A Note On Ingham's Summation Method

Ingliam's (nonregular) summation method (I) is closely connected with prime number theory. An easy limitation theorem for (I) (observed by Hardy) is if Xc, is summable (I) then c " = o(loglog n). We show this result to be best possible. Ingham's summation method (1) [2] (also discovered independently by Wintner [6]) may be defined as follows : A series Ecn will be said to be summable (1) to A if lim 1 Y dc, = lim 1 cd [ x l = lim i mc,n = A, x-x nz x din x-x d x dcx m(x/d where [x] as usual is the greatest integer in x and the three forms of the limit are clearly equal. (I)-summability is closely connected with prime number theory, and was used by Ingham to give an original proof of the Prime Number Theorem (for further details of such connections see [2 ; 1 ; Appendix IV ; and 5]). Let I(x) = 1 I Y cdc d. as observed by Hardy [1, Theorem 265]. We show this is best possible. 95