Mellin Transforms And Fourier-ramanujan Expansions

Mathematische Zeitschrift

Math. Z. 193, 515-526 (1986)

9 Springer-Verlag 1986

Mellin Transforms and Fourier-Ramanujan Expansions Dieter Klusch Tanneck 7, D-2370 Rendsburg, Federal Republic of Germany

1. Introduction

Ramanujan's trigonometrical sums cq(n)=

l<=h<~qZ

e-2~inh/q=

d~[nd#(d)

(1.1)

(h,q) ~ 1

(q, n ~]N; # (.) is M6bius' ~-function) are periodic arithmetical functions satisfying the orthogonal relations M (cq cp) = ,fop(q)' q=p (1.2) ~0, q*p' where (p(.) is Euler-s (p-function and (if the limit exists) M ( f ) = lira N - ~ ~ f(n) N.-+m

n<=N

denotes the mean-value of an arithmetical function f : N~IE. Hence in analogy to the Fourier theory of real functions one expects Ramanujan-expansions f ( n ) ~ Y', aq(f)Cq(n)

(1.3)

q>l

for arithmetical functions f with the Fourier-coefficients 1 a, (f) = ~ ( ~ M ( f Cq)

(1.4)

(if the limits exist). Point-wise convergent expansions of the form f(n) = f, aq cq(n) for some well-known number-theoretic functions f were first given o>1

by S. Ramanujan ([16], pp. 259-76) and G.H. Hardy ([8], pp. 263-71), where in general the aq do not coincide with the Fourier-coefficients (1.4). General criterions on the existence of (i.3) with (1.4) (including the cases when f is multiplicative or additive) are due to e.g. Wintrier [22], Delsarte [7], Delange [6], Schwarz [18], Schwarz-Spilker [19], Tuttas [20] and Hildebrand [9].

516

D. Klusch

In the present paper we do not refer to these results. We here investigate some relations between the theory of Fourier transforms and Fourier-Ramanujan expansions of arithmetical functions f~(n) = ~ d 1 -~ w(d) din

(neN, R e e > 0),

(1.5)

where w (x)= M - 1 {F(s)} (x e N +, Re s > 1) is the inverse Mellin transform of F (s) belonging to L ( - o% + oo). By means of S. Ramanujan Fourier expansion [16] a l - , ( n ) = ~(s) ~ q-* %(n)

(Re s > 1),

(1.6)

q_>_l

where a~(n)= ~ d ~ (heN, sell2) and {(s) is Riemann's zeta-function we prove that din f~(n) has an absolutely convergent Fourier-Ramanujan expansion (1.3) with (1.4) given by an integral of Mellin's type taken along the vertical line c = Re s 1 q~' aq (s = ~7~ i ~ F (s) ~ (s + ~) q -S d s. (c)

(1.7)

Conversely we establish a representation of aq(s by a modified Ramanujanseries. By suitable special choice of F(s) in (1.7) the aq(f=) are easily computed by means of known Mellin transforms. Thus in the applications of our general results we treat the logarithmic derivate ~9 of Euler's F-function, some hyperbolic functions and the logarithms of Jacobi's elliptic theta-functions. In addition we give some point-wise convergent expansions using Ramanujan's outstanding identities ([16] ; [8], p. 263) 0~- ~, q-1 cq(n) (1.8) q>l

and - d ( n ) = Y, q- 1 logq %(n)

(1.9)

q>l

(d(n)=~ro(n)) , which are "equivalent" to the prime number theorem ([,14], pp. 568-9) and which do not fit in the popular theory of Ramanujan expansions ([-13], p. 216; [11], pp. 32-6).

2. Theorems

Denote by Ks the class of all arithmetical functions L(n) = Z d I ~' w(d), aln

(2.1)

where heN, Re a > 0 and w (x) is real-valued and piece-wise continuously differentiable on N +,

(2.2)

Mellin Transformsand Fourier-RamanujanExpansions

517

GO

F(s) = ~ x s- 1 W(X) dx absolutely convergent in the strip 0

(2.3)

51
F(s)~L(--oo, +oe),

~ ]F(a+it)[dt
i.e.

