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References
1. Vinogradov I. M. On the distribution of quadratic rests
and non-rests of the form p + k to a prime modulus. Rec.
Math. Moscow, n. Ser., 1938, vol. 3, no. 45, pp. 311?319
(in Russian).
2. Vinogradow I. M. An improvement of the estimation
of sums with primes. Bull. Acad. Sci. URSS. Ser. Math.
[Izvestia Akad. Nauk SSSR] 1943, vol. 7, pp. 17?34 (in
Russian).
3. Vinogradov I. M. New approach to the estimation of a
sum of values of ?(p + k). Izvestiya Akad. Nauk SSSR.
Ser. Mat., 1952, vol. 16, pp. 197?210 (in Russian).
4. Vinogradov I. M. Improvement of an estimate for the
sum of the values ?(p + k). Izvestiya Akad. Nauk SSSR.
Ser. Mat., 1953, vol 17, pp. 285?290 (in Russian).
5. Vinogradov I. M. An estimate for a certain sum
extended over the primes of an arithmetic progression.
(Russian) Izv. Akad. Nauk SSSR Ser. Mat., 1966, vol
30, no. 3, pp. 481?496 (in Russian).
6. Karatsuba A. A. Sums of characters, and primitive
roots, in finite fields. Doklady Akademii Nauk SSSR,
1968, vol 180, no. 6, pp. 1287?1289 (in Russian).
7. Karatsuba A. A. Estimates of character sums. Math.
USSR-Izv., 1970, vol. 4, no. 1, pp. 19?29.
8. Karatsuba A. A. Sums of characters over prime
numbers. Math. USSR-Izv., 1970, vol. 4, no. 2, pp. 303?
326.
9. Rakhmonov Z. Kh. On the distribution of values of
Dirichlet characters. Rus. Math. Surv., 1986, vol. 41,
no. 1, pp. 237?238. DOI: 10.1070/RM1986v041n01ABEH
003232.
10. Rakhmonov Z. Kh. Estimation of the sum of
characters with primes. Dokl. Akad. Nauk Tadzhik. SSR,
1986, vol 29, no. 1, pp. 16?20 (in Russian).
11. Rakhmonov Z. Kh. On the distribution of the values of
Dirichlet characters and their applications. Proc. Steklov
Inst. Math., 1995, vol. 207, no. 6, pp. 263?272.
12. Fridlander Dzh. B., Gong K., Shparlinskii I. E.
Character sums over shifted primes. Math. Notes, 2010,
vol. 88, iss. 3?4, pp. 585?598. DOI: 10.1134/S00014346
10090312.
13. Rakhmonov Z. Kh. A theorem on the mean value
of ?(x, ?) and its applications. Russian Academy of
Sciences. Izvestiya Mathematics, 1994, vol. 43, no. 1,
pp. 49?64. DOI: 10.1070/IM1994v043n01ABEH001558.
14. Rakhmonov Z. Kh. A theorem on the mean-value
of Chebyshev functions. Russian Academy of Sciences.
Izvestiya Mathematics, 1995, vol. 44, no. 3, pp. 555?569.
DOI 10.1070/IM1995v044n03ABEH001613.
15. Vinogradov A. I. On numbers with small prime
divisors. Dokl. Akad. Nauk SSSR, 1956, vol 109, no. 4,
pp. 683?686 (in Russian).
16. Burgess D. A. On character sum estimate with r = 3.
J. London Math. Soc., 1986, vol. 33, no. 2, pp. 219?226.
DOI: 10.1112/jlms/s2-33.2.219.
??? 511.325
????? ???????????? ???????? ???????????????????,
?? ?????????????? ?????????? ?? ?????? ???????
?. ?. ????????
???????? ??????-?????????????? ????, ?????????? ??????????????? ??????????? ??. ?. ?. ??????????,
parviz.msu@gmail.com
???????? ????? ??????????????? ???????? ???????????????????, ?? ?????????? ?????????? ??????????????? ?? ??????
???????.
???????? ?????: ??????????? ????????????? ?? ?????? ???????, ????? ?????????, ??????? ???????, ??????-????????
???????.
?????? ? ??????????? ????????????? ??????? ???? ??(x), ??? ? ? ?????????????? ?????, ?(x) ?
???????, ??????????? ????? ????????, ???????? ?. ??????. ?? ?????? [1] ???????, ??? ?????
??? ???? ? ?????????????????? {??n } ?????????? ????????????, ??? ? > 1 ? ?????????????? ?????.
?????? ?????????? ??????? ????? ? ?? ?? ???? ?????????. ? ??????? ?. ?. ??????????, ??????,
??????? ????????? ??????? ????? ?, ??? ?????????????????? {??n } ?????????? ???????????? ??
?????? ???????, ? ? 2 ? ????? ?????.
????????????? ?????????? ?????????????: {fn }?
n=0 ? ?????????????????? ?????????: f0 = 1,
?
f1 = 1, fn = fn?1 + fn?2 ??? n ? 2, ? = (1 + 5)/2 ? ??????? ???????.
