-limit of convergent series operators $\sum_{n=0}^{\infty}a_ne^{-sb_n}$ exists, and the following holds: \[ im_{moıfty}U_m\bigg[T^{-z}\Big(um_{n=0}^{ıfty}a_ne^{sb_n}\Big)\bigg]=\bigg\{\frac{z^{k+1}}{ȁmma(k+1)}ıt_{0}^{ıfty}e^{-zt}F^{(k)}(t)dt \bigg\} \] where \[ F^{(k)}_{(t)}=um_{b_n}a_n(t-b_n)^k,\qquad k>0. \]