Some asymptotic formulae involving powers of arithmetic functions

SOME ASYMPTOTIC FORMULAE INVOLVING POWERS OF ARITHMETIC FUNCTIONS V. Sitaramaiah §i.

Introduction.

and

M.V° Subbarao

S. Ramanujan was probably the first mathematician to consider

asymptotic formulae for sums of powers of certain arithmetic functions.

For

example, in 1916 in his paper [I0] he generalized the classical Dirichlet divisor problem and gave estimates, without proof, for

~ TS(n), where r(n) denotes the n

depends only on

Bk(P), Plr

By Lemma 2.12, Theorem 3.1

We assume (3.1) for some

k > i and all

r.

is true for

k = 1

and for all

We have, since

~(m)m -I = ~dl m ~(d)d -I ,

d 2.

For each fixed integer

m ~ (r~(m)~ m ) ~k

(~.21) where

r.

We may mention here that in view of Remark 2.1, in [20] they would

get (3,20) (k=l) with error term 0 (xl(x)a ~(r))

THEOREM 3.4.

m.

k ~ I,

= ~kBk +~X(~)(log

x)) N+k-l ,

(k=l) is clearly

224

(k~2)(k-l) + i

,

if

k

is even

k ((k+i/2))(k-l) + I

,

if

k

is odd ,

(3.22)

N =Nk=

and A k = ~ {I + (P-l,,)k((p+l)k-pk) } , p pk+l(p+l)kBk(p)

where

PROOF:

Bk

and

Let

Bk(P)

are as given in Theorem 3.1.

g(m) = mk/(~(m)) k

for any

m.

g(m) =

for any

We write

Z f(d) , dm]

m, so that by the M'obius inversion

Therefore,

if

(3.23)

p is a prime and

a

(3.24)

formula, we get

is a positive integer,

f(m) =

[ ~(d)g(mld). d| m

we have

f(p~) = g(p~) _ g(p~-l) g(p)-I

,

if

~ = i

0

,

if

~ _>

(3.25) ~

since

g(pa) = g(p) for any prime

]f(p)[

= Ig(p)-ll = i -

p

2

,

and

~ > i.

We have

k

)k pk = (p+l (p+l) k (p+l) k

p

+ C )p' +

. . .

+ ck _ )p

(p+l) k i + (k-l)Mkpk-i