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TRANSACTIONSOF THE AMERICANMATHEMATICALSOCIETY Volume 213. 1975 BINARYDIGIT DISTRIBUTIONOVER NATURALLY DEFINED SEQUENCES BY D. J.NEWMAN AND MORTONSLATER ABSTRACT. In a previous paper the first author showed that multiples of 3 prefer to have an even number of ones in their binary digit expansion. In this paper it is shown that in some general classes of naturally defined sequences, the probability that a member of a particular sequence has an even number of ones in its binary expansion is A. I. Binary digits contained proved that in counting in arithmetic progressions. In [4] Newman the number of ones in the binary expansion multiple of 3 that count will come out even more often than odd. striking result in that superfically distinguishable sequence about multiples would be expected there seems nothing either of 3 in the binary system. to oscillate in "odds" of a This is a special Certainly or. "evens" or a random being more numerable. In the case of multiples of 3, the probability have an even number of digits. odds is o(n), in fact about A natural question which exhibit proportion That is, the preponderance here would be: Are there other natural and further is there a natural of evens over odds or vice versa of the number of elements in the sequence? preponderance of evens sequence does limN_#0O(E(N)/i4(N)) = 0 ? In this paper classes of natural Theorem. ing function of evens over nlos3/1°«4. this odd behavior which the preponderance is l/2 that the multiples isa sequences sequence positive If E(N) gives over odds and A(N) is the counting in function the of the we shall prove that the above limit is zero over various sequences. Let A be the sequence of this sequence. {M • n\°°_x. Let A^(N) be the count- Then F ( V) lim -r^j-r-0 /V-»ooA.AN) and \EM(N) - %A iN)\ < C/Vl0«3/lo8 4. Received by the editors June 4, 1974. AMS(MOS)subject classifications (1970). Primary 10L10, 10H20, 10H30. Key words and phrases. abundant Binary digits, arithmetic progressions, sieving, numbers. Copyright © 1975, American Mathematical Society 71 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 72 D. J. NEWMANAND M. SLATER Proof. Let 0(72) be the number of ones in the binary expansion for n. Let y,v)= Z (-i)D<"-M'->. 0<y<N/M Thus we want to prove that when N is a multiple of M, |SM(iV)| < CNln3,/ln4. As in [4] it follows that, if N = 2Kl + 2K2 + • • • + 2Ki, Kx > K2 > - -. > K. > 0, then SM{N)= S{2Kl-Pl)where S{2K2 - p2) + ... + (- 1)/- lS{2Ki - p ) + (-1)' p¿ is the small positive integer so that p¿ + 2 '+l + 2 «*2 + ••• + 2 J is divisible by M. If it can be shown that S(2K'' - p.) < C2K'(ln3/I,l4) - j y then the result yn3/ln4 V'V> < Ci Z 2K,0nVln 4) < Ci Z 2Kí = CiNla3/'n " ¿=l i=l follows. «2*-,) This = /,, JLLu .„ t.'»-^» m. ok-b \x\='A (j _ xM)x2k-P r can be evaluated , w M .. ,. _ Z . . by the method . £J • an Mth root of unity of residues n*-iu-a>f) -iS~i-r— 2"-p+l we shall Jfe-1 max J] (1-22")| and is equal to fe-1 <max II d-«? - tú, l\ CO. In the lemma which follows dx. + +l 1 »2=0 l >• show that ,fcln3/ln4 <CX2* H = 1 72=0 Thus pending proof of the lemma, the theorem is established. Lemma. i_ *-l 1 ax J] Ün3/ln4 tl-*2 >")| <C12Asl ,= 0 The proof of this lemma is quite interesting. idea of where the maximum occurs the maximum Although we have no or what the value is, we can show that is quite close to the value at z = e Proof. If z = ei0, |l-z2"| max = 2|sin 2"-1ö|, k-l k-1 2k n a-*2 ) = max e n=0 Ul=i B=0 , so II sin2"-1Ö License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 73 BINARYDIGIT DISTRIBUTION Substituting 26 for 6 to simplify notation we rewrite the above product as follows: (|sinl/30| *n |sin^32"ö Since the max of the product be treated separately: sin1/3 2" +1Ö|j |sin2/3 2fe"10| . < the product of the maximums, each term may |sin1/30| |sin2/32A:- !0| < 1. If /(0) = sin2/32"0sinl/32" +10, /'(ö)=2Cisin-l/32«öcos2"öSinl/32" + 2^isin-2/3 at a max, we have +1ö 272+ lö cos 222+ 10 sin2/3272ö tan 2"+ 6 = - tan 2"0. The only solutions of this are 2"0= 77/3, 277/3, 477/3, 57r/3, 0, 77. A quick check shows then that maxe|sin2/3 2n0sin1/32"+10| = V3£. Thus fc-1 Jyß\k-1 = | V3 C2ln3/ln4) n d-22") <2 max 22=0 and the lemma is established. II. a Suppose that we had a general ••• and we wanted to determine sequence the relative number of ones in its binary expansion. of integers frequency a y «2, ••• , that a. has an even Let A(n) be the counting function for the sequence. If the following number of ones. integral Choose is 0(^(22)) then k so that r 2 M of the a.'s 11* nU-z2 ) / «»-J|.|..-=ST— (, dz. a;<N We will establish Theorem. that the square Let \a.\ free numbers be an arithmetic have an even < N < 2 + , divide evenly. progression. Then CN-S~ log/V. Proof. |pov)| < /H=1 ¡nnfe=0i- z2"\ \\i<Nz-a'\ In-§I we showed that on the unit circle nd- „2"i I 22=0 dB. the max of <C|Vlog3/log4 logJ_< log 4 So |PQV)|<CN-*-f\2.z-"i\d6. