On Expansions Of Arithmetical Functions

On Expansions of Arithmetical Functions R. D. Carmichael PNAS 1930;16;613-616 doi:10.1073/pnas.16.9.613 This information is current as of March 2007. This article has been cited by other articles: www.pnas.org#otherarticles E-mail Alerts

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VOL. 16, 1930

MA THEMA TICS: R. D. CARMICHAEL

613

8 R. L. Moore, "Concerning Upper Semi-continuous Collections," Monatshefte fir Mathematik und Physik, 36, 81-88 (1929). See especially Theorem 2. 4 R. L. Moore, Transactions, loc. cit., Theorem 1. 6 Bull. Amer. Math. Soc., 29, 289-302 (1923). 6 R. L. Moore, Monatshefte, loc. cit., Definitions, pp. 81-82 and Theorem 6. The set described above does not satisfy Moore's condition (b), p. 81-a condition which is not essential under any circumstances. 7 G. T. Whyburn, "A Generalized Notion of Accessibility," Fund. Math., 14, 311326 (1929). 8 G. T. Whyburn, "On Certain Accessible Points of Plane Continua," Monatshefte fur Mathematik und Physik, 35, 289-304 (1928).

ON EXPANSIONS OF ARITHMETICAL FUNCTIONS BY R. D. CARMICHAEL DEPARTMUNT OF MATHUMATICS,

UNWmRSITY

OF

ILLINOIS

Communicated July 24, 1930

1. The remarkable expansions of arithmetical functions obtained by Ramanujan in his notable memoir (Collected Papers, No. 21) on certain trigonometrical sums and their applications are contained as special cases of much more general expansions which have also other special cases of particular interest. The purpose of this paper is to present these generalizations and to draw from the expansions some conclusions of importance obtained by means of a hitherto unnoticed fundamental property of the Ramanujan sums c,(n), namely, that expressed by the relations qp

E cq(n)cp (n) n=l

q

=

0 if p $ q,

L n=1

cq(n)

=

qp(q),

where s(q) denotes Euler's sp-function of q. This has led to the notion of orthogonal arithmetical functions analogous to the notion of orthogonal functions in analysis. 2. By x(a), x1(a), x2(a), ..., we denote any characters modulis k, kj, k2, . . , respectively, and by xo(a), x1o(a), X2o(a), . . . we denote the principal characters for these moduli. If a symbol for a character is written with an argument which is not an integer it may have for this argument any conveniently assigned value. The character which is equal to unity for all arguments will sometimes be replaced by 1. We use MA(a) to denote the Mobius function. By 7q(n) we denote the sum of the flth powers of the qth roots of unity; then 77q (n) is q or 0 according as q is or is not a factor of n.

614

MATHEMATICS: R. D. CARMICHAEL

PROC. N. A. S.

We introduce the functions

c(X)(n, Xi,Xl 2) = E (d) X (d) xi(d)x2 (d) 7'(fn) where the sum is taken for the divisors d of q and where X is positive. This reduces to Ramanujan's function cq(n) when X = 1 and x(a) xi(a) X2(a) = 1. Let F(u, v) denote a function of u and v and write

D(n, Xi, X2)

E xi(d)x2 () F (d, d)-

=

Then we have the following expansion:

D(n, X1, X2)

-

c(

X,

XI, X2) E x(v)F(v 1/21t

+

c(X)(n, x. xi, X2) E 2

+

1

c(nx,

(2v1 2v/

(2v)

/)xx~)

V"'s1

)

X2) E 3v) x(v)F

3. If we take F(u, v) = us then XI, X2),

2n\

+ +...y t

3v

.

