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Показатель сходимости среднего значения полных рациональных арифметических сумм.

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??????????? ???????
??? 16. ?????? 4.
??? 511.9.
?????????? ?????????? ????????
???????? ?????? ????????????
?????????????? ????1
?. ?. ????????? (?. ??????)
?????????
? ?????? ?????? ??????? ?????? ???????? ?????????? ?????????? ???????? ???????? ?????? ???????????? ?????????????? ???? ??? ?????????????? ???????, ??????????????? ??????????????? ????????? ???????? ????.
? ?????????, ?????????? ???????? ????????????? ????? ?????????.
???????? ????????? ?????????? ??? ?????? ???????????? ?????????????????? ???? (??? ??-???, 1952).
????? ????????? ?????????? ?????? ?????????? ???????????? ???????. ?? ??????? ?. ?. ??????????? ?? ???????????? ???????????? ??????????? ? ?????? ????? ??????.
?????? ???????????? ?????????????? ????? ???????? ?????????
????????????? ?????????? ?? ????????????? ???????, ????????, ?????????????????? ???????. ? 1978 ?. ???? ???????? ???????? ??????????
??? ??????? ???????? ?????????? ?????????? ??????????????????? ????????? (?. ?. ???????, ?. ?. ????????, ?. ?. ?????????).
??? ??????????? ???????? ? ????????? ????? ??????? ???????? ?????? ??????? ? ?????? ?????? ?????????? ?????????? ???????????????
???? ? ??????????.
???????? ?????: ??????? ?????? ????????? ??? ?????-??????? ??????, ?????? ???????????? ?????????????? ?????, ?????????????? ????????? ?? ?????? ??????? ??????? ?? ?????? ???????????? ?????, ?????????? ????????.
????????????: 19 ????????.
THE RATE OF CONVERGENCE OF
THE AVERAGE VALUE OF THE FULL
RATIONAL ARITHMETIC SUMS
V. N. Chubarikov (Moscow)
1
?????? ????????? ?? ?????? ???? ? ?? 13-01-00835
304
?. ?. ?????????
Abstract
In this paper the exact value of a index of convergence for the mean-value of
the complete rational arithmetical for the arithmetical function, satisfying the
functional equation of Gaussian type, is found. In particular, the Bernoulli?s
polynomials satisfy for this functional equation.
A similar result holds for the complete rational trigonometric sums (Hua
Loo-keng, 1952). The deduction of the main result of the paper leads of the
elementary method. We owe to I. M. Vinogradov for the demonstration of
fruitful results and profit of it.
The complete rational arithmetic sums are the analogue the oscillatory
integral of a periodic function, for example, trigonometric functions. In 1978
similar results for the exact value of the index of convergence of the trigonometric integral were obtained (G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov).
In nowadays for a multivariate problem there are successful to get only
upper and lower estimates for the index of convergence of appropriate sums
and integrals.
Keywords: the Gauss theorem of a multiplication for the Euler gammafunction, complete rational arithmetical sums, a functional equation on a
complete system of residues by modulo of natural number, the Bernulli polynomials.
Bibliography: 19 titles.
1. ????????
????????? ?????? ????? ????????? ????????? ????????? ????????? ? ???
???????????????????.
????? s ? ??????????? ?????, F (x) ? ????????????? ??????? ? ????????
1, ??????? ??? ?????? ???????????? ????? m ? a ? ??????? ???????? ? m, ???
????? ???????????? x ????????????? ?????????? ??????????????? ?????????
(
)
(
a)
a(m ? 1)
1?s
m F (mx) = F (x) + F x +
+ иии + F x +
.
(1)
m
m
???????? ????????? ???? ??????? ??????? ??? ?????-??????? ?????? ? ??????????????? ??????????? ???? ??????? ???????? ?????????. ??? 0 ? x < 1
??????????? (1) ??????????? ??? ??????????? ???????? Bs (x), ??????? ???
s ? 1 ????? ?????????? ???:
B0 (x) = 1, Bs? (x) = sBs?1 (x),
?1
Bs (x) dx = 0,
0
??????
s! ? e2?i?x
Bs (x) = ?
, Bs (1 ? x) = (?1)s Bs (x),
s
s
(2?i)
?
|?|?1
?????????? ?????????? ???????? . . .
