DISTRIBUTION OF VALUES OF ARITHMETIC FUNCTIONS IN RESIDUE

CLASSES

B. M. Shirokov UDC 511.37

We give a criterion for weakly uniform distribution of integral multiplicative

functions ~(~) of the class C$ modulo N , generalizing a result of

Narkiewicz (W. Narkiewicz, Acta Arithm., 12, 269-279 (1967)). We obtain an

asymptotic formula for N(~ml~(~)-e(m0dN)) . We consider particular

cases for the function ~(~) : ~(K) the number of integral points in the

circle ~+~t~, ~9(~ ).~ ~9 and others.

i. Introduction. Narkiewicz [3] introduced the concept of weakly uniform distribution

of a sequence of integers modulo a natural number. As in [3] we will write N(~IP(n)) to

denote the set of natural numbers ~ for which the property Q holds, and N denotes

a fixed natural number. A sequence ~m of natural numbers is called weakly uniformly dis-

tributed modulo N if for all natural numbers ~ and ~ such that (~,N)=(~,~)=~, we

have for ~ .o~

provided the set {0%l(~,N) =J } is infinite.

~(~) will denote an integral multiplicative arithmetic function. We will say that

~(~) is weakly uniformly distributed modulo ~ , if this is the case for the sequence of its

values. We will say that the function # (~) is polynomiallike if for every natural number j

there exists a polynomial ~j(~) such that ~(pJ)=Fj(p),p a prime.

For polynomiallike functions Narkiewicz [3] proved a criterion for weakly uniform distribu-

tion modulo N and gave conditions on N which are necessary and sufficient for the func-

tion ~[~I (the number of divisors of ~ ) and the function ~(~ (the Euler function) to

be weakly uniformly distributed modulo N . Moreover, he showed that if ~ (~ is weakly

uniformly distributed modulo N and (~,N)=~, then we have for ~--,o~ and ~=n~m~-]

where ~ is a prime,

(1)

where CI) 0 is a constant which is independent of ~ �9 A similar result is also given

for q (~).

Sliva [4] gave necessary and sufficient conditions for ~C~) to be weakly uniformly

distributed. Earlier, Rankin [3] showed that if %>/5 is a prime number, 9 a natural

number and ~=(a-O/(~,~,-~) , then for ~----~

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 176-186, 1983.

1356 0090-4104/85/2903-1356509.50 �9 1985 Plenum Publishing Corporation

where the positive constants O~ and C 3 are represented as infinite products. For ~(~),

i.e., for ~=0, this result was given with remainder term by Sathe [6].

Let ~m(~) be the number of representations of the number ~ as a sum of two squares

of integers. For this function Scourfield [7] showed that

C~ ~ (4) N $ where C~>O is a constant which depends only on the prime number ~ �9 Fomenko [i] gave

necessary and sufficient conditions for weakly uniform distribution of the function ~-a%z(~)

modulo a prime ~ and showed that if these are satisfied for ~=~,..-)~-I then

,

where the constant C~ is the same as in (4). The formulas (5) imply (4). Fomenko also

considers the functions g~ ~n) and

where % (~) is a real nonprincipal character for some modulus. His paper contains further

bibliographical references on these questions.

2. Notation and Results. We note that the functions ~9~,%) and ~-Z%zC~) are not

polynomiallike.

Definition. We will say that # (m) belongs to the class CS ("congruence similar"

to ~ ), if there exists a natural number k such that whatever N~0(,~0~ k) the con-

g~uence pm~(~o~N) implies the congruences ~CpJ)=-~(~ ~) (nzo~N), J= 4,g,... Here

p and ~ are prime numbers.

The smallest integer k for which the property in the above definition is satisfied for

the function ~(~) will be denoted by k (~) �9 It is clear that every polynomiallike

function belongs to C~ , with k(~)={ �9 For ~(~)=~(~) we find in view of the repre-

sentation �88 where %~ is a nonprincipal character modulo 4, that k~) ~-

If the modulus of the nonprincipal character % is Z, then we find for ~(~)=~C~,%) that

We introduce some further notation. As above, p and ~ are prime numbers, k, J, ~,

�9 , ~, ~, ~ are natural numbers. For a function ~ (m) of the class ~S the number ~ de-

notes the l.c.m, of the numbers N and k (~), R ~N) the ring of integers modulo N , G (N)

the multiplicative group of the ring R (N) . Further, we denote by ~j (~) the mapping