(c51
(2.4)

-oo

For the class K~ we prove Theorem 1. Let f ~ K ~ . Then the Fourier-coefficients aq(f~) exist and are given by

where (c) denotes the vertical line ( c - i ov, c+iov), c rel="nofollow"> 1. For n ~ N the FourierRamanujan series ~ aq(f~) G(n) converges absolutely to f~(n). q>l

Corollary 1.1. Let f.EK~. Define (2.6)

aln Then n 1-~ w(n)= Z aq(f~) bq(n).

(2.7)

q>l

The Ramanujan-series in (2.7) is absolutely convergent. For the Fourier-coefficients aq (f,) we prove Theorem 2. Let f ~ K ~ . Define by (2.5) g~ (n) = Z d a~ (L).

(2.8)

din

Then g~(n) ---- Z q-~ w (q) eq (n)

(2.9)

q>=l

with eq(n) = ~ d (m) Cq/m(n).

(2.10)

mlq

The Ramanujan-series in (2.9) is absolutely convergent.

Corollary 2.1. Let f~eK~. Define by (2.10) hq (n)= Z # (d) eq (d)"

(2.11)

drn

Then n a, (f~) = Z q- ~ w (q) hq (n). q>l

The Ramanujan-series in (2.12) is absolutely convergent.

(2.12)

518

D. Klusch

3. Proofs To prove Theorem 1 we first show that for every n e N the series with a o ( c ~ ) = ~ / ~ V(s) ~(s+o0 q-~-~ds

aq(~)Cq(n) q~l

(c> 1)

(c)

is absolutely convergent and equal to fi,(n). We consider a fixed n. By (2.2) and (2.3) Mellin's inversion theorem ([-5], p. 88) furnishes that w(x) is the inverse transform of F(s). Thus 1

w ( x ) = M - l { F ( s ) } = ~ i ~,F(s) x-~ds

(c> 1, xMR+).

(3.1)

(c)

Hence for Re e > 0 f~(n)=~/

~ F(s) al_,_,(n)ds.

(3.2)

(c)

By (1.6) we have for R e ( s + e ) > 1 o'l-~-~(n)=~(s+~) ~ q-~-~ cq(n).

(3.3)

q>=l

Thus we get from (3.2) 1

f~(n)=~/

y F(s) ~(s+7) ~ q-~-~ co(n) ds. (c)

(3.4)

q >_ t

s=(r+it, ~>1. By (1.1) we have ]Co(n)l<(rt(n) and since ~(s) is bounded on any fixed line (c), c > 1 we get by (2.4)

L e t ~---~a q-i~2, ~ 1 ~ 0 and

-boo

IF(c +it)] [((c +cq +i(t +ce2))[ Z q-C-~, [co(n)] dt < ~ -oo

(3.5)

q> l

and

aq(e)=O(q -c)

(q--*~, c> 1).

(3.6)

Hence by Lebesgue's dominated convergence theorem it is permissible to invert the order of summation and integration in (3.4). Finally we prove that aq(o0 coincides with the Fourier-coefficient aq(fi,) defined by (1.4). For N e N we have by (3.4) on the line (c), c > 1

In view of (1.2), (2.4) and by Lebesgue's dominated convergence theorem we get lim N -t ~', f~(n) cq(n)= q)(q~) f F(s) ((s+~) q-S-~ ds. N~oo ,,<~N 2zci (~)

(3.7)

Mellin Transforms and Fourier-Ramanujan Expansions

519

1

Hence aq(e)=aq(f~) with a l ( s F(s)((s+e)ds and Theorem 1 is proved. (r Corollary 1.1 is a simple application of M6bius' inversion formula. By (2.1) we get the inversion

and (2.7) follows immediately from Theorem 1. We now prove Theorem 2. By (2.5) we have

q aq(f~)= ~---~i(! F(s) ((s +~) q~-~-~ ds

(3.8)

Hence by (2.8) and (1.6) - 1 g~n()-2~i~i~!) F(s) (Z(s+e)q>=tZq-~-~ cq(n) ds. But

d(q) q-S

(2(s)= ~

(3.9)

(Re s > 1).

(3.10)

d(q) q-S-~. Z q-~-~ cq(n) ds.

