General Dirichlet series

In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of where a n {displaystyle a_{n}} , s {displaystyle s} are complex numbers and { λ n } {displaystyle {lambda _{n}}} is a strictly increasing sequence of nonnegative real numbers that tends to infinity. A simple observation shows that an 'ordinary' Dirichlet series is obtained by substituting λ n = ln ⁡ n {displaystyle lambda _{n}=ln n} while a power series is obtained when λ n = n {displaystyle lambda _{n}=n} . If a Dirichlet series is convergent at s 0 = σ 0 + t 0 i {displaystyle s_{0}=sigma _{0}+t_{0}i} , then it is uniformly convergent in the domain and convergent for any s = σ + t i {displaystyle s=sigma +ti} where σ > σ 0 {displaystyle sigma >sigma _{0}} . There are now three possibilities regarding the convergence of a Dirichlet series, i.e. it may converge for all, for none or for some values of s. In the latter case, there exist a σ c {displaystyle sigma _{c}} such that the series is convergent for σ > σ c {displaystyle sigma >sigma _{c}} and divergent for σ < σ c {displaystyle sigma <sigma _{c}} . By convention, σ c = ∞ {displaystyle sigma _{c}=infty } if the series converges nowhere and σ c = − ∞ {displaystyle sigma _{c}=-infty } if the series converges everywhere on the complex plane. The abscissa of convergence of a Dirichlet series can be defined as σ c {displaystyle sigma _{c}} above. Another equivalent definition is