305
??? ??? s = 1 ????? ???? ?????????? ??? ??????? ???????? ?? ????.
????? ????, ??? s = 1 ????????? (1) ????????????? ??????? F ????
(
)
F (x) = ln 1 ? e2?ix = ln |2 sin ?x| ? i??(x),
??? ?(x) = 0, 5 ? {x}. ? ?????????? ????? ???????????? ???, ??? F (x) ? ???????????? ???????, ?.?. ???????? C > 0 ?????, ??? ??? ???? ???????????? x
????? F (x) ? C. ????? ????? F (x) ???????? ???????-??????????? ? ????????????????? ?? ???????, ?.?. F (x) ????????????? ???????? ??????????? ???????
??????? ? ?????????? ??????? ? ?????????? ??? ?????.
?????, ?????, p ? ???????? ??????? ?????, l ? ??????????? ?????. ???????
?????? ???????????? ?????????????? ?????? S(F ) ????? ????
(
S(F ) = S
f (x)
;F
pl
)
p
?
(
l
=
F
x=1
f (x)
pl
)
,
(2)
??? f (x) = an xn +и и и+a1 x+a0 ? ????????? ? ?????? ?????????????? an , . . . , a1 ,
????????? ??????????? a0 ? ???????????? ????? ? (an , . . . , a1 , p) = 1.
???????
?????????
?p = ?p (2k) ?????? ???????????? ?????????????? ???(
)
?? S
f (x)
;F
pl
??????? ????????? ????
?p = 1 +
+?
?
A(pl ),
l=1
???
pl ?1
A(pl ) =
?
1
(
)2k
? ? f (x)
?l
d?0 ,
иии
p
S
;
F
pl
pl ?1
f (x) = an xn + и и и + a1 x + ?0 .
an =0
a1 =0 0
(an ,...,a1 ,p)=1
? ????????? ?????? ?? ??????? ????????? ???????????.
??????? 1. ??? ?p = ?p (2k) ???????? ??? 2k > 0, 5n(n+1)+1 ? ??????????
??? 2k ? 0, 5n(n + 1) + 1.
2. ??????????????? ???????????
????? 1. ????? p ? ??????? ?????, g(x) ? ????????? ? ?????? ??????????????, a ? ?????? ????????? m ????????? g(x) ? 0 (mod p), ? ????? u ?
?????????? ??????? ????? p, ??????? ??? ???????????? ?????????? h(x) =
= g(px + a). ????? ????? ?????? ?????????
p?u h(x) ? 0 (mod p)
? ?????? ?? ?????????? ?? ??????????? m.
306
?. ?. ?????????
??????????????. ??., ????????, ? [6], ?. 55, ????? 2.
????? 2. ????? p ? ??????? ?????, f (x) = an xn + и и и + a1 x + a0 ? ????????? ? ?????? ??????????????, (an , . . . , a1 , p) = 1, ? ????? u ? ?????????
??????? ????? p, ??????? ??? ???????????? ?????????? g(x) = f (?+px)?f (?).
????? ????? 1 ? u ? n.
??????????????. ??., ????????, [6], ?. 56, ????? 3.
????? p ? ??????? ?????, l ? 2 ? ??????????? ?????, f (x) =
n
?
ak xk ,
k=0
(an , . . . , a1 , p) = 1. ?????????? ??????? ??????,
( ?????
) p ??????????? ???????
f (x)
?????????? f, ?.?. p > n. ?????????? ????? S pl ; F ? ????
(
S
f (x)
;F
pl
)
p
?
=
(
p
?
l
S? ,
S? =
F
x=1
x?? (mod p)
?=1
????? ????????? ???????????.
????? 3. ????? f ? (?) ?? 0 (mod p). ????? S? = F
(
f (x)
pl
f (?)
p
)
.
(3)
)
.
??????????????. ?????
p
?
l?1
S? =
y??
p?1
?
(
F
y=1
z=0
(mod p)
f (y + pl?1 z)
pl
)
p
?
p?1
?
l?1
=
y??
(
F
y=1
z=0
(mod p)
f (y) f ? (y)z
+
pl
p
)
. (4)
????????????? ?????????????? ?????????? (1). ??????
(
p
?
l?1
S? =
y??
F
y=1
(mod p)
f (y)
pl?1
)
.
???????? ??? ?????????, ???????
