It is shown that, asymptotically near ,? = oo , we have
and
log R p (.:') = ( - ,r)p log z + ( - K 1 -$ log z - ~ log 2
F (*//>)
+ 2' =-P
where the numbers F are substantially Biemann {-functions of nega- tive argument.
It is then shown that these expansions can be generalised so as to apply to wide classes of functions, notably those with algebraic sequence of zeros. General formulae are given which symbolically contain the expansion of all integral functions of the type considered, which are such that the Maclaurin sum formula can be applied to the function expressing the dependence of the nth zero upon n. And as an example of a function with transcendental sequence, the asymptotic expansion of
00 r
n
= i L
1+-
is obtained completely.
Part IV deals with the asymptotic expansion of repeated integral functions of simple sequence. It is necessary to take an extended definition of " order " in the case of such functions, and then actual expansions are obtained for standard functions
(1.) Of finite (non-zero or zero) order less than unity ;
(2.) Of finite non-integral order greater than unity ;
(3.) Of finite integral order greater than or equal to unity.
Subsequently symbolic formulae are given for all integral functions of the prescribed type ; and, as an example, the asymptotic expansion of a repeated function with a transcendental index is considered. The formulae are verified in the case of the G-function.
Part V is devoted to applications of the previous expansions to the problems mentioned in the introduction.
A knowledge of the asymptotic expansion of a function serves to determine the number of roots which it possesses inside a circle of given large radius. If the function is of order p, the number of roots within a circle of large radius r is to a first approximation
sin Trp
