Lune, J. (Jan) van de

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stichting mathematisch centrum AFDELING ZUIVERE WISKUNDE J. van de LUNE SOME THEOREMS CONCERNING THE NUMBER fHEORETICAL FUNCTIONS w(n) AND ~(n) zw 15/73 ~ MC DECEMBER 2e boerhaavestra_at 49 amsterdam P«n:ted cit .the Mcithema.Uc.ai. Cen,t,c.e, 49, 2e Boe1r.ha.a.ve6.tJta.a;t, Am6.teJr.dam. The Mcithema-t-i.c.ai. Cen,t,c.e, 6ou.nded .the 11-.th 06 FebJr.u.aJty 1946, -l6 a. non-pJr.o6U in6titu.ti..on cum,i.ng cit .the pll.omo.tion 06 pu/l.e mcithema.Uc.6 a.nd Lt6 a.pp..U.c.a.Uotv.. .. I.t -l6 .6pon6oJr.ed by .the Ne.theJr.1.a.nd6 GoveJr.nmen:t .thll.ou.gh .the Ne.theJr.1.a.nd6 0Jr.ga.n,i.za.Uon 6oJr. .the Adva.nc.emen:t 06 Pu/l.e Ruea.Jr.c.h (Z.W.0), by .the Mu.n,i.cipa...u..ty 06 Am6.te1r.dam, by .the Un,i.veMUy 06 Am.ti.teJr.dam, by .the FJr.ee Un,i.veMUy cit Am6.te1r.dam, a.nd by indu..6:tlue6. AMS (MOS) subject classification scheme ( 1970): I OA20, I OHOS Some theon~ms concerning the number theorectical functions w(n) and Q(n) *) J. van de Lune Abstract The functions wand Qare defined as follows: w(I) = Q(I) = 0 and if el e2 er n = p 1 Pz •••Pr is the canonical factorization of the natural number n, then w(n) ==rand Q(n) = e 1 + e 2 + ••• + e. It is known that l (-l)Q(n)= r n~x = o(x), (x-:,-oo). There seems to be no corresponding result in the literature for w(n). In this report it is shown that I< (-I)w(n) = o(x), (x-+oo), Fur-n_x ( ) thermore, it is shown that the series 1:=I (-I)w n /n converges to zero. Finally, the remarkable duality relation = I d/n and some of its consequences are discussed. *) ' . f . . . f bl' . . 1 This paper 1.s not or review; it is meant or pu ication in a JOurna • -·----Introduction. As usual, let w(n) denote the number of distinct prime divisors and Q(n) the total number of prime divisors of the positive integer n. That is, w(l) = Q(l) = 0 and if is the canonical factorization of n, then w(n) =rand Q(n) = e 1+e2+ ••• +er. It is known [1, p. 123], [4, II, p. 617], [5, p. 74] that l (-l)Q(n) = o(x), n~x (x + ~). There seems to be no corresponding result in the literature for w(n). In this report we will prove that S(x) d~f l (-l)w(n) = o(x), n<x We will also establish the convergence of I n=l (-l)w(n) n (x + oo). and show that the sum of this series is zero. For the corresponding result for Q(n) see [4, pp. 617-621]. The above results can be sharpened con-siderably, but we will not take the effort here to do so. Finally we will prove a remarkable duality relation between wand Q and discuss some of its consequences. I. Proposition 1.1. The function 00 ¢(s) d~f l n=l (-I)w(n) s n 2 which is obviously analytic for Res= cr > I, has an analytic continuation up to cr > I. Proof. From the definition of w(n) it follows inm1ediately that for cr > I Thus 00 ¢Cs)= I n=1 s n ~(s)¢(s) = n--P I s p TT (I p I ------s 2s p p • 11(1 --1-) = p s p -I ... ) , (p prime) . Since ps - I has all its zeros on the imaginary axis, it follows that P(s) is analytic for o > ~- Furthennore, it is well-known that ~ls) is analytic for o > I and it follows that I ,~ (s) = ~(s) . P(s) is also analytic for o > I. This completes the proof. D THEOREM I.I. S(x) def I (-l)w(n) = o(x), n,;,X (x • oo). Before proving this theorem we state the following special version of the well known [2, p. 124] WIENER-IKEHARA Tauberian theorem: Let F(x) be non-negative and non-decreasing for x 2:. 0. Let f CX) -sx f(s) = e F(x)dx 0 converge for o > I. If f(s) is analytic for o > 1, except for a simple pole at s = I with residue A, then lim F(x) = A. X x• 00 e Proof of' Theorem 1.1. Note that I + (-I)w(n) ~ 0 for n = 1,2,3, .... Hence F(x) d~f [x] + S(x) is a non-negative, non-decreasing function not exceeding 2x. It follows that for cr > I, 00 (-I) w (n) ;; (s) + Hs) I I + = = s n=l n 00 f00 x-sdF(x) = I F(n) - F(n-1) = = s n=l n 1-0 -s I ~-0 + s I: -s-1 = x F (x) F(x)x dx = Joo Joo -s-1 -s X = s 1 F(x)x dx = s O e ~(e )dx. Thus (cr > I). Recall that ;;(s) is a meromorphic function with only one simple pole at s = I with residue A= I. From Proposition I.I it now follows that the function ;;(s) + cp(s) s s-1 is analytic for cr > I. Now, applying the WIENER-IKEHARA theorem, we obtain A= F(ex) = lim ---'--- = X e lim F(x) = X 3 4 Thus, = lim [x] + S(x) = X I + 1 . S (x) 1.m --. X lim S(x) = 0 or S(x) = o(x) X and this proves the theorem. D THEOREM 1.2 .. The series CX) I n=l (-l) w (n) n converges and its sum is zero. Before proving this theorem we state the following well known result [I, p. 124]. Let CX) g(s) = I n=l a(n) s n be absolutely convergent for a> 0 and suppose that g(s) is analytic for a~ 0. If in addition, a(n) = o(l), then CX) l a(n) n=l converges and its sum is g(O). Proof of Theorem 1.2. Define a(n) (-l) w (n) = -=-~--n Then we have g(s) = ; (-l)w(n) = l s+l n=l n <l>(s+l). Since ~(s) is analytic for cr > 1, we see that g(s) is analytic for cr > 0, and it follows that the series appearing in theorem 1.2 converges. Moreover, g(O) = <1>(1) = lim P(s) = 0 s+l r; (s) and the theorem is proved. 