New Möbius inversion formulas

PHYSICS LETTERS A

Physics LettersA 164 (1992) 1—5 North-Holland

New Möbius inversion formulas Shang Yuan Ren’ Department ofPhysics and Astronomy, Arizona State University, Tempe, AZ 85287, USA Received 16 January 1992; accepted for publication 10 February 1992 Communicatedby J.P. Vigier

More Möbius inversionformulas may be derived from severalinfinite summation identities involving the Möbius function and the relevant function F(x). Some of them have complete new formalisms. A simple example of applications of one of the new inversion formulas is also discussed.

Recently, since Chen published his pioneer work [1] showing that the Möbius inversion in number theory may be successfully used in solving physics problems, the Möbius inversions have attracted interest in the physics and mathematics community [1—6].Several Möbius inversions have been given, each may have their special applications. We notice that several infinite summation identities involving the Mäbius function and the relevant function F(x) may have even more fundamental importance, because: (1) more new Möbius inversion formulas can be derived from these identities; (2) these identities may be combined to give more identities from which more Möbius inversion formulas can be derived. Here we show several examples. (1) The simplest but also the most fundamental infinite summation identity involving the Möbius function and a relevant function F(x) is [3], for a real function F(x) of a real variable x,

>~n)F(n~~max)=F(x), (1) where a is a non-zero real constant, F(x) is an ordinary real function of a real variable x, and ~u(n) is the Mäbius function, defined as [71 m>~I

ifn=l =

(— 1) r

=0,

if n is the product of r distinct primes, otherwise.

(2)

The condition for eq. (1) to be true is that the relevant summations are convergent. Then from eq. (1) we should have ~ ~,i(n),j(l)F(nc~~ma1PkPx)= ~ >Jtt(l)F(I~k~x)=F(x), m=1 n=I k=I 1=1

(3)

k=I l~1

where a and $ are two non-zero real constants. Therefore from eq. (3) we can have a new inversion formula, that is, if G(x)= ,n=1 ~ k=I ~ F(m”~k8x), Onleave from the University of Science and Technology of China, Hefei 230026, China. 0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

(4a)

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6 April 1992

then F(x)= ~ >~/i(n)~z(1)G(n”~1~x).

(4b)

n=1 1=1

The way for further extending eq. (4) is obvious. (2) If a, $ and y are three non-zero real constants, then we should have for a function F(x) of a realvariable x, ~ ~(n)(_l)m_I[ym_iF(am_mnflkPx)+ymF(amnflkflx)] ,n=1 n=1 k=l

~ ~(n)

=

n=i k=I

~ (_l)m_l[ym_IF(am_mnflkflx)+ymF(amnskflx)] m=l

~ ~(n)F(n~k~x)=F(x),

=

(5)

n=l k=l

if the relevant infinite summations are convergent. Therefore from eq. (5) two inversion formulas can be obtained. They are, if G(x)= ~ [F(k~x)+yF(ak~x)]

(6a)

,

then F(x)=~/i(n)~(_l)m_Iym_IG(am_mnPx),

(6b)

and, if G(x)= ~

~

(_l)m_Iym_LF(am_lkflx),

(7a)

k=l ,n=I

then F(x)= ~ ~t(n)[G(n~x)+yG(an~x)] .

(7b)

Several MObius inversion formulas used in ref. [5] can be considered as special cases of eq. (6). (3) In the process of proving their new Möbius inversion formula, if g(x)= ~ (—l)m~’f(mx),

(8a)

then ~ ~(n)2/~_lg(2k_mnx),

(8b)

n=l k=1

by simple algebra, Chen and Ren [4]have shown that ~ ~(n)(_l)m2k~lf(2k-lmnx)=f(x).

(9)

n=I ,n=I k=I

We can easily see that based upon eq. (9), two new Möbius inversion formulas can be obtained. The are, if 2

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G(x)= ~F(2”—’x),

6 April 1992

(lOa)

then F(x)=~>/.t(n)(_lY~~’G(mnx)I(mn),

(lOb)

and, if G(x)=

m~l n~I

(_~m+~2n_lmx(m)’

(lla)

then F(x)=>~~u(n)G(nx)/(n) .