~(~) ~ R(N) defined as follows. Define ~(~) by the implication p~(r~oa~)- >

~(p~)~__~(~)CF~o~) , and let ~ be the natural epimorphism of R(~) onto R(N) �9

Then we put c~j ~) -- ~(~(~)) �9 R~ denotes the image of ~(~) in C~N) under the mapping

~, ~ is the smallest j for which R ~ and ~,~- is the subgroup of G CN) gener-

1357

ated by the set gm, �9

N , and cO and u0 o

Let ~ and ~o be an arbitrary and the principal character modulo

an arbitrary and the principal character modulo ~,

i ~ - X Co%(~)) ~ (~), �9 z (~ , , .o )= 4Ca) .~c,~a)

~ a nonnegative integer (different from the one used in the proof of Theorem I), ~ 0, Z2f:7(~oo )- ~/, P~(%,%) a polynomial of degree ~, whose coefficients depend on the

function ~(~) , the character ~ and the modulus N.

THEOREM i. If N>/$~eCS~ n~i~-o~ and ~eGCN), there exist a constant C~>0 ,

such that if ~--~oo we have for all

Og i.+ ~ c - C s v ~ ] (6)

where A~ ~(N/~ ~N(N a.y, and fl.~ a r e c o n s t a n t s depending on f ( A ) and on the c h a r -

a c t e r .N , e q u a l l i n g 1 on -/~-m~"

THEOREM 2. The f u n c t i o n ~(~)~ C S i s weakly u n i f o r m l y d i s t r i b u t e d modulo N i f and

on ly i f f o r eve ry n o n p r i n c i p a l c h a r a c t e r /~ which equa l s 1 on .Am, , t h e r e e x i s t s a prime

number p such t h a t e i t h e r P I N or p..~2 ~ and

~D k./.. ~ ' = 0 ( 7 )

Note that if An~----~C~) , then ~ ( ~ ) is weakly uniformly distributed modulo N.

As applications of these theorems we give some corollaries.

COROLLARY i. Let ~ECS, N >/$ and n~<+o~. Then for every ~ , if ~=7/~o#i, then

(8)

if ~= i~ then

(9)

where A~ and C 6 are some positive constants.

COROLLARY 2. The function ~z(n) is weakly uniformly distributed modulo

only if one of the following holds:

a) N=4,

b) N=~ ~ , ~ >/~, ~ >/

c) N=go~,or~5 , ,$ ~' I Moreover, if gl~ (~)

and 2 is a primitive root modulo ~

and 3 is a primitive root modulo ~.

then

N(~ml~z~(~)-~(~0~4))= ~ + 0~v~ e-c~-~) ;

N if and

( i 0 )

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if ~>/~) and (0.,~)=~, then we have for every

if 3 is a primitive root modulo ~ , then we have for every k

+ 0 (~ (~ ~)~-~-~), ~ ~ A is a positive constant which does not depend on �9 E ~ (.p_Q~). where ~L = ; ~ # ; % ~ , 5

COROLLARY 3. I f ~ >/3 we have for every ~b

-| If ~ is odd, then COROLLARY 4. Let ~>5 and ~= (~'~-{)

where ~ and 6 7 are positive constants; if ~ is even, then we have for every

COROLLARY 5. If N>/~ is an odd number then ~"~,(1r is weakly uniformly distributed

modulo N.

There are similar asymptotic formulas also for the functions q(~] and ~), which

sharpen the corresponding formulas of Narkiewicz (cf. (I)). Formulas (14) and (15) sharpen

Rankin's results (2) and (3), formula (13) sharpens the result of Scourfield (4), and Corol-

lary 2 generalizes the corresponding result by Fomenko and sharpens formula (5).

3. Proof of Theorem i. Denote by ~ (~) the domain in the complex S-plane: $ = ~ +~,

,Zo,~(Z+l-~l) /> ~ > O, runs through the real numbers, ~,~ and T are positive constants. Let ~($) be a

function defined in the domain ~(&). For ~=~p ~ ) and ~o=o~+ ~--~ we put

i 1%+~m F(s) ~,s 3 (~)-~.~ ~-~T ---r- ~s. The basis of the proof is the following lemma.

LEMMA. Assume that the function ~[5~ satisfies in t h ~ ( ~ the following conditions:

for some constant C a >0

for the complex number Z

(16)

G (s) (s) = (-~:7_~)z,

(17)

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where ~ (S) is analytic in

such that

a) if ~=0~-~,-~,.., ,

(a) and ~(a)~ 0 �9 Then there exists a constant Cs > 0

then

J (~:) = 0 (~" e -C~- -~ ) ,

b) if Z={ , then

i - "-'/ -C9 _ c~{~ ~ + Ok~ e ) 3 (~c)- ~

c) if Z

where p~(~)

tion P (S).

