By Kubota T.
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Extra resources for An elementary theory of Eisenstein series
Example text
E s s T ) and [ 1 -$)+p++&) SMALL ADDITIVE FUNCTIONS c m = - t C y 1s + Cp w 1= l +l ; %=-Iw+l-1) tw (_,)V+l p w=l vp 43 w(ps-l)w ;t w p w=2 w ( p p v=2 wp ws t + 1 ; (-l)v+l t v + pI pS(pS-l) p w=2 v(ps-l)w which converges uniformly for u 2 1+ 2 E , for any preassigned E > 0 Therefore g(s, t ) = elogg(s-'t) represents a holomorphic function of 1 for u > 7 (see Apostol C11 p. 394). 22) . : uniformly for It1 s 1 . 4. We first have to verify that D ( t ) all a . 21). 4, we obtain E E CmCO,ll C"C0,lI 1 for n f , from the representa- , and hence g(l,t)/r(t)~c~rO,lI.
2 . e x i s t constants bk, l,. 31) 1 l/dk(n) = bk, n a Proof. For every k z 2 and every fixed natural . b k, fl x logl'k-l depending only on k xt . 11) t h a t N there such t h a t 3: + O(x log1/ k-N-1 x) RECIPROCALS OF MULTIPLICATIVE FUNflIONS For DlIk ( x / n ) . 31) after collecting terms . x logl/k-i2 for i = 1 ,2,.. ,N For another application of the convolution method we now consider a(n) , the number of non-isomorphic abelian groups with n elements. 32) and in particular a ( p ) = 1 Dirichlet series of l/a(n) t p-s P -2s + ++ t l t ____ t t a (PIP" -4s -3s t , = 2 for every prime p n (1 n=l n(1 P is the number of (unres- .
1, with ~ ( t= )g(l,t)/r(t) E . c~+~co,~I From this definition, we observe that if (g,f) uniformly for It I s 1 . 3. now state the following theorem. 8. Let @,f) E S; .