(3.11)

q>l

Thus (3.9) becomes 1

j" F(s) Z

g~(n)=~/

(c)

q> l

q> l

Since for c > 1

~, q -c

d (q) < ~ ,

q>l

~ q-C [cq (n)] < c~. q>l

Dirichlet's multiplication rule ([15], p. 375) and (2.10) yield n)

1

where the last series again converges absolutely for Re(s + c0 > 1. Hence by (2.4), Lebesgue's dominated convergence theorem and (3.1) we get (2.9) with (2.10). Thus Theorem 2 is proved. Corollary 2.1 follows from (2.8) and (2.9) using M6bius' inversion formula. 4. Applications

4.1. We here treat some examples from class Ko. By (2.1) and Theorems 1 and 2 fo(n)= • d[n

dw(d)= Z aq(fo) Cq(n)

(4.1.1)

q> l

and go(n) = Z din

dad(fo)= Z w(q) eq(n) q>=l

(4.1.2)

D. Klusch

520

with 1_ aq(fo) = 2 ~ (!I F(s) ~(s) q-~ ds

(c> 1).

(4.1.3)

(a) Let F(s)=~ cosec(rcs). Then we have the Mellin transform ([3], p. 345) 1 x(x+l)

w(x)=

M-l{rcc~

(l
Since Icosec (7c(~-+ i t)) I= sech (n t) condition (2.4) holds. Hence fo (n) = - Z (d+ 1)-' sKo din

and by (4.1.3)

aq(fo): ~--~ (f rcc~

~(s) q-S ds

(c :-~),

which is ([3], p. 355)

a,(fo) = - ~

1) _~_~},

{0(1 + q-

r'(s)

where ~ is Euler's constant and 0 (s) = F ~

(Re s > 0) is the logarithmic derivative

of Euler's F-function satisfying the functional equation ([2], p. 16) O (s + 1) = O (s) + s - 2.

(4.1.4)

Now (4.1.1), (4.1.2) and (4.1.4) yield the expansions Z ( d + 1) -~ = Z q-2{0( 1 + q - 1 ) + 7 } cq(n), din

(4.1.5)

q>- I

d(n)+ Ta_l(n)+ ~ d -1 ~p(d-a)= ~ din

q>~l

1

q(q+ 1)

eq(n).

(4.1.6)

If we use Ramanujan's identities (1.8) and (1.6) (with s-= 2) and the functional equation (4.1.4) then (4.1.5) reduces to the point-wise convergent expansion 6

rc2 7a-,(n)+~, ( d + l ) - 1 = ~ q-2 0 ( q - l ) Cq(n). din

(4.1.7)

q>__l

(b) Now let Re a > 0 , Re s > l and F(s)=sF(s)(2a) -~-~. We then have the Mellin transform ([3], p. 312) w(x)= xe- 2~

M - ~ { s r ( s ) ( 2 a ) - ~ - 1}.

Since for a > 0 and It [--*oo

IF (a + it)[=e-~l t [ l tl~-~ / ~ condition (2.4) holds.

{ l + O(t- ')}

(4.1.8)

Mellin Transforms and Fourier-Ramanujan Expansions

521

Hence fo(n)= ~ d 2 e-2"a~Ko din

and by (4.1.3) aq (f0) = g ~1

~sF(s)~(s)(2a)_S_lq_Sd s

(c> 1),

which is ([3], p. 323) aq (fo) = 88q csch2 (a q). Now (4.1.1) and (4.1.2) yield the expansions. d2 e - 2 a d - = l d[n

2 q csch2(aq) cq(n),

(4.1.9)

q>- i

88~ d2 cschZ(ad)= ~ qe -2aq e~(n). din

(4.1.10)

q>=l

Similarly one can prove

2 dZ e-2a'f{ 1-4e-2"a} =882 q sechZ(aq) cq(n), din

(4.1.11)

q> l

88 d2 sech2(ad) = E qe-2"~{ 1-4e-2aq} eq(n). din

(4.1.12)

q>_l

These identities fit in the class of similar "Ramanujan-formulae" like 7Z3

( _ 1)q q- 3 csch (n q) = and

1 csch2~q=6

1 2~'

360

1 1 r4(88 ~ s e c h 2 r c q = - 2 - + 2 ~ --t 162c3

q=>l

q>=l

or

zc co(z)+co(z-l)=~-;