D(n, Xi, X2) becomes

the function

as(ni

as(n, Xi, X2)

X1(d)X2

= E

Here s denotes the complex variable or + i t where u- and t this function the foregoing expansion takes the form

a-s(n. Xi,X2)

= C )(n,

are

real. For

(V)

X, Xi, X2) E

1/21

+

C(

(n,

X, XI,

x.v) X2) V=i E (2v)s~+ )) +

t _ nn

If a + X > 1 we may let t become infinite and so obtain the expansion: C)(n, X, X1, a_'S(nXI, X2) = L(s + X, X) E c( S+X X2) co

where L(s, X) denotes the L-series 00

L(s, X)

=

x(a)

E a=l as

We have also the expansion as XI,X2) =

SL (s + X, x)

c

(n,

X, X2,

Esi

XI)

following

VOL. 16, 1930

615

MA THEMA TICS: R. D. CARMICHAEL

4. Putting X = 1, X2(a) expansion of o--, we have

_,(n, X, 1)

=

1 and XI(a)

L(s + 1, x)

x(a) in the second foregoing

x(q),(n),

> O.

From this relation it may be shown that -A, c,(n)cr-.(n, x, 1) = L(s + 1, x)p-l x(p)(p(p) + 0

muM

, o > 2.

=1

This general asymptotic relation is of considerable interest. Special cases of it have appeared in several papers (see Dickson's History of the Theory of Numbers, vol. I, pp. 291-4, 301, 322). Putting x(a) 1, we have imE

-

cp~nafAn, 1, 1)

+ 1+ s +3+l =S+1 1 VP(P) (1+1 2s/1 /\

+0

, a >2.

If X is an integer greater than unity and we put s = 2X - 1, we have

-n (n, 1, 1) - 'o(P) 72X 22xlBx + 0 1 f ki, Mn X=1 (2X)! P2 where the B's are the Bernoulli numbers B1 = '/6, B2 = 1/30. For the case p = 1 the last formula leads to the following theorem: When X is an integer greater than unity, the sum of the (2X - 1)th powers of the reciprocals of the divisors of n is in the mean (on the average) equal to ~2)

(2X)! the error term of the average for n = 1, 2, ..., m being 0(1/m). When x(a) is the principal character xo(a) we have c-s(n, Xo, 1) equal to the sum -y -s(n, k) of the sth powers of the reciprocals of those divisors of n which are prime to the modulus k. Then we have 1m

m

l

s+ El1+l + c,(n),y-,.(n, k) = XoW(P)(P)

>2

where the prime in Z' indicates that q ranges over the integers which are prime to k. From this result we have the remarkable relation 1im

lim - a cd,,(n) -..sf(n, k)

m= X

n

=

1

=

0,

(d, k) > 1,

o-

> 2.

616

MA THEMA TICS: R. D. CARMICHAEL

PROC. N. A. S.

If x(a) is replaced by the non-principal character R(a) modulo 4, then from the first two equations of this section we have

u~(n, R, 1)

1

1

-

+ 5S+1

.

..*) (ci() -C3(2) +C5(n)

)I

S+1-

in

..}.

> o

1 ~R(p)(p(pp1 ()(P(1 1+ 1+-**)

m as c(n) a_(n, R. 1) =

+0

o >2.

5. Incidental to the investigation of these a-functions it was proved that E c(X)(n, Xo, X1, X2) = 0. co

q=1 q

Taking x2(a)-1 and xi(a) 3 xo(a) we find that Z' c(X)(n) = 0. q

where the sum is taken for all positive integers q which are prime to a given positive integer k. The case k = 1 = X of this remarkable relation is due to Ramanujan (loc. cit., p. 185). 6. The expansions of a-functions employed in this paper lead to remarkable expansions of functions relating to the representation of integers as sums of squares. We shall content ourselves with stating three particular results. Using N[n = X2 + y2] to denote the number of ways of representing n as a sum of two squares and employing similar notations for other cases, we have

N[n

=

x2 + y2] = 2ir {cl(n) + -c5(n) +-c9(n) + *

N[n

=

X2 + 2Y2]= r 2{cl(n) + 3 c3(n) + - c9(n) +

Acli(n) +--.

N[2am - x2 + y2 + Z2 + t2, m odd] = g(ct)r2m {cl(m) + +

3(2

c2(r) 52

+

where g(a) = 1 or 3 according as a = 0 or a > 0 and where in the second formula the subscripts appearing are those of the forms 8x + 1 and 8x + 3.