(
p
?
l?2
S? =
F
y=1
y?? (mod p)
f (y)
pl?2
)
(
p
?
= иии =
F
y=1
y?? (mod p)
f (y)
p
)
(
=F
????? ????????. 2
????? 4. ????? f ? (?) ? 0 (mod p). ?????
p
?
(
l?1
S? =
y??
y=1
(mod p)
pF
f (y)
pl
)
p
?
(
l?2
=p
y=1
F
f (? + py)
pl
)
.
f (?)
p
)
.
?????????? ?????????? ???????? . . .
307
??????????????. ???????
p
?
S? =
(
p?1
?
l?1
F
y=1
z=0
y?? (mod p)
(
p
?
l?1
=p
F
y=1
y?? (mod p)
f (y)
pl
f (y) f ? (y)z
+
pl
p
)
p
?
(
l?2
=p
F
y=1
)
=
f (? + py)
pl
)
.
????? ????????. 2
3. ????? ??????? ?? ?????? ????? ??? p > n.
??????????? ????? S(F ) ?? ??????? (3). ??????
(
S(F ) = S
f (x)
;F
pl
)
=
p
?
Sv = ?1 + ?2 ,
v=1
???
?1 =
?
S? ,
?2 =
p
?
Sv ,
v=1
v?=?
?
?????? ? ????????? ?? ???? ????????? ???????? ????????? f ? (?) ? 0 (mod p).
???????? ????? 3 ? 4, ???????
(
)
f (? + py)
F
?1 = p
,
l
p
? y=1
(
) ?
(
)
p
p
?
f (v)
f (v)
?2 =
F
=
F
+ ?Cn,
p
p
v=1
v=1
p
??
l?2
|?| ? 1.
v?=?
??????????? ????? ?1 . ?????
p
??
(
l?2
?1 = p
?
y=1
F
f (? + py) ? f (?) f (?)
+ l
pl
p
)
.
??? ??? f ? (?) ? 0 (mod p), ?? ?? ????? 2 ??????? f (?+py)?f (?) = pu1 f1 y ? u1 ?
2, ?????? ???????????? ?????????? f1 (y) ? ???????????? ?????? ? p. ????
308
?. ?. ?????????
?? l < u1 , ?? ????? ?1 ?????? ?????????? ?????? ?????????: |?1 | ? Cnpl?1 .
?????????????, ??? l ? u1 ??????????? ???????????
?1 = p
u1 ?1
p
??
S?,v ,
S?,v =
v=1
?
(
?
pl?u1
y?v
F
y=1
(mod p)
)
f1 (y) f (?)
+ l
pl?u1
p
.
????? ? ?????? ????????? f1? (?) ? 0 (mod p) ? f1 (? + py) ? f1 (?) = pu2 f2 (y),
??? ????????? f2 (y) ????? ???????????? ? ???????????? ??????? ? p. ???
u1 ? l ? u1 + u2 , ???????? ?????????? S?,? , ?????? |?1 | ? npl?2 .
?????, ??????, l > u1 + u2 . ????? ?????????? ?1 ? ???? ?1 = ?11 + ?12 , ???
?
?11 = pu1 ?1
S?,? ,
?,?
u1 ?1
?12 = p
p
??
?
(
F
v=1
v?=?
f1 (v)
f (?)
+ u1 ?1
p
p
)
u1 ?1
=p
p
?
v=1
(
F
f1 (v)
f (?)
+ u1 ?1
p
p
)
+ ?1 Cnpu1 ?1 ,
+
|?1 | ? 1.
??????????? ????? ?11 . ?????
u1 ?1
?11 = p
pl?u1
?,?
y=1
y?? (mod p)
(
l?u ?1
u1 ?1
=p
1
?p?
?,?
F
z=1
=p
1
?p?
?,?
(
F
z=1
f2 (z)
pl?u1 ?u2
(
l?u ?u2
u1 +u2 ?2
=p
1
?p ?
?,?
F
f1 (y) f (?)
+ l
pl?u1
p
)
=
f1 (? + pz) ? f1 (?) f1 (?) f (?)
+ l?u1 + l
pl?u1
p
p
l?u ?1
u1 ?1
(
?
?
F
z=1
f1 (?) f (?)
+ l?u1 + l
p
p
f2 (z)
pl?u1 ?u2
)
=
)
=
f1 (?) f (?)