- 1)q-x sech {(2q- 1)~zz} (Re -c> 0), co(z)-- Z ( 2q-~-iq>-i

due to S. Ramanujan ([17], Chapt. 14), Kiyek and Schmidt ([10]) and the author ([-12]), respectively. For further results of this kind see also B.C. Berndt ([4]). 4.2. We here establish the inverse Mellin transforms of the logarithms of Jacobi's theta-functions. Let H={z~ffr Then Dedekind's eta-function is defined for zeH by q(z)--q i~ I-[ (1 _q2,), q=e,~i~. (4.2.1) n>=l

The Jacobi theta-functions Oi(zJO)=O~(z)(i=2, 3, 4) of zero argument, -c~H are given by ([21], pp. 469-70) 02 (z)= 2 q+ I~ (1 -q2")(1 + q2,)2 n>l

(4.2.2)

522

D. K l u s c h

qZ.- 1)2

(4.2.3)

04(~) = I~ (1 -- q2n)(1 -- q2,- 1)2.

(4.2.4)

0a (z) = ]--1(1 -- q2")(1 + n>_l

n>l

Taking logarithms we get by (4.2.1) ~i

z+l

logO2(~)=log2+-~+51ogrl(z)-21ogrl(T)--21ogrl(2) log 03 (r) = 5 log t/(z)- 2 log q (2) - 2 log t] (2 z) log Oa (z) = ~ + 5 log t/(r)-- 2 log q gig

where log r/(z) = ~ - +

-- 2 log q (2 z),

(4.2.5) (4.2.6) (4.2.7)

o(1)(z--+ioo).

N o w take v=ix, xslR +. Set, for brevity, q(ix)=q(x) and Oi(ix)=ffi(x ) (i = 2, 3, 4). Consider the Hurwitz zeta-function defined by the series (s, a) = Y', (n + a)- ~, n>O

0 < a < 1, Re s > 1 and its Mellin transform ([3], p. 355) exp {(1 -a)x} exp (x) - 1

-M-l{r(s)r

(Re s > 1, xelR+).

(4.2.8)

Define

(a(s)= r(s) ~(s)(2~) -~

(Re s > 1).

Then by (4.2.8) (4.2.9)

w 1 (x) = - 1 {coth (n x) - 1} = M - 1 { _ q~(s)} w2(x) = 89{coth(Tr x ) - 1} - {coth(2 r c x ) - 1} =

M -1

{(1 --21 -~) ~b(s)}

(4.2.10)

Wa(X) = 89{1--5 eoth(rcx)} + coth ( ~ ) + coth(2rcx) = M-1

{ _ (1 - 21 -*)(1 -

21 +*) q5(s)}

(4.2.11)

w4(x)= 89

~+1)q~(s)}.

(4.2.12)

By (4.2.1) we have the Lambert series log0(x)=

7~X

12

~ n-l(e2 . . . .

1)-1"

n>l

Thus by (4.2.9) log(e ~x/12 q(x)) = M -1 {-- qS(s) ~(s+ 1)}

(4.2.13)

Mellin Transforms and Fourier-Ramanujan Expansions

523

and by (4.2.5)-(4.2.7) we get from (4.2.13) log (1 e . x/4 if2 (x)) = M -1 {(1 - 21 - ~) ~b(s) ( (s + 1)}

(4.2.14)

log ga (x) = M -1 { - ( 1 -- 21-s)(1 --2 *+1) q~(s) ~(s+ 1)}

(4.2.15)

log g,~(x) = M -1 { ( 1 - 2 *+1) ~b(s) ~(s+ 1)},

(4.2.16)

observing Lebesgue's dominated convergence theorem and (4.1.8). 4.3. We turn to some examples from the class K~. Now by (2.1) and Theorems 1 and 2 A ( n ) = Z w(d)= Z aq(fl) c,(n) din

(4.3.1)

q~l

and gl(n)= ~ da~(A)= Z q- ~ w(q) e,(n) din

(4.3.2)

q>=l

with

qa.(fl)=~--~z (~ F(s);(s+llq-~ds

(c> 1).

