On the Dirichlet Series and Asymptotic Expansions of Integral Functions of Zero Order

Abstract

IF an integral function F(z), of finite or zero order, has -Oj, -a 2 , ... for its sequence of zeros, the Dirichlet series 2 a~s converges when the real part of s is greater than some finite number k, not less than zero, and, for this range of values of s, defines an analytic function of s.If the sequence Oj, a^, ... is perfectly general, the line 3&$ = k is a barrier of essential singularities of the function.If, however, a n is an analytic function of n, it will usually happen that the function defined by the Dirichlet series can be continued across the line 311 s = k, and that it gives rise to an analytic function S (s), whose finite singularities are isolated points.M. Mellin has shown that when this is the case there is an intimate relation between the function S(s) and the asymptotic expansion of the function log F(z).*The typical function of finite non-zero order has -n p for itsn-th zero.The Dirichlet series Sw" 1 " then leads to Riemann's function £ (ps), and the properties of this function and its generalisations play an important part in the theory of Dirichlet series and asymptotic expansions of functions of finite order, tIn the integral functions of zero order, we have, for all values of p, | cin | > n p when n is sufficiently great.There is no single functional form for a n that is typical in the sense in which n p is typical in the case of finite non-zero order, but an unlimited number of such forms.Thus, starting from e n , we have among possible forms of a n , e n \ exp(e* P ), exp exp(e'O, ..., (k > 1, p > 0),

Locations

• Proceedings of the London Mathematical Society - View
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