Proof.

tion path in

the path consisting of the two segments

contains two segments ~=• ~ ot

at the point S==L, for which ~eS ~

bounded by 0 (~e -cgV~---~) �9 using (16).

is not a rational integer we have for every

:] ( ~ ) - ( ~ ) ~ - z _ + 0 (~0~),~+~_~ ,

is a polynomial of degree ~ whose coefficients are determined by the func-

Parts a) and b) are proved by the well-known method of transforming the integra-

~ L 4 be (~)- Only part c) requires some clarification. Put ~=2g~-~" Let

~)~6~,~=• b~ the line~J~l~T, ~=~(%)~ L~

and ~$ is a semicircle of radius ~ with center

The integral along ~4 and L~ can be easily

Let ~ be the radius of convergence for the analytic

function ~ iS) at the point S=i If ~ is sufficiently large and ~ sufficiently

small the curve L5 U L~ will be inside this circle. Then one can use Taylor's theorem for

S ~,~UL U and we obtain for all

~-4 k

k=O

where 0~ k are the Taylor coefficients for the function G(S) in the point s = ~, s

I R,, (s }J ~ I s-4".,~,~, ~" Using t h i s , we f i n d t h e n

= LsUh ~

Now we complete the curve L 5 U J,~ to a Hankel curve in the integral under the summation

sign, and then estimate the integrals over the completed curves:

.z ~ ~ lz-O...~-k~,o(~-~-F). I ( ~ - 4 ~. ~,s=(~,~#~_z. r (z)

~%~ L3UL ~

This is substituted into (18) and then the integral containing ~m(S) is estimated, leading

to the result under c), where it is evident that

The lemma is established.

" ~-t)(z.-~), �9 (z-~) f k . P~ (,~): ~ ~K ~ : o F (z)

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Since

"t'CN) the proof of the theorem reduces to the estimation of the sum of the coefficients in the

Dirichlet series

for some character

Here Q~ (S)

%(~).# oI,

. This function can be represented in the form

r'~ (~) :%, Csl@~(s) .~ (s), is analytic and bounded for every g > 0 in the domain ~ ~/~ + E and

(19)

k ) ,

where 5 / (~) and

is bounded and analytic for ~ > 0 and ~iC----~)=~=O , ~ 0 is an integer,

~i (s)= ~ (i-/(E(P~))'~ -~ PSN p~i,s /

Since ~ O~ , it is convenient to group the prime numbers in the product into residue

classes modulo ~. Using the properties of the functions in the class C~ , we obtain

(21)

where ~ (S) is a function which is analytic and bounded in the domain ~ >/~-------T + $ for

> 0 , with ~%(_i) # 0 , and ~(~0) is the Dirichlet ~ -function corresponding to

the character OO �9 We now rewrite the function ~ (5~ in another form which is con-

venient for the application of the lemma:

Formulae (19), (20) and (21) show that the function ~/[~) has a representation

and does not vanish at S~-- | By suitable choice of the constant ~, we can achieve that -~.

~(-~) lies in the half-plane ~ >/~--~+ +~ for sufficiently small ~ . Furthermore, we

shall choose the number ~ in such a way that the ~, -functions I,(~$.o0) do not vanish in

~(J~). This is possible in view of the well-known theorems about the zeros of Dirichlet

I. -functions together with Siegel's theorem (note that the modulus ~[ of the character o~

is fixed). Moreover, in this domain we have the following estimates: if I%1>i~ then

L+t(,~s, ~o) = 0 (.Bi~ C~ + I~!)), ,.,o -#-- ~ o +~.

=

1361

(cf. [2]). All these results about the properties of the functions I,(~S u0) in the domain

(~ and the properties of the functions ~/[5I,~(S),~/~CS) , and therefore also of

~/(S), enables us to apply the lemma to the integral in the formula

This formula follows from Perron's formula for ~----[~] + /-~ If we apply the lemma we ob-

tain the theorem for the given values of ~ �9 The passage to arbitrary values of 05 can

be done using the result of the theorem for values of ~ which differ from natural numbers

by i"

4. Proof of Theorem 2. A proof of this theorem of Narkiewicz for polynomiallike func-

tions can be based on the theorem we just proved for functions of the class C~.

5. Proof of Corollary i. We note that

t Ci(+O.

To obtain formulas (8) and (9) we put ~ =/o for all characters / in formula (6).