Е ???????? ?. ?., 2013
117
???. ?????. ??-??. ???. ???. ???. ??????????. ????????. ???????????. 2013. ?. 13, ???. 4, ?. 2
????? 1. ???????????? ??????????????? ??????? ? ????????????? ?????????????? a,b,c,d
? c +d2 6= 0 ?? ? ????? ??????????? ???? ???????? ??????? ? ????????????? ??????????????
?? ?:
bc ? ad
ac ? bc ? bd
a? + b
= 2
?+ 2
.
c? + d
c ? cd ? d2
c ? cd ? d2
2
??????????????. ????????? ?????????? ?2 = ? + 1, ?????????? ??????-???????? ??????? ?
????????????? ?????????????? ?? ?, ???? ???????? ??????? ?? ? ? ????????????? ??????????????:
А А
б
б
(a? + b) c ? ? 12 ? 2c ? d
a? + b
a? + b
б
б
= А
=
= А
2
c? + d
c ? ? 21 + 2c + d
c2 ?2 ? ? + 41 ? c4 ? cd ? d2
ac?2 + (bc ? ac ? ad)? ? bc ? bd
(a? + b)(c? ? c ? d)
=
=
2
2
c ? cd ? d
c2 ? cd ? d2
ac(? + 1) + (bc ? ac ? ad)? ? bc ? bd
(bc ? ad)? + ac ? bc ? bd
=
=
=
2
2
c ? cd ? d
c2 ? cd ? d2
ac ? bc ? bd
bc ? ad
?+ 2
.
= 2
2
c ? cd ? d
c ? cd ? d2
=
????????, ??? ??????????? d2 + cd ? c2 ?? ?????????? ? ???? ??? ???????????? ???????????? c, d,
??????????????? ??????? c2 + d2 6= 0.
????? 2. ????? lim (xn ? yn ) = 0. ?????, ???? ?????????????????? xn ? yn ????????????
n??
??????????, ???? ?????????????????? xn ? yn ?? ???????????? ??????????.
??????????????. ????? ?????????????????? xn ?????????? ???????????? ? l 6= 0 ? ????? ?????.
???????? ???????? ?. ????? ???????????? ?????????????:
n
1 X 2?ilxk
e
= 0,
n?? n
lim
(1)
k=1
e2?ilxk ? e2?ilyk = cos(2?lxk ) ? cos(2?lyk ) + i(sin(2?lxk ) ? sin(2?lyk )).
?? ??????? ????????
cos(2?lxk ) ? cos(2?lyk ) = ?2?l(xk ? yk ) и sin(2?l?k ),
sin(2?lxk ) ? sin(2?lyk ) = 2?l(xk ? yk ) и cos(2?l??k ),
?k ? (xn , yn ),
??k ? (xn , yn ).
???????
e2?ilxk ? e2?ilyk = ?2?l(xk ? yk ) и sin(2?l?k ) + i и 2?l(xk ? yk ) и cos(2?l??k ).
???????? | sin(2?l?k )| ? 1, | cos(2?l??k )| ? 1, ???????:
????? ???????,
?
»
» 2?ilx
k
»e
? e2?ilyk » ? 2? 2|l| и |xk ? yk | ? 0
(k ? ?).
lim e2?ilxk ? e2?ilyk = 0.
k?+?
???? ?????????????????? e2?ilxk ? e2?ilyk ????? ??????, ?????? 0, ?? ?????????????????? ???????
??????????????:
n
1 X 2?ilxk
e
? e2?ilyk
n
k=1
????? ????? ??????, ?????? ????, ???????
n
1 X 2?ilxk
e
? e2?ilyk = 0.
n?? n
lim
k=1
118
??????? ?????
?. ?. ????????. ????? ???????????? ???????? ???????????????????
??????? ?? (1) ?????????, ???
n
1 X 2?ilyk
e
= 0.
n?? n
lim
k=1
??????, ?? ???????? ????? ?????????????????? yn ?????????? ????????????.
?????????? ????? ????????, ??? ???? yn ?????????? ????????????, ?? xn ????? ?????????? ????????????.
???????. ?????????????????? {?fn }?
n=0 , ?? ???????? ?????????? ??????????????.
??????????????. ????????? ???????? ????? ????????? fn+1 /fn ???????? ??????????? ???????
??? ???????? ??????? ? ??? ?? ?????????? ? ??????????? ?????. ??????? ?? ??????? ???????
?=
fn+1
?
+ 2,
fn
fn
|?| ? 1.
?????
?fk =
х
fn+1
?
+ 2
fn
fn
Х
fk =
fk fn+1
? и fk
.