(4.3.3)

Define by (4.2.9)-(4.2.12) the arithmetical functions

f(~a)(n)= ~ Wk(d) ( n e N ; k = 1, 2, 3, 4), a[n

which are obviously elements of the class Ka, since by (4.1.8) +oo

y IF(s) ((s)(2rc)-sl dt
(ges>l)

--o0

and condition (2.4) holds. We first consider f~l)(n). Then by (4.3.3)

qa qkdl ~4"(lh-1~ J-2rci (c)Jf --F(s) ((s) ((s+ 1)(27cq)-Sds which is by (4.2.13)

q aq(fl(1)) = log {e€

(c>1),

s

Hence (4.3.1) and (4.3.2) yield the expansions fl(1)(n) = Z q-l log{ e'~q/12q(q)} cq(n)

(4.3.4)

q>l

and ~2 al (n) + • log fl(d) = Z q-1 wl (q) eq(n). din

(4.3.5)

q>=l

Similarly we get from (4.2.10)-(4.2.12) and (4.2.14)-(4.2.16) the expansions fl(Z)(n) = ~. q-~ log{ le~q/4 ~2(q)} cq(n) q_>-i

(4.3.6)

524

D. Klusch fa(3)(n) = Z q - i log ~3(q)

cq(n)

(4.3.7)

cq(n)

(4.3.8)

q>l

f~4)(n)= y, q-1 log g,,(q) q>l

and

7Z

-d(n)log2+~al(n)+ ~,logff2(d)= ~ q-1 w2(q)eq(n) din

~ q-1 wa(q) eq(n)

~, logffa(d)= din

(4.3.9)

q> l

(4.3.10)

q>--_l

~ q-1 w4(q) eq(n).

log ff,~(d)= din

(4.3.11)

q>_l

Formula (4.3.7) has been first given by the author ([11], p. 30). By means of (1.8) we can eliminate the factor 89in the Fourier-coefficient of (4.3.6) and the resulting expansion f(2)(n ) = ~ q-1 log{e~q/4 ff2(q)} Cq(n)

(4.3.12)

q~l

is point-wise convergent. 4.4. We now treat some further examples which follow directly from the previous results. (a) Since the first theta-function 01 (z Iz) has a zero at z = 0 we consider Jacobi's relation (E21], pp. 470-2) for

a01(zlz) z=o=0i(z) ~

G (~)= 02(~).G(~).o~(~) ( ~ ) . Define O'l(ix)=ff'l(x ) and by (4.2.9) (4.3.6)-(4.3.11) yield the expansions

ws(x)=3w~(x)=-3{coth~x-1}. Then

f~5)(n) = ~, ws (d) = ~, q-1 log {89e~q/4 g'~(q)} c, (n) d[n

(4.4.1)

q>=l

and

-log2.d(n)+~al(n)+ ~logff'l(d)= ~ q-~ ws(q) eq(n). din

(4.4.2)

q>=l

Again using (1.8) in (4.4.1) we get the point-wise convergent expansion f~5)(n)=

~ q- i log(e~q/4 ff'l(q)} cq(n).

(4.4.3)

q>-i

(b) The discriminant d(z)(zeH) is defined in terms of the invariants g2, ga of the WeierstraB p-function ([17, p. 14) A(z) = g~ (z) -- 27 g3 (z). The connection with Dedekind's eta-function and Klein's modular function is given by ([-17, pp. 21, 51) A(z) = (2 u) i 2 qz4 (z) ----g~ (z) {J(z)} -1

J(z)

Mellin Transforms and Fourier-Ramanujan Expansions

525

Define A(ix)= i(x) (xGR +) and by (4.2.9) w6(x)=24wl (x)= - 1 2 { c o t h z c x - 1}. Then by (4.3.4) and (4.3.5) we get the expansions f~6)(n) = Z w6(d)= ~, q-~ log {(2zc)- ~2 eZra i(q)} din q>__1

ca(n)

(4,4.4)

a~ad - 1 2 d ( n ) l o g 2 7 : +2rc
~', q-1 w6(q)eq(n).

(4.4.5)

q> l

By means of (1.8) we can eliminate the factor (27:)-12 in the Fourier-coefficient of (4.4.4) and the resulting expansion fl(6)(n)= ~

q-l log{e 2~q i(q)} c~(n)

(4.4.6)

q=>l is point-wise convergent. (c) Finally we mention a connection between Ramanujan' point-wise convergent expansion (1.9) for d(n) and transformation formulae for Jacobi's thetafunctions, Dedekind's eta-function and the discriminant. The behaviour of Oi(z), q(z) and A(z) under the generator S z = - z -1, zGH of the modular group is given by ([-21], pp. 475-6; 1-1], pp. 48-50)

Oa(z)=(--iz) -890 a ( - - z - : ) A(z)=(--iz) -12 A ( - - z - 1).