6. Proof of Corollary 2. As we already pointed out above, if i(m)=+Zm(~) we have

k ( ~ )=4, Q= [~,N] and

i+{ , if " "6-~(VVt.o(~4) O i , ( - -= �9

One s e e s f r o m t h i s t h a t i f

I~G(N). Therefore, if N

Assume that ~ is odd.

--~-~ "~,(I~) is weakly uniformly distributed modulo N if and only if 2 is a primitive root

modulo N . For odd numbers ~ this is only possible for N =%~, ~ >/~, ~ >/ ~.

If ~ is even one verifies readily that as before condition (7) cannot be satisfied for

any nonprincipal character. Since ~ is generated by the numbers i and 3 if ~$ N, we

therefore have that +Z~(m) is weakly uniformly distributed modulo N if and only if 3 is

a-primitive root of the even number N , i.e., N=~ or N=~ ~, ~ >z5 , ~ >/~ �9 The

asymptotic results follow from Theorem i.

7. Proof of Corollaries 3 and 4. These corollaries follow from Corollary i. To obtain

formula (13) one has to put N =~ and ~=~ in Corollary i. Formulas (14) and (15) are

obtained from (13) by putting N =@ and ~----{ to obtain (14) and ~=~-~ for (15).

8. Proof of Corollary5. Firstof all, ~=N ,o~(~)=~+~ ~ and oL~(1)=%e~N). Conse-

quently, rr~=~. Let ~ be an arbitrary prime factor of N~ ~ >/5 �9 We distinguish two

cases.

a) ~--_--~(~oa~). In this case the congruence ~---0(~) has no solution. In

[~,) we consider the mapping ~(~)=0L+{. Let G~) be the set of quadratic residues

in G (~) The equation T(m)=~ has no solution in ~(c~) , and consequently the set

NQ[N) 4 A is even then ~4=I0;ZIA~[N)=~ = [~5~ ~ , since

is even we have ,~=~, and if it is odd then ~-i.

Condition (7) cannot be satisfied for any character. Therefore,

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{I} UT[G~(~)) and therefore also A I contain at least ~ distinct elements. But

this means that A I =GC~) , i.e., R4 generates G(~)

b) ~(~o&4), i.e., -le~[~) �9 Since T61)=~ we see that T(G~(~)) con-

tains ~ elements. But this set must contain quadratic residues, since otherwise all

elements of G(~) would be quadratic residues, seeing that for every k E~ (~) we have

k=T~(4) and ~E~(~). Now we can apply Sliva's lemma [4]:

SLIVA'S LEMMA. Let ~ be a primitive root modulo ~ and ~ >? If A={~k~ ~k~...,

~k~} contains ~ elements such that at least one k~ is odd then A generates ~ C~)-

Application of the lemma for ~ >T and direct computations for ~=~ show that in

both cases ~ 4 generates G (q')- Now let N=~,$>/~,~>/5

further reasoning follows [4].

congruences

and assume that ~ is a primitive root modulo ~ . The

If Ze~ (~), then g~0 [n~o~). Therefore, the

and

are equivalent.

of

by

k �9

k I , k z .... , k~ �9 Choose natural numbers [L~ such that

(22)

I + ~Z k -- F- 0 ~"~o~ ~) (23)

As we have shown already, congruence (23) has a solution for some odd values

For the same powers the congruence (22) has also a solution. Denote these numbers

The last requirement can be satisfied by changing ~j by multiples of ~--~.

=~

and the powers generate G

As usual, this result can be extended to arbitrary odd modulus by representing ~ ~N)

as direct sum of groups of the form ~C~ ~) �9 since ~4 generates ~(N), the function

~(~ is weakly uniformly distributed modulo N.

In this case

.

2. 3.

4.

5.

6.

7.

LITERATURE CITED

O. M. Fomenko, "Distribution of values of multiplicative functions modulo a prime number," Zap. Nauchn. Sem. LOMI, 93, 218-224 (1980). K. Prachar, Primzahlverteilung, Springer-Verlag, Berlin (1957). W. Narkiewicz, "On distribution of values of multiplicative functions in residue classes," Acta Arithm., 12, No. 3, 269-279 (1967). J. Sliwa, "On distribution of values of o(n) in residue classes," Colloq. Math., 27, 283- 291 (1973). R. A. Rankin, "The divisibility of divisor functions," Proc. Glasgow Math. Assoc., 5, No. i, 35-40 (1961). L. S. Sathe, "On a congruence property of the divisor function," Am. J. Math., 67, 397- 406 (1945). E. J. Scourfield, "On divisibility of r2(n)," Glasgow Math. J., 18, No. i, 109-111 (1977).

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