0 Remark. Actually, ~(s) has a zero of order 2 at s = 1 because, lim <l>(s) 2 s+l (s-1) = lim { 1 s+l (s-1) r; (s) = 4 log 2. TT (1 -p>3 TT p>3 I 2) fa 0. (p-1) Note by contrast that the corresponding function I (-l)n(n) = n=l ns r; (2s) r; (s) only has a simple zero at s = 1. 5 6 2. From their definitions it is hardly to be expected that there is much of a relation between w(n) and Q(n). However, we will exhibit below a re-markable duality relation between the two functions and discuss some of its consequences. We first note that [3, p. 355] w(n) = O( log n ) log log n' from which it is easily seen that if Res= a> 1 then 00 I n=l w(n) z s n is an entire function of z. Also, for a> I and lzl < 2°, it is clear that 00 Q(n) I _z __ = TT (1 s p 2 +~+-z-+ s 2s ••• ) = TT ---2 .... p n=l n p p is an analytic function of z. THEOREM 2.1. If lz] < 2° and a> I then 00 (2. I) < I n=l w(n) z ) s n 00 < I n=l (1-z)Q(n) s ) = r;; (s). n f>l>oof. 00 w(n) I z TT (1 + ~ + _z_ + _z_ + ... ) = = s p s 2s 3s n=l n p p p 1 s s = TT ( I + z p ) ;::: TT p + z -p 1 1 p s 1 p -s p s p 1 and 00 (1-z) SI (n) 2 3 I ( 1 1-z (1-z) (1-z) ... ) = TT + -- + + + = n=l s p s 2s 3s n p p p 1-z P.s s = TT ( 1 + ) = TT p p 1-z p s --- p + z - 1 s p Hence 00 w(n) < I z ) n=l ns 00 < I n=l (1-z) SI (n) s ) = n s s nF + z - p 1'; (s) = TT = TT = p s 1 pp s 1 p 1 - _I p - + z -s p and the theorem follows easily. D Performing Dirichlet multiplication in (2.1), equating coefficients and changing z into I - z we obtain: I SI (E.) zw(d) (1-z) d = I n w(-) zSl(d)(l-z) d d/n d In a remarkable duality relation between wand SI. We now study the analytic continuation of 00 g <s) <l~f I z n=l w(n) z ---, (z fixed, cr > I) s n as· a function of s. We first have Proposition 2.1. If 11-zl < 2 and cr > I then = 1 , 7 8 log p (2.2) g' (s) z g (s) z r;'(s) \' = z r;(s) + z(z-1) l ------p (ps+z-I)(ps-1) (where all derivatives are taken with respect to s). Proof. From (2.1) we obtain which can be written as 1-z +-+ s p 2 (1-z) 2s p s g (s) = r;(s) • TT P + z - 1 z p s p Taking logarithmic derivatives, we find g' (s) r;' (s) s z = + I < :e + ••• ) = r; (s) , - t)log p = g (s) z r;(s) p s p + z - I r;' (s) + I - z log p = = r; (s) s p p + z -r;, (s) + I ( - z 1-z log p + = --) r;(s) s s p p + z - p -1 = z • r;, (s) + z (z-1) I lo~ p Z: (s) s s , p (p +z-1) (p -1) which completes the proof. D I 1-z --log p = s pp -1 If I 1-z I < h , then g' (s) is regular for cr > ½. It then follows that gz(s) is z except at s = 1 and at the (possible) zeros of s(s) 1 of the line cr = 2. we get Integrating the formula g' (s) z ---= z g (s) z s'(s) + z(z-l)R (s) s(s) z z g (s) = s (s) exp(P (s)), z z where Pz(s) is analytic for cr > ½· 1 regular for cr > 2 situated at the right Thus g (s) is analytic in the shaded region below, where the p's z stand for the (possible) zeros of s(s) which lie at the right of the line 1 a = 2· 0 9 10 Hence, if (for example) z is irrational and lt-zl ~ ✓2, we find, surprising-ly enough, regardless of which z is chosen subject to the above conditions, that the set of singularities of gz(s) in the halfplane cr >!always con-sists of the same points, namely s = 1 and the zeros of ~(s) lying in the halfplane cr > !· This seems to lend credence to the Riemann-hypothesis. References [ 1 J R. AYOUB, An introduction to the analytic theory of numbers., A.M.S., Providence, Rhode Island, 1963. [2] K. CHANDRASEKHARAN, Introduction to analytic number theory., Springer Verlag, 1968. [3] G.H. HARDY and E.M. WRIGHT, Introduction to the theory of numbers., 4th edition, Oxford, 1968. [4] E. LANDAU, Handhuch der Lehre van der Verteilung der Primzahlen., Chelsea, New York, 1953, [5] K. PRACHAR, Primzahlverteilung., Springer Verlag, 1957. 1 1 

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