+ l?u1 + l
p
p
)
.
?? ????? 1 ?????????? ??????? ?????? ?, ?, . . . ?? ??????????? n ? 1 ? ??? ??????? ?????? ?????? ?????? ?????????? ????????? ????? ??????????? (u1 , u2 , . . . )
? ????? t = t(?, ?, . . . ) ?????, ???
l ? u1 ? u2 ? и и и ? ut > 1,
l ? u1 ? u2 ? и и и ? ut+1 ? 1.
?????????? ?????????? ???????? . . .
309
?????? ???????
(
S(F ) = S
f (x)
;F
pl
)
=
?
pu1 +иии+ut ?t T (u1 , u2 , . . . , ut ) + R,
?,?,...,?
???
(
pl?u1 ?u2 ?иии?ut
T (u1 , u2 , . . . , ut ) =
?
F
y=1
f (?) f1 (?)
ft (?)
+ l?u1 + и и и + l?u1 ?u2 ?иии?ut +
l
p
p
p
ft (? + py) ? ft (?)
+
pl?u1 ?u2 ?иии?ut
R=
t?1
?
h=0
u1 +иии+uh ?h
p
p
?
(
F
y=1
+
t?1
?
f (?)
pu1 +иии+uh
+
?h npu1 +...uh ?h ,
f1 (?)
pu1 +иии+uh?1
fh (y)
+ иии +
p
)
,
)
+
|?h | ? 1.
h=0
????? s ?????????? ?????????? ????? t = t(?, ?, . . . , ?). ?????
|S(F )| ? 2Cnpl?s .
????? ??????? ???????? ????????? ???????????.
??????? 2. ????? n ? 3, l ? 1 ? ??????????? ?????, p > n ? ???????
?????, f (x) = an xn + и и и + a1 x + a0 ? ????????? ? ?????? ??????????????,
(an , . . . , a1 , p) = 1; ? ????? s ? ?????????? ????? t = t(?, ?, . . . , ?), ??????????
??????? ??????????? (u1 , . . . , ut ). ????? ??????????? ??????
(
)
f
(x)
|S(F )| = S
; F ? 2Cnpl?s .
l
p
4. ??????? ???????????
?????? ?????????? ?????? ????? ??????? ????? p. ??? ????????? ??????????? ?????????? ????????????.
????? n ? 3, l ? 1 ? ??????????? ?????, p ? n ? ??????? ?????, f (x) =
= an xn +и и и+a1 x+a0 ? ????????? ? ?????? ??????????????, (an , . . . , a1 , p) = 1.
?????? ??? ???????????. ????????? ?????????? ?0 ?? ???????
p?0 ?(nan , . . . , a1 ).
??????? w = [ln n/ ln p]. ????? ?0 ? w. ?????,?????, ? ? ?????? ?????????
p??0 f ? (?) ? 0 (mod p).
310
?. ?. ?????????
??? ????? ??????? ? ??????????? ????????? ?????????? ????????? f1 (y) =
n
?
= f (y +?) =
bs y s . ????? f (x) = f1 (x??). ?????? ???????,??? ????????????
s=0
???? ??????????? ????????????? ????????????
bn = an ,
(n)
bn?1 = an?1 +
an ?,
1
...
...
...
...
(
)
(n)
s+1
bs = as +
as+1 ? + и и и +
an ? n?s ,
s
s
...
...
...
...
( )
(
2
n)
b1 = a1 +
a2 ? + и и и +
an ? n?1 ,
1
1
(5)
???
an?1
an = bn ,
(n)
= bn?1 ?
bn ?,
1
...
...
...
...
)
(n)
s+1
a s = bs ?
bs+1 ? + и и и + (?1)n?s
an ? n?s ,
s
s
...
...
...
...
( )
(n)
2
b1 = a1 ?
a2 ? + и и и + (?1)n?1
an ? n?1 .
1
1
(
?? ???? ??????????? ?????, ??? p?0 ?(nbn , . . . , 2b2 , b1 ). ??????????, ??? ???????? ?0 ???? ? ?? ?? ??? ??? ????????????? an , an?1 , . . . , a1 , ??? ? ???
bn , bn?1 , . . . , b1 .
?????? ??? ??????????? ??????????? ???, ??? ??????????????? ??? ?????