(llb)

The significance ofthese two new formulas is that for the function F under the summation, now the coefficient in front of its variable x has the summation index n as an exponential rather than as a factor as in the Möbius formulas given before [1—5].It means, these two Möbius inversion formulas can be used to get the inversion of a completely new type of infinite summations. We know in science that some problems may have the integer powers oftwo involved, for the simplest examples in static electric interactions, we have Coulumb interactions, dipole interactions, quadrupole interactions, and so on. We may notice that, if we define eq. (8a) then eq. (9) gives eq. (8b). This is the inversion formula given by Chen and Ren [4]. But if we define g(x)=

~

2k_If(2k_Ix),

(12a)

k= I

then eq. (9) gives f(x)= ~

~jL(n)(_1)m+lg(mnx).

(l2b)

It can be used to prove eq. (10). If we define 12/c_lf(2k_Imx), m=I k=I ~

(l3a)

(_l)m+

then eq. (9) gives ~ ~u(n)g(nx).

(13b)

n= 1

This is another new inversion formula which can be used to prove eq. (11). If we introduce F(x)=xf(x)

(14a)

and G(x)=xg(x),

(14b)

then from eq. (12), we have, if 3

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(l5a)

~ F(2”~x), G(x)=xg(x)= ~ 2’~xf(2’~1x)= then F(x)=xf(x)=x ~

~ ~(n)(_1)m~g(mnx)

n=1 m=1

(15b)

~ This is the proof of eq. (10). Similarly, from eq. (13), we have, if G(x)=xg(x)=

~

~

(_l)m

(sk_lmx)f(2k_Imx)/(m)=

,n=I k=I

~

~ (_l)m+lF(2n~~Imx)/(m), (16a)

m~1 n=I

then F(x)=xf(x)=x(n)g(nx)=~j~(n)nxg(nx)/(n)=~(n)G(nx)/(n).

(l6b)

This is the proof of eq. (11). (4) From eq. (9), we obtain ~ i= 1

j= I

l),u(n)(_l)m+3+l2k_1f(2~~_1_Imnj1x)

k= I 1=1 tn= 1 n= 1

~

=

~(n)(_l)m2k~lf(2k~lmnx)=f(x).

(17)

n=I ,n=I k=I

Therefore, we can obtain the following inversion formulas: if g(x)= ~

~ (_l)m+~J~f(mjx),

(18a)

j=I ,n=I

then ~ ~(1)~(n)2k_lf(2k_~nlx),

(l8b)

i=I k=I 1=1 ,~=I

and, if G(x)= ~ ~ F(2k_~i_lx)

(l9a)

1=1 k=I

then F(x)= ~

~ ~

(19b)

j=I 1=1 ,n=l n=l

and, if G(x)= ~ i=lj=I

4

~ (_l)~+JF(2k_~_Imjx)/(mj), k=1 m=l

(2Oa)

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then F(x)= ~

~ ~u(1)~u(n)G(n1x)/(nl).

(20b)

1=1 n=I

The one-dimensional inversion formulas eqs. (4), (6), (7), (10), (11), (18), (19), (20) described here may be easily extended further to higher dimensions. (5) As a simple example of applications of the new inversion formulas, let us assume that in eq. (10) F(x)=xI_a,

Re(a)>1.

(21)

Then from eq. (lOa), we have (22a)

~

Eq. (lOb) means x~=

~ ~ n=I m=l

or ~ n=I

,?,=

~(n)(_1)m+l(1_2I_a)_I(mn)_a=l.

(22b)

I

But we know that [3,4,6] (23) Eqs. (23) and (22b) give 1 m>~I (_Um(m)_~

(24)

This is a property of the Riemann zeta function [8]. We just proved it by another way. Of course, there could be more possible applications of the new Möbius inversion formulas. The author is grateful to the Air Force Office of Scientific Research (Contract No. AFOSR-9 1-0418) and Professor John D. Dow for their generous support. He also thanks Dr. N.S. Chen for sending preprints of his papers and for helpful discussions.

References [1] N.X. Chen, Phys. Rev. Lctt. 64 (1990) 1193. t2]J. Maddox, Nature 344 (1990) 377. [3]S.Y. RenandJ.D. Dow, Phys. Lett. A 154 (1991) 215. [4] N.X. Chen and G.B. Ren, Phys. Lett. A 160 (1991) 319. [5] N.X. Chen and G.B. Ren, Carlsson—Gelatt—Ehrenreich technique and MObius inversion theorem, to be published. [61 D.N. Hughes, N.E. Frankel and B.W. Ninham, Phys. Rev. A 42 (1990) 3643. [7] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, 5th Ed. (Oxford Univ. Press, Oxford, 1979) pp. 235—237. [8] I.S. Gradshtyen and I.M. Ryzhik, Table ofintegrals, series and products, corrected and enlarged edition (Academic Press, New York, 1980) p. 1073.

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