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use g_ P(N) < 74 D. J. NEWMANANDM. SLATER Now if fß(i is an arithmetic n terms progression, with difference a, containing < N, / -J" 11- J-n a;<N Set Q= a =/_n T=T dd * C(Jo*nde+L f) * <*«* - lo*«■ Differentiating, /|Sa<N2-a^l9 of course we find the expression minimized for k = ifri so < C log N and for \a\ an a.p. |P(/V)| < CN-8""logW. This, shows that P{N) = o{A{N)). We might remark that the above argument is quite general. We have P{N) < N-8~L1{AN). For equal frequency we need P{N) = o{A{N)) so lHan) o{A{N)/N-8~) is sufficient. Returning to the question above that the L of arithmetic progressions of many arithmetic with respect to the digits in the binary expansions. N progressions we mean the set of integers principle on the collection progressions If our set we see from the norm of an a.p. is only log N. Thus a set which is a combination a.p.'s, would have to be equidistributed By a combination of formed by using the inclusion-exclusion of a.p.'s. Under these conditions we can combine and our set will be equidistributed. ÍC¿!, C. ¿ C, i 4 7» is made up of a.p.'s in this way, then .2 z~*\+...<N'' f\T.*c\<A\L,"<\,-A\Z We note in passing and has positive square-free = that if a set is a complement logN, of one of the above sets density, then»it also has the equidistribution property. The numbers are such a set. We first form the set of numbers containing combine multiples of 2 and 3 then subtract a square factor as follows: off multiples of 6 etc., or if Sj is the set of multiples of d < N we form - %x< . ßj u{d)S¿. Now while each integer we have too many arithmetic We observe progressions, a square factor occurs exactly namely however that our set of progressions sets of progressions: one set containing which our theorem applies o{N) integers containing once, A/"', to use our estimate. can be divided fewer than Af into two progressions and the other set of progressions to which contains all together. We note that this approach formula for evaluating is valid because the variance in distribution, instead of using the integral we can simply use as an License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 75 BINARYDIGIT DISTRIBUTION upper bound the number of integers in a set. progressions S¿ as follows: progressions plus the remaining integers With this in mind we divide the S¿, d < k, and S ,, k < d < yN. i<d<\ß \k< Thus the total estimate iß J/ This gives k which total \<N/k + yß. for the variance this is minimized for k = N' /yjlog N. is N'8k log N + N/k + y/Ñ Thus we find that the variance distribution is < /V9 \A°g N. This establishes the equidistribution square-free numbers provided that numbers containing and in for a square factor have density < 1. In order to establish -£ this, .we must evaluate p(d)dW2K-N Z i^ +o(VÑ) sU) and since the sum d>yJN d £(2) = 77/6 the number of numbers < N containing is (n2 - 6)/V/t72 and this is of course a set of density The set of square-free integers follows that the set of square-free Theorem. The probability is the complement integers that a square-free of one's in its binary expansion set of square-free integers arbitrary of the above set. density It and also integer has an even number In the previous section out all multiples more general those integers set, are the remaining Of course set. by sieving we pose the following from the set of all integers < 1. is %. III. Some general sieved sets. In this section has positive a square factor integers of the perfect question. which are multiples equidistributed the answer to this is no if no restriction we formed the squares. If we sieve out of some in the binary system? is made on the sieving For example we might sieve with the set of all odd primes. In that case the remaining set would be only powers of 2. We will prove the following Theorem. Given a sequence < oc then the set of all integers equidistributed of integers sieved of any of the a.'s l/a. is in the binary system. We will prove the theorem in two steps. set has positive progression a , a., ••• such that 2~« that are not multiples estimates lower density. Step 1 will establish that our Step 2 will show that the arithmetic of §11 apply. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 76 D. J.NEWMAN AND M. SLATER Lemma 1. Let A{n) be the counting We note that of the sieved Il°^0 (l — l/a¿) > 0 since the infinite if and only if ^°10 ^-la- converges that the density function and this was given. of such a set actually set. product converges Erdös [2] has shown exists. Proof of Lemma 1. The lemma will follow if we can show that the density of a set sieved by the first by the convergence Nli>Ml/a M of the a¿'s is > Il^Kl of 1.a., sieving < eN integers by the remaining by proper choice The proof is by induction. a.'s removes for at most of M. If we sieve with only aQ then DQ > (l - l/ßn) and we need to show that Dk > (l - l/a¿)ofe_j sieving - l/a); where Dk is the density after with k + 1 integers. We note that there exists a number not being divisible an exact probability or density by any of k integers associated with namely k-l P.1-TÍ+ Z i=0 where I I denotes ' —-(-D*- iV<*-llflf the least ay! i<Vßfe-l! common multiple. Thus we can phrase the argu- ment in terms of probability. We want to establish P{n not divisible that for n < N, by any aQ, ... , afe) > Í1 - — JP{n not divisible by any aQ, ... , ak_x) or — P{n not divisible ak by any aQ, ... , ak_)) > P(n not divisible by aQ, ... - P{n not divisible , a, _j) by flQ, ... , a^) > P{n divisible by a^ but not aQ.ak-? > P{ma, not divisible 1 {a., a) ... , «¿.j) aQ a = ——P In not divisible by ;--, ..., is the g.c.d. step follows since ak where / by aQ, ... \ The last {aa, a.) — {a k-l' a^ma^ -*>) if and only if a\m{a., a.)fz —* a./{a., a,)\m and also the number of numbers < N is a,tet' i i i te, times as much as the number of numbers < N/ak and the probability that a License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 77 BINARYDIGIT DISTRIBUTION number < N/a, probability is divisible by a member of a certain of a number < ¿V being divisible sufficiently large. set is the same as the by a member of the set for N But our last inequality P(n not divisible by any an, ... > P/72 not divisible , «,) a0 by ak- (aQ,ak)" is true by inclusion. Incidentally Thus reversing we may rewrite the logic we arrive at the desired the statement k i-Si+LA—»(-o k j 2.7 'iak_vak)j 1 ana, 0 1 • • • a,k >i-Z¿+ Z >a¿' rAv--*-»' i ' i.i;i<i ay> \an, an inequality which would seem quite difficult Proof of theorem. lower density. tion property Thus if we can show that its complement with the first may be formed in two parts ¿-integers more than.once remaining set has positive has the equidistribu- sieving adds so we subtract etc. set. This we subtracted we have integers First puts 2 we k progres- have been placed off the multiples since Altogether Ai£. ,1/a. estimate as follows. some integers of \a., a., ak\ too many times equidistribution in our sieving However, Then add back multiples numbers ,ak\ directly. We have now shown that our sieved sions into our complement. complement to establish ax, ... the theorem will follow. The complement sieve lemma. of Lemma 1 as in the of \a{, a.\. off some progressions. to the complement. The Thus the is 2kN-8logN+NZr i>k ' and we want to show that this is o(N). If we pick k = (2/11) log N we need only pick N so large that £.>Al/a. < { by convergence, One application integers of the above theorem is of course that the square-free are equidistributed. A second deficient and the proof of the theorem is complete. application numbers would be that both the abundant have the equidistribution is a number which is exceeded number is abundant any multiple if it is abundant numbers An abundant by the sum of its proper divisors. is also abundant d\A, Bd\BA and BA will be abundant. abundant property. and the number If a for if A is abundant and A number is said to be primitive but not a multiple of another License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use abundant number. D. J. NEWMANAND M. SLATER 78 Erdös [2] has proven that the sum of the reciprocals abundant numbers converges. our theorem as the sieving the sieved buted. Thus set. the primitive Thus the abundants set composed of deficient of the primitive abundants and perfect can serve in are equidistributed and numbers is also equidistri- Since the perfect numbers are rare (Euler proved that all even perfect numbers are of the form 22 PN ) we have equidistribution - 2*-1 and all odds have the form for the deficient numbers also. A further example would be that the set of all numbers not divisible any twin prime has the equidistribution from Brun's result property. by This follows at once, that the sum of the reciprocals of the twin primes con- verges [3]. BIBLIOGRAPHY 1. L. Dickson, History of the theory of numbers. Vols. 1, 2, 3» Publ. no. 256, Carnegie Inst., Washington, D.C., 1919, 1920, 1923; reprint, Steche«, New York. 2. P. Erdös, (1934), 278-282. 3. E. Landau, On the density Elementare of abundant Zahlentheorie, numbers, J. London Math. Soc. 9 Teubner, Leipzig, 1927; English transi., Chelsea, New York, 1958. MR 19, 1159. 4. D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. S0c. 21 (1969), 719-721. DEPARTMENTOF MATHEMATICS,BELFER GRADUATESCHOOLOF SCIENCE, YESHIVAUNIVERSITY,NEWYORK, NEWYORK 10033 DEPARTMENTOF MATHEMATICS,CITY COLLEGE (CUNY), NEWYORK, NEWYORK 10031 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use