? = ?1 , . . . , ?m , m ? n ??????????????? ?????????
p??0 f ? (?) = p??0 b1 ? 0 (mod p).
????? ??? ??????? ????? ? ???????????? ???? ?????????? u1 = u1 (?) ?? ?????????? ??????? ?????? ?????????
pu1 ?(pn bn , . . . , p2 b2 , pb1 ).
???? l?u1 > 2w+1, ?? ????????? ?? ??????? ????. ? ????????? ?????? ???????
???????????, ?????????? ????? ? ????? ???????? ?? ?????? ????? u1 = u1 (?).
?????? ???. ??????????? ?????? ? ? ?????????? ??? ?????????? u1 = u1 (?).
?????
f (py + ?) ? f (?) = pu1 f1 (y), f1 (y) = cn y n + и и и + c1 y,
?????????? ?????????? ???????? . . .
311
???
(cn , . . . , c1 , p) = 1,
pu1 cs = ps bs ,
s = n, . . . , 1.
????? p?1 ?(ncn , . . . , 2c2 , c1 ). ????????? ?????????
n
?
g1 (y) = f1 (y + ?) =
ds y s .
s=0
???????
p?1 ?(ndn , . . . , 2d2 , d1 ).
?????, ??????, ? = ?1 , . . . , ?r ????? ?????????
p??1 f1? (?) = p??1 d1 ? 0
(mod p).
??????????? ?????? ? ? ?????? ?????????? u2 = u2 (?, ?) ?? ???????
pu2 ?(pn dn , . . . p2 d2 , pd1 ).
???? l ? u1 ? u2 ? 2w + 1, ?? ??????? ???????????, ?????????? ?????? ?????? ?, ? ????? ???????? ?? ???? ????? u1 , u2 . ? ????????? ?????? ????????? ?
?????????? ????.
????, ?????? ?????? (?, ?, . . . ) ???????? ??????? ??????????? (u1 , u2 , . . . , ut ),
??? ????? ??????? t = t(?, ?, . . . ) ????????? ?? ???????
l ? u1 ? и и и ? ut?1 > 2w + 1,
l ? u1 ? и и и ? ut ? 2w + 1.
????? 5. ????? p ? ??????? ?????, f (x) ? ????????? ??????? n ? ?????? ??????????????, ??????? ???????? ? ???????????? ? p. ????? ??????????
??????? ??????????? (u1 , u2 , . . . ) ?????????? f (x) ?? ??????????? n.
??????????????. ??., ????????, ? [6], ?. 63, ????? 5.
????? 6. ????? n ? u1 ? u2 ? ut ? 2.
??????????????. ??., ????????, ? [6], ?. 63, ????? 6.
????? 7. ????? n ? 3, l ? ??????????? ?????, p ? ??????? ?????, f (x) =
= an xn + и и и + a1 x + a0 ? ????????? ? ?????? ??????????????, (an , . . . , a1 , p) =
= 1, ? ????? s ? ?????????? ????? t = t(?, ?, . . . ) ??????? ???????????
u1 , . . . , ut , ???????????? ????? ????????????? ????? 5. ????? ??? ?????
(
S(F ) = S
f (x)
;F
pl
)
p
?
(
l
=
x=1
????? ??????
|S(F )| ? npl?s .
F
f (x)
pl
)
312
?. ?. ?????????
??????????????. ????? ?????? ????? 7 ?????? ? [6], ?. 63, ??????? 3.
????? p? ?(nan , . . . , 2a2 , a1 ), ?.?. p? ? ????????? ??????? ????? p, ??????? ?????????? ????? ???????? ????? nan , . . . , 2a2 , a1 . ????? ? ? w = [ln n/ ln p]. ????
l ? 2w + 1, ?? ????? S(F ) ?????? ?????????? |S(F )| ? Cpl . ???? ?? l > 2w + 1,
?? ?????????? ????????? ????? ? ?????????
p?? f ? (?) ? 0 (mod p),
0 ? ? < p.
(6)
????? ?????????? ????? S(F ) ? ????
S(F ) =
p
?
(
p
?
l
Sv ,
Sv =
v=1
F
x=1
x?v (mod p)
f (x)
pl
)
.
??????????? ????? Sv ? ??????? ???????????
x = y + pl?? ?1 z,
??? ?????????? y ????????? ??? ?????? ?? ?????? pl?? ?1 , ? z ? ??? ?????? ??
