CC BY
282. Ваш L.E., Billingsley P. Asymptotic distributions for the coupon collector's problem // Ann. Math. Stat. 1965. 36. t*. 1835-1839.
3. НвОиев И. Локальная предельная теорема с оценками остаточного члена в задаче времени ожидания // Исследования по вырождающимся дифференциальным уравнениям и предельным теоремам теории вероятностей. Ташкент, 1985. С. 43-47.
4. Фомин А.С. Локальная теорема для серий независимых, разно-распределенных в серии, целочисленных случайных величин // Вероятностные задачи прикладной математики. Петрозаводск, 1964. С.70-75.
5. Петров В.В. Суммы независимых случайных величин. М.: Наука, 1972.
Труда Петрозаводского государственного университета Серия "Математика" Еыпуск 1, 1993 г.
УДК 511.45 Широков Б.М.
ОМ A LENGTH OF THE CONTINUED FRACTION'S PERIOD
В статье дана оценка количества чисел d из отрезка натурального ряда, для которых непрерывная дробь для имеет большой период.
Let L(d) be the length of the continued fraction’s period of Vd for d that isn’t a square of an integer. Let p be a prime number, N, P,Q,k,r, я -natural numbers.
FYom the resulte of E.V.Podsypanin [1] it follows
L(d) = 0{Vd log d).
From the results of E.P.tiolubeva (2) it follows
where x w a character of the quadratic field Q(\/3), u(d) is a number of prime divisors of d.
In the paper [3] has been proved that for any real К > 0 and for
sufficiently great N the number of integers d, N < d < 2N, for which
L(d) > Ky/d don’t exceed cN/\ogK witb absolute constant c.
The irrational
where P anu Q sutisfy to the congruence
P7 = d (mod <7) (2)
is called reduced if 0 < ( < 1 and conjugate number £' < —1,
С = (P - Vd)/Q. By other words, £ is reduced if
0 <P<Vd, \fd-P<Q<Vd + P
I/Ct'8 note that the continued fraction's remainders have form (1) and sattdfy to conditionb (2) and (1).
Let H(d) denote a number of the all reduced irrationals of form (1) with the same d. The quantity H (d) equel to the lengthes sum of the not equivalent in pairs quadratic irrational's periods.
In this paper will be proved the following theorems.
Theorem 1. For any N
n JN
— N3/* + 0{N) < £ H(d)<Ay/2N^7
* *=N+\
with the absolute constant into the symbol O.
Theorem 2. For any Nand real K > 0
4/5
#{d | N < d < 2AT, L(d) > Ky/d) <
K
Proof of Theorem 1. The quantity H(d) equal to a number of all pairs (P,Q) of P and Q satisfying to inecmalitinn (3) and t.o rnnpnienr* (2).
I-et p(d, Q) denote the number of solutions of congruence (2). Then 3 N a n
E »»< E E owe).
i^N+t d=N+iQ <Q<aV3
Let's change the order of summing in the right part:
3 N
E E E *<<*.«) (“)
*=N+\ o<g<3v^TN,1=w+1
Wc have increased the region of the variation of d for values Q > y/2N. This permutation of the sums os a matter of fact is using of the quadratic reciprocity law.
Let‘s consider the interior sum of the right part of (4).
IN IN/91 + 1 N + kQ
E pM)< E E <<<*.«) (*)
d-N+1 t=» <f=N+(fc-l)Q+l
f
where [x) 18 an entire of real x. Let S(Jfc,Q) be the interior aum of the right part of (5). Let ft = (d,Q) = raJ, d\ = d/6, Qj = Q/6, where r is a square-free integer. We arc need in the following lemma now.
Lemma. Congruence (2) ia solvable if and only if d is a quadratic residue modQi and (r,Q,) = 1. Jn this case there exist s solutions of congruence (2)for every solution of the congruence
rxa= d\ (modQ\) (6)
Proof. An existance of a solution of congruence (0) is nesessary and sufficient for a solvable of congruence (2). Let (6) is solvable. Then ir\Qi) — 1* If r' i" an inverse element for r in the multiplicative group in the ring of residue classes modQ* then dir* is a quadratic residue modQ\. But r' is a quadratic residue if and onli if r is a quadratic icbiuuc. Ilcacc d,r je 3. residue. Thus d is a quadratic residue
modQi.
Inversly, if d is a quadratic residue modQ\ then d/a3 is the same thing. Let's suppose that (r,Qj) = 1. Then there exists r' that rr' = l(modQi) and dir' is a quadratic residue mndQ\. Therefore, congruence (6) is solvablcs and congruence (2) is solvables too. The first statement of the lemma has proved.
