FUNCTIONAL EQUATIONS WITH PRIME ROOTS FROM ARITHMETIC EXPRESSIONS FOR. BARRY BRENT Efmhurst, New York 11373

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1 FUNCTIONAL EQUATIONS WITH PRIME ROOTS FROM ARITHMETIC EXPRESSIONS FOR BARRY BRENT Efmhurst, New York 373. I this article, a geeralized form of Euler f s law cocerig the sigma fuctio will be obtaied ad used to derive expressios for ^ T% which cotai just fuctios ivolvig additio ad multiplicatio. These will be substituted i the equatios () < ^() - a - = 0 to obtai equatios with classes of solutios idetical with the class of prime umbers. 2. Let F < > = S f ( d ) d Propositio. If D Ft) x =l coverges o some iterval about 0, the (2) 0 = R() + V F(a)E( - a) a=l where _ ^ f()/ (3) 2 ^ E()x = II ( - x ) =0 = The proof mimics Euler's for the case f = idetity, which is the recursive expressio for sum of divisors he obtaied by describig E. [l] 99

2 200 FUNCTIONAL EQUATIONS WITH PRIME ROOTS [April Proof. f ( ) J^ f()x /(l - x ) = 2] =l k x k f(l)x + f(l)x 2 + f(l)x 3 + f(l)x 4 + f (l)x 5 + f(l)x f (2)x 2 + f (2)x* + f (2)x f (3)x 3 + f (3)x f (4) x f(5)x f (6)x OO OO = X > f ( d ) = F()x. =l d =l That is, (4) J ] f()x /(l - x ) = Yl FWx. =l =l Suppose f()/ (5) o < ( i - x ) < «> o some iterval about 0. We show that (2) holds uder (5) ad the that (5) holds whe =l coverges o some iterval about 0. Let (5) hold. We have the idetity: 00 f()/ log ( - x ) = J2 f()/log ( - x ) =l x Differetiatig, ad substitutig from (3) as (5) permits:

3 974] FROM ARITHMETIC EXPRESSIONS FOR E-WVtt-^) =_L sf? (i - - f()/ TCJ75T ( - x ) ^E»tm)^)/EBto)^ 2 mrdjx " / ^ R(m)x m. 0 0 Hece, by (4), ( 6 ) m " 2 R ( m > x m / R ( m ) x m = 2 «) x / ( - x ) = J^ F()x 0 0 ad Eq. (6) gives: = I 2 F()xJ J{ 2 R ( m ) x m J + 2 ^w^ So, for each > 0, the coefficiet x is 0: 0 = 2 F ( a > R ( - a ) + R < ) a=l It remais to show that (5) holds whe OO 2 F() * =l coverges o some iterval about 0. By Eq. (6), E F()x = -xd/dx log P(x)

4 202 FUNCTIONAL EQUATIONS WITH PRIME ROOTS [April where Therefore, (7) Hece P(x) = 0 iff iff f()/ P(x) = (l. x ) P(x) = exp OO f - E FdDx " J -, I E F()x ~\lx = <*> * x dx. F()x ad P(x) =oo iff E (l/)f()x = -oo. Thus (5) holds iff E (l/)f()x o some iterval about 0, ad this is the case whe E F()x < oo o the same iterval. Q.E.D. Now it is ecessary to show that the coditios of Propositio apply to ^ we show a little more. Propositio 2. Let. Actually, The, E d f(d) = F() E F ( ) x < -

5 974] FROM ARITHMETIC EXPRESSIONS FOR 203 o some iterval about 0 if ad oly if E f ( ) x < - o some iterval about 0. Proof. 5 > ( ) x < - -» ^ f ( ) x / l - x < - - * ^ f ( ) x < by (4) ad compariso. For the other directio, let I E fwx < By the root test, lim sup f() < That is, sup L. <, where o some sequece {a., }. Defie {c, } by: L. = lim f(a ) k a ik f(c ) = max f(d) for a sequece {a, }. For each k, c, is oe of the divisors of a,. The f lim f(c, ) < sup L < oo, ad over all sequeces {a, } the {c, } are bouded by: That is, sup lim f(c. ) < sup L. < k { a k } ' sup lim max f(d) dla, < sup L..