?????? p? +1 . ???????
pl?? ?1
Sv =
y?v
(
p? +1 ?1
?
?
y=1
(mod p)
F
z=0
f (y) zf ? (y)
+ ? +1
pl
p
)
???????? ??? ????????: 1) v ?= ?, ?.?. v ?? ????? ?? ?????? ?? ?????? ? ????????? (6); 2) ??? ?????????? ????? ? ????????? (6) v = ?.
?????????? ?????? 1). ????? v ?= ?, ?.?. p?? f ? (v) ?? 0 (mod p). ????? ???
y ? v (mod p) ???????
(
p? +1 ?1
?
z=0
F
f (y) zf ? (y)
+ ? +1
pl
p
)
(
?
=
F
t=1
?????? ?????
(
p
?
l?1
Sv =
p
?
F
x=1
x?v (mod p)
f (x)
pl?1
f (v + pt)
pl?1
)
.
)
.
???????? ??? ????????? s ??? ???, ????? s = l ? 2w. ???????
p
?
(
2w
Sv =
x=1
x?v (mod p)
F
f (x)
p2w+1
)
.
?????????? ?????????? ???????? . . .
313
???????? ? ???????????? ?????? 2). ????? ?????? ?????????? y = ? + pz
?????
(
)
pl?1
?
f (? + pz) ? f (?) f (?)
S? =
F
+ l
.
l
p
p
z=1
???????
f1 (z) = p?u1 (f (? + pz) ? f (?)),
??? f1 (z) = cn z n +и и и+c1 z ? ?????????, ???????????? ???????? ? ????????????
??????? ?????? ? p, ???????
?
(
pl?u1
u1 ?1
S? = p
F
z=1
f1 (z) f (?)
+ l
pl?u1
p
)
.
???? l ? u1 > 2w + 1, ?? ????????? ????????? ?????. ?????
S? =
p
?
(
?
pl?u1
S?,v1 ,
S?,v1 = p
u1 ?1
v1 =1
z?v1
F
z=1
(mod p)
f1 (z) f (?)
+ l
pl?u1
p
)
.
???????? ??? ????????? ????????? ??? ?? ??? ??? ????
l ? u1 ? и и и ? ut > 2w + 1,
l ? u1 ? и и и ? ut ? ut+1 ? 2w + 1.
?????? ??????? ????????? ???????????.
??????? 3. ????? n ? 3, l ? 1 ? ??????????? ?????, p ? ??????? ?????, f (x) = an xn + и и и + a1 x + a0 ? ????????? ? ?????? ??????????????,
(an , . . . , a1 , p) = 1; ? ????? s ? ?????????? ????? t = t(?, ?, . . . , ?), ?????????? ??????? ??????????? (u1 , . . . , ut ). ????? ??????????? ??????
(
)
f (x)
? 2Cnpl?s .
|S(F )| = S
;
F
pl
5. ?????????????? ??????? 1
?????????? ???????
(
)???????? ?p = ?p (2k) ?????? ???????????? ????????f (x)
?????? ????? S pl ; F ????
?p = 1 +
+?
?
A(pl ),
l=1
???
pl ?1
l
A(p ) =
?
1 (
)2k
?? f (x)
?l
иии
; F d?0 ,
p S
l
p
pl ?1
an =0
a1 =0 0
(an ,...,a1 ,p)=1
f (x) = an xn + и и и + a1 x + ?0 ,
314
?. ?. ?????????
?????? ?????? ???????????? ?????????????? ????? S(F ) ????? ????? ???
(
S(F ) = S
f (x)
;F
pl
)
p
?
(
l
=
F
x=1
f (x)
pl
)
,
??? f (x) = an x +и и и+a1 x+?0 ? ????????? ? ?????? ?????????????? an , . . . , a1 ,
????????? ??????????? ?0 ? ???????????? ????? ? (an , . . . , a1 , p) = 1.
?????? ?????? ????? A(pl ). ???????????? ?????????? f (x) ??????????
? ???????? 0 ? an , . . . , a1 < pl ? ????????????? ??????? (an , . . . , a1 , p) = 1.
???????? ??? ?????????? ?? ?????? Ks , ?????????? ????? ??????? ????????
??????????? ????? s ??????? ??????????? u1 , . . . , us ??? ???????? ?????????
u1 , . . . us . ?? ???????? 2 ? 3 ?????