02 (T) = ( - - ii7)--8904(--7--1);

tT(z)=(--iz) -~ ~/(--z- a);

Hence for : = ix(xe]R +) we get by (1.9), (4.3.4), (4.3.7) and (4.4.6) the point-wise convergent expansions

89

- Z q-l log{ :q/12 0(q-l)}

(4.4.7)

q>l

89

= f : 3 ) ( n ) - ~ lo8

q>l

12d(n)=f~6)(n) - Z

ffa (q- 1) cq(n),

log{ ez"~ if(q-')} c&).

(4.4.8)

(4.4.9)

q___l Similar results hold for the other theta-functions. The expansion (4.4.8) has been first proved in f i l l ] , p. 32).

References 1. Apostol, T.M,: Modular functionsand Dirichlet series in number theory. New York-HeidelbergBerlin: Springer 1976 2. Batemann, H., Erd61yi,A.: Higher transcendental functionsVol. I. New York-Toronto-London: McGraw-Hill 1953 3. Batemann, H., Erd61yi,A.: Tables of integral transforms. Vol. I. New York-Toronto-London: McGraw-Hill 1954 4. Berndt, B.C.: AnalyticEisensteinseries, theta-functionsand series relations in the spirit of Ramannjan. J. Reine Angew. Math. 303[304, 332-365 (1980) 5. Courant, R., Hilbert, D.: Methoden der Mathematischen Physik I. Berlin-Heidelberg-NewYork: Springer 1968 6. Delange, H.: On Ramanujan expansions of certain arithmetical functions. Acta Arith. XXXI, 259-270 (1976)

526

D. Klusch

7. Delsarte, M.J.: Essai sur l'application de la th6orie des fonctions presque p6riodiques a 'larithm6tique. Ann. Sci. Ec. Norm. Super. (3) 62, 185-204 (1945) 8. Hardy, G.H. : Note on Ramanujan's trigonometrical sum ca(n ) and certain series of arithmetical functions. Proc. Cambr. Phil. Soc. 20, 263-271 (1920/21) 9. Hildebrand, A.: Dber die punktweise Konvergenz von Ramanujan-Entwicklungen zahlentheoretischer Funktionen. Acta Arith. XLIV, 109-140 (1984) 10. Kiyek, K., Schmidt, H.: Auswertung einiger spezieller Reihen aus dem Bereich der elliptischen Funktionen. Arch. Math. Vol. XVIII, 438-43 (1967) 11. Klusch, D.: Funktionalgleichungen fiir die Riemann'sche, die Hurwitz'sche und die LipschitzLerch'sche Zetafunktion. Dissertation, I. Math. Inst. d. Freien Universit~it Berlin 1978 12. Klusch, D.: On the approximation of analytic functions in a strip. Math. Proc. Cambr. Phil. Soc. 97, 381-384 (1985) 13. Knopfmacher, J.: Abstract analytic number theory. North Holland Libr. Vol. 12, 1975 14. Landau, E.: Handbuch v o n d e r Lehre der Verteilung der Primzahlen II. Leipzig-Berlin: B.G. Teubner 1909 15. Prachar, K.: Primzahlverteilung. Berlin-G6ttingen-Heidelberg: Springer 1957 16. Ramanujan, S. : On certain trigonometrical sums and their applications in number theory. Trans. Cambr. Phil. Soc. 22, 259-276 (1918) 17. Ramanujan, S.: Notebooks vol. II. Bombay 1957 18. Schwarz, W.: Uber die Ramanujan-Entwicklung multiplikativer Funktionen. Acta Arith. 27, 26979 (1975) 19. Schwarz, W., Spilker, J.: Mean Values and Ramanujan-Expansions of Almost Even Functions. Coll. Math. Soc. J/tnos Bolyai, Debrecen 1974, S. 315-357, 1976 20. Tuttas, F.: l~Iber die Entwicklung multiplikativer Funktionen nach Ramanujan-Summen. Acta Arith. 36, 257-270 (1980) 21. Whittaker, E.T., Watson, G.N.: A course of moder analysis. Cambr. University Press 4th edn. 1927 22. Wintner, A.: Eratosthenian averages. Boston Waverly Press 1943

Received July 2, 1985