Let xo be any solution of congruence (6). Put
P = raxo + krsQu 0 < Jfc < s. (7)
It's clear th«tt these numbers are not congruenced modQ. Furthermore
P3 = (rax0)a + 2 rxoQk + k^rioQQi = (rax0)2 = d{modQ).
Thus for every solution x0 of congruence (6) formula (7) gives the a distinguish solutions of congruencc (2). The lemma has proved.
Using the lemma we'll find the quantity n(6,Q) of the numbers d with the fixed 6 = (d}Q) for which congruence (2) ;s solvable»).
Owing to the square-free factor r of the integer d is fixed the number such d is defined by the number of quadratic residues or non-residues modQi provided that r is a quadratic residue or non-residue correspondency.
For every piime factor p of Qi there are such numbers modp and there are such numbers modpa. There are <M,Qi)/2u^<^'^ such numbers modQ\ for an odd integer Q|. Here <^(n) is the Euler's fonction and u>(n) is a number of prime factors of an integer n.
If Qi container 2k with maximum kt then the correspondent conditions have the form;
d = r(mod2m), m = m»n{fc,3}.
Hence
if Q\ isoddorQi = 4(mod8),
n(6,Q) = *(Qi) 2-^‘>+1 if Q\ = 2(mod4), n(6,Q) = ^(Qi)2~‘^^l^~I if Qt = 0(mod8).
If for given d and Q congruence (2) is solvable then owing to the lemma for a number of the solutions we shell have:
p(d,Q) = 2ta’*<*t*« if Qj H 1 (mod2) or Q\= 4 (rnodH),
p{d,Q) = 2*Qi)-'s if Qi = 2(niod4), p(d,Q) = 2w(<,i)+1ji if Qt =0(modS).
F>om this we get:
S(*,Q)= £ ,«£).
r.JlQ.(r,g/r.»)=i
Let's denote u the production of the primes p which divide Q but p* don't divide Q and v = Q/u. Then
*(*.<?> = ££<) £ ««£> =
**l* r|. <|^ M
(•.•/•■Hi
-E E ’•x^) s«E*<5> = «■> = <?.
•>|. W|. a
Owing to (4) and (5) we find:
a N
H(d)<N £ (l + %) < 4v/2Af,/3.
*=N+1 Q<iy/TR
The right part of the inequality of theorem 1 has proved.
For the the proof of the left part inequality we «hell take into account only that pairs (P,Q) in the reduction region that satisfy- to the conditions
0 < Q <* y/d, v3 - Q < P < y/d.
In this case
3 N
E E E
«I=W+1 0<Q<y/iN*=N+iA4,Q)=\
The interior sum is estimated similarly to the same thing in (4): i*,N)
E mq)>w«w«).
N<d<iN
Therefore
3 N
E E W<3W«) = -^w1/’ + o(N).
*=w+l 0<Q<V^W
The theorem haf» proved.
Proof of theorem 2. Owing to l\d) < 11(d) theorem 1 implies
3 N
Y, < 4%/2N*'7. (9)
*=/V + l
I<et*8 choose any real K and denote n for a number of d in (9) for that L(d) > K Vd. Then inequality (9) implies *
«
nKy/N < Ay/2N3/1.
FVom this inequality it follow» the Rtatemcnt of theorem 2.
HI DLIOG11A PH Y
1. Podsypanin E.V. On a length of the quadratic irrational's period // Zap.nauch.sem.l/OMI. if»7H. V.82. P.95-09.(Rus).
2. Oolubcva E.P. On a length of the quadratic irrational's period// Math.Sb. 1981. V.I23. No.l. P.I20-129.(Rua).
3. Rockett A.M., Sztisx P. On the length of the Period of the continued fraction» of Square-Roots of Integers//Forum Math. I990. V.2. No.2. P. 119—423.
СОДЕРЖАНИЕ
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Заика Ю.В. Интегральные операторы идеального наблюдения динамических систем.....................................................
Иванов А. В. Миксеры, функторы и мягкие отображения.........22-
Моисеев Е.В. О пространстаанствах с А#-структурой...........27
Ыосягин В. В. К теории операторов в локально-выпуклых пространствах с конусом................................................35
Нилник П.П., Аккаев Р.И. Анаморфова декартова абака в составных номограммах...................................................41
Павлов Ю.Л. О случайных деревьях............................47
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Платонов С.С. О взаимно однозначном соответствии между инвариантными подпространствами в некоторых пространствах.............54
Старков В. В. Гармонические локально квазиконформные отображения ..............................................................61
Степанова Е.Н. О пространстве частичных селекций............70
Стреколо»ская Н.С. О ретракте экспоненты ехргХ бикомпакта X..............................................................77
Фоыин А.С. Локальная предельная теорема в одной задаче о времени ожидания.....................................................80
Широков Б.М. On a lenght of the continued fraction's period............................................................85