(
)
?l
f
(x)
p S
? 2Cnp?s
;
F
pl
n
?????? ?????? ?????? ?????????? ???????????, ????????????? ?????? Ks . ??
???????? (5) ? ??????????? ?????? ??????????????? ????????? ??? ??????????? f (x) ?? ?????? p ???????
(us ? 1)us
(u1 ? 1)u1
? иии ?
,
2
2
??? l ? 2w ? 1 ? l1 = u1 + и и и + us ? l.
???????????????? ??????? 5 ? 6, ???????
|Ks | ? pA ,
A = ns ?
(n + 1)n
(n + 1)n
+ n(l ? l1 ) ? s
+ n(2w + 1).
2
2
?????, ?????????? ???????????, ??????? ???????? ????? ???????????
(u1 , . . . , us ), ?? ??????????? ln ps . ?????????????,
{
}
?
l ? 2w ? 1
l
2k n n(2w+1) s(?2k+0,5n(n+1)+1)
A(p ) ?
(2Cn) l p
p
, s0 = max 1,
.
n
s?s
A?s
0
?????? ???????, ??? ?p ???????? ??? 2k > 0, 5n(n + 1) + 1.
??? p > n ??? ?p ?????????? ??? 2k ? 0, 5n(n + 1) + 1. ??? 0 ? an < pn ,
(an , p) = 1, . . . , 0 ? a1 < p, (a1 , p) = 1, ?????
(
)
pn
?
an n
a1
S(F ) =
F
x + и и и + x = F (0)pn?1 .
n
p
p
x=1
????????? ?p > ?(p) ? ???
pn ?1
?(p) =
?
an =0
(an ,p)=1
)2k
p?1 p?1 pn
?
? ? ( an
a
1
(x + c)n + и и и + (x + c) ...
F
n
p
p
a =0 c=0 x=1
1
(a1 ,p)=1
???????? ????????????, ??? ?p ????? ??????????.
?????????? ?????????? ???????? . . .
315
6. ??????????
? ????????? ?????? ?????????? ???????????? ?? ?????????? ??????? ???????? ?????????? ?????????? ?????? ???????????? ???? ? ????????, ??????????????? ??????????????? ????????? ???? (1). ??? ????????, ???????
??????????????? ????????? ????????????? ?????????? ????????. ?????????????? ?????????? ?????? ? ?????????? ???? ??????? ????? ???????????????
????????? ? ???????? ?????????????? ??????? ??????????????? ???????? ?
????? ???????? ?????????????? ????.
???????, ??? ???????? ????????? ????? ????? ??? ???????????? ??????????? ????
am xm + ar xr + и и и + an xn
,
pl
(am , ar , . . . , an , p) = 1, 0 ? am , ar , . . . , an < pl ,
f (x) =
??? 1 ? m < r < и и и < n, m + r + и и и + n < n(n+1)
.
2
?
??????? ???????? ?p ????? ??????????? ?????????????? ? ?????
?p?
=1+
+?
?
l=1
pl ?1 pl ?1
l
A1 (p ),
l
A1 (p ) =
? ?
)2k
(
?
f (x)
?l
.
иии
;
F
p
S
pl
pl ?1
am =0 ar =0
an =0
(am ,ar ,...,an ,p)=1
??????????? ????????? ???????????, ??? ?p? ???????? ??? 2k > m + r + и и и + n
? ?????????? ??? 2k ? m + r + и и и + n.
??? ????? ?????????? ???????? ?????? ?? ???????? ??????????? ???????
??? ??????? ???????????? ?????????????? ????. ???????, ??? ??? ??????
????? ????? ????? ??????
(
)
G(x
,
.
.
.
,
x
)
1
r
S
; F ln,r q r?1/n (? (q))r?1 ,
q
??? G(x1 , . . . , xr ) ? ????????? ? ?????? ??????????????, ? ???????????? ???????? ? q, ?????? ??????? ??? ?? ?????? ?????????? ?? ??????????? n. ??????
????????? ?????? ?????? ??? ?????????? ?????????? ???????? ???????? ??????? ?????? ???????????? ????. ??? ????? ??? nm/2, m = (n + 1)r . ?? ???
??????, ?????? ? ?????? ???????? ??????? ?????????? ?????????? ????????????