6 204 Now FUNCTIONAL EQUATIONS WITH PRIME ROOTS [April max d a, f(d) max f(d) dla, So: sup Hm { a k } max I f (d) J d! \ ^ sup L. < That i s, Now, we demostrate below that lim sup max f(d) < sup L. <. d * X > ( ) x < o some iterval about 0, where T is the umber-of-divisors fuctio. The demostratio below is valid but clearly circuitous. Thus, by the root test, ad lim sup T() < Thus, lim sup ~T() max f ( d ) ] = lim sup r ( ) [ m a x f(d) l L 4 J Ld J < lim sup T() lim sup [ m a x f ( d ) Ld J < OO o some iterval about 0. The, 2 > T ( ) max f(d) < d q. e.d. 5 > ( ) x < j ^ F() x < E E f(d) x dj < E T W max f(d) x < o di We repair the gap i the proof of Propositio 2, the assertio without demostratio that E r ( ) x coverges o some iterval about 0, by comparig this sum with aother. obvious o comparig T() with the idetity fuctio: The result is

7 974] FROM ARITHMETIC EXPRESSIONS FOR IE** < o (-,). Oe more propositio is eeded to fiish the backgroud for a demostratio that Propositio applies to 9/. Propositio 3: J ] l / F()x coverges o some iterval about 0 iff ^ F ( ) x coverges o some iterval about 0. Proof. Uder the hypothesis that X ; i / F ( ) x coverges we have by the root test: That is, lim sup F ( ) ( l / ) <». a, sup lim F ( a k ) ( l / a k ) < <*>.. Now, clearly whe F ( a k ) ( l / a k ) k coverges, its limit is - a, K lim F ( a k ) Also, it is clear that coverges if ad oly if a. k F ( a k ) too 5 coverges e So F ( a k ) ( l / a k ) ) a k a, a, lim sup j F() = sup lim F(a fe ) = sup lim F (a- M l / a, ) = lim sup F()(l/)

8 206 FUNCTIONAL EQUATIONS WITH PRIME ROOTS [April So X>() x coverges o some iterval about 0. The other directio is similar, or by compariso, q. e.d. Now we prove that Propositio may be applied to C Z^. Propositio 4. coverges o some iterval about 0. Proof, y^x coverges o [0,). Apply Propositio 3 iductively: for each a, E a x coverges o some iterval. The., by Propositio 2, E CZP i () f \ x a coverges, q. e. d. Propositio! ow yields a recursive relatio o ^ i terms of the coex. efficiets of the power series for P(x) with f() =. P(x) is a ifiite product ad, i order to determie a expressio for ^ which is recursive i additio ad multiplicatio, we express the coefficiets of the power series for P(x) as the coefficiets of the expasio of a fiite product. Propositio 5. 0 = R() + ^ ( a ) R ( - a), a=l where R(k) = coefficiet of x i k (i - x ) =l Let (Defiitio). Proof. Applyig Propositio, to The a-l (i - x ) = SW* =l =0 k *' (i - x ) = E R k ()x k+, - Lk+lJ 5 3 \ + ( ) x = ( - x ) = ( - x K + ) x R k ()x =l E ] 7 r=0 \ [ k + r l ] a _ ) ( - l ) r x ( k + ) r x E \ ( ) x r /

9 974] FROM ARITHMETIC EXPRESSIONS FOR <& 207 = E R k ()x* + [ k + ^ _ ( f k + I]"" ) (-D'x^'x E H k ()x r=l > ' = E W)** Noe of the terms i the secod summad have expoets ^k. Thus R k (D = R k + (D for all i ^ k Ideed, R* k (i) = R^i) for all i ad such that i < k <. Thus k E s ( ) x = lim ( - x ) = lim V R, ()x = T lim R, ()x = E B Wx, k l k V k R k ad R: () = S(), q.e.d. It is ow possible to defie a fuctio, which turs out to be ^, which is expressible i terms of just additio ad multiplicatio, ad which leads to the equatio metioed i the title. Defie F () = ad, supposig F defied o, 2, o e o, - l f let F () satisfy 0 = R() + ] P F (a)r( - a), a=l where R is defied as i the statemet of Propositio 5. The, by Propositio 5, F = ^ 9 a ad F satisfies 0 = F () - - just whe is a prime umber. REFERENCE, Euler, Opera Omia, Series, Vol. 2, pp , Discovery of a Most Extraordiary Law of the Numbers Cocerig the Sum of Their Divisors^"