+
fn
fn2
????????? A(k, n) := fk fn+1 ? fk+1 fn ,
A(k, n) = fk fn+1 ? fk+1 fn = fk (fn + fn?1 ) ? (fk + fk?1 )fn = fk fn?1 ? fk?1 fn =
= ?(fk?1 fn ? fk fn?1 ) = ?A(k ? 1, n ? 1) = . . . = (?1)k A(0, n ? k),
A(0, n ? k) = f0 fn?k+1 ? f1 fn?k = fn?k+1 ? fn?k = fn?k?1 .
??????? A(k, n) = (?1)k fn?k?1 ??? k < n ?
fk fn+1 = fk+1 fn + (?1)k fn?k?1 ,
????? ???????,
?fk =
?????? (?1)k
? и fk
fn?k?1
? и fk
fk fn+1
+
= fk+1 + (?1)k
+
,
fn
fn2
fn
fn2
k < n.
(2)
fn?k?1
? и fk
+
:
fn
fn2
fk
1
<
?0
fn2
fn
(n ? ?).
??? ????????????? k:
fn?k?1
fn?k?1
fn?k
fn?1
=
и
иии
.
fn
fn?k
fn?k+1
fn
?????? ????????? ? ???? ???????????? ????????? ? ??1 ??? n ? ?, ???????
fn?k?1
? ??k?1
fn
(n ? ?).
????????? ? ????????? (2), ???????, ??? ??? ?????? k
А
б
?fk ? fk+1 + (?1)k ??k?1 ? 0
????????? k ? +?:
?fk ? fk+1 ? 0
(n ? ?).
(k ? +?).
(3)
??? ??? ?????????????????? {fk+1 }, ????????, ?? ???????? ?????????? ??????????????, ??, ???????? ????? 2, {?fk } ?? ???????? ?????????? ??????????????.
??????????
119
???. ?????. ??-??. ???. ???. ???. ??????????. ????????. ???????????. 2013. ?. 13, ???. 4, ?. 2
?????????. ????? ? ???? ???????? ???????????? ??????? ?? ?, ?? ???? ? = f (?)/g(?), ???
f (?), g(?) ? ?????????? ? ?????? ?????????????? ?? ?. ????? ?????????????????? {?fn } ??
???????? ?????????? ??????????????.
??????????????. ????????? ???????? ?2 = ? + 1, ??????? f (?), g(?) ? ???? ???????? ???????
?? ?. ????? ????? ????????, ??? ? ???? ???????? ??????-???????? ??????? ?? ?. ?????, ????????
????? 1, ???????? ??????-???????? ??????? ?? ? ????? ???????? ? ???? ???????? ??????? ?
????????????? ?????????????? ?? ?. ????? s, t ? ???????????? ?????.
???????? ?????? (3)
(s? + t)fk ? (sfk+1 + tfk ) ? 0
(k ? ?).
?????????????????? sfk+1 +tfk ?? ???????? ?????????? ?????????????? (??? ??? s, t ? ???????????? ?????). ???????, ???????? ????? 2, ?????????????????? (s? + t)fk ????? ?? ???????? ??????????
??????????????.
????????????????? ??????
1. Weyl H. U?ber die Gleichverteilung von Zahlen mod. Eins. // Math. Ann. 1916. Vol. 77. P. 313?352. DOI:
10.1007/BF01475864.
On the Class of Exponentially Growing Sequences
that are Not Uniformly Distributed Modulo One
P. Z. Rakhmonov
Moscow State University, Russia, 119234, Moscow, Leninskie Gory, 1, parviz.msu@gmail.com
The paper presents a family of exponentially growing but not uniformly distributed sequences modulo one.
Key words: uniform distrubtion modulo one, Fibonacci numbers, golden ratio, homographic transformation..
References
1. Weyl H. U?ber die Gleichverteilung von Zahlen mod. Eins. Math. Ann., 1916, vol. 77, pp. 313?352. DOI:
10.1007/BF01475864.
??? 511.9
??????? ???????????????? ?????????
? ??????? ??????????? ??????? ?. ?. ??????????
?. ?. ????????
????????? ??????? ???????, ??????????????? ??????? ? ?????????, ???????? ??????????????? ?????????????? ??????????? ??. ?. ?. ????????, rodionovalexandr@mail.ru
? ?????? ??????????? ????????? ?????????? ??????????????? ?????????? ????????????? ??????? ??????? ????????
?????????, ??????????????? ???????????????????? ??????.
???????? ?????: ???????????????? ????????? ? ??????? ???????????, ?????????-???????? ?????.
? 1961 ???? ?. ?. ????????? ? ?????? [1] ????????? ????????? ????? ??????? ?????? ???? ???
?????????? ?????? ???????????????? ????????? ? ???????? ????????????:
х
Х
?
?
?u
=Q
,...,
u(t, ~x),
0 ? t ? T,
?? < x? < ?
(? = 1, . . . , s),
(1)
?t
?x1
?xs
u(0, ~x) = ?(~x),
Е ???????? ?. ?., 2013
~x = (x1 , . . . , xs ),
(2)
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