????? ??????? ???????.
?????? ???????????? ??????????
1. ?????????? ?. ?. ????? ?????????????????? ???? ? ?????? ?????, 2-?
???., ?.: ?????, 1980, 144 ?.
316
?. ?. ?????????
2.
Hua L.-K. An improvement of Vinogradov?s mean-value theorem and several
applications// Quart. J. Math. 1949. V.20. P. 48?61.
3.
??????? ?. ?. ??????? ? ??????? ???????? ?????? ??????? ?????????????????? ?????// ???. ???????. 1975. ?.17. ?. 84?90.
4.
??????? ?. ?. ????????? ?????. ????: ???-?? ?????????? ???. ??-??, 2013.
464 ?.
5.
??????? ?. ?., ????????? ?. ?. ??????? ?????????????????? ?????//???.
?? ????, ???. ???. 1976, ?.17, ?1. ?.209?220.
6.
??????? ?. ?., ???????? ?. ?., ????????? ?. ?. ?????? ??????? ?????????????????? ????. ?.: ?????, 1987. 368 ?.
7.
Arkhipov G. I., Chubarikov V. N., Karatsuba A. A. Trigonometric Sums in
Number Theory and Analysis. De Gruyter expositions in mathematics; 39.
Berlin, New York, 2004. 554 c.
8.
Franel J. Les suites de Farey et le probleme des nombres premiers// Go?ttinger
Nachrichten. 1924, S. 198?201.
9.
Landau E. Vorlesungen u?ber Zahlentheorie. Leipzig, 1927 V.2. 240 c.
10. ??????? ?. ?. ?????? ????? ? ?????????????? ??????: ??????? ??????.
?????: ???-?? ???. ??-??, 2013. 478 ?.
11. Greaves G. R. H., Hall R. R., Huxley M. N., Wilson J. C. Multiple Franel Integrals// Mathematika, 1993. V.40. P.51?70.
12. ????????? ?. ?. ?? ????? ??????? ?????????????????? ?????????// ????. ?? ????. 1976. ?.227, ? 6. ?. 1308?1310.
13. ????????? ?. ?. ? ??????? ???????????? ?????????????????? ?????? ?
??????? ??????????// ???. ???????. 1978. ?.20, ? 1. ?. 61?68.
14. ????????? ?. ?. ? ?????????? ?????????? ??????? ????????? ????? ??????????? ?????????? ????????// ????. ?? ????. 2015. ?.463, ? 5. ?. 530.
15. ????????? ?. ?. ?????????????? ????? ? ???????? ??????? ?????????//
??????????? ???????. 2015. ?.16, ???. 2(54). ?. 231?253.
16. ????????? ?. ?. ???????????? ????? ?????? ?????? ???????????? ?????????????? ????? ?? ??????????// ??????????? ???????. 2015. ?. 16, ???.
3(55). ?. 452?461.
?????????? ?????????? ???????? . . .
317
17. ????????? ?. ?. ?????? ???????????? ?????????????? ?????// ???????
????. ??-??. ???. ???., ???. 2016. ???. 1. ?. 60?61.
18. ????????? ?. ?. ?????????????? ????? ?? ???????? ???????????// ????.
???. 2016. ???. 466, ? 2. ?. 1?2.
19. ?????????? ?. ?. ?? ????? ?????? ???????????? ??????????, ???????
????. ??-??. ???. ???., ???. 2015. ???. 5. ?. 61?63.
REFERENCES
1. Vinogradov, I. M. Metod trigonometricheskikh summ v teorii chisel.(Russian)
[The method of trigonometric sums in the theory of numbers] Second edition.
?Nauka?, Moscow, 1980. 144 pp.
2. Hua L.-K. 1949, ?An improvement of Vinogradov?s mean-value theorem and
several applications?, Quart. J. Math. vol. 20, ??. 48?61.
3. Arhipov, G. I. 1975, ?A theorem on the mean value of the modulus of a multiple
trigonometric sum?, Mat. Zametki vol. 17, ??. 143?153.
4. Arkhipov, G. I. 2013, ?Izbrannye trudy?, [Selected works], Orl. Gos. Univ., Orel,
464 p. (Russian).
5. Arkhipov, G. I. & C?ubarikov, V. N. 1976, ?Multiple trigonometric sums?, Izv.
Akad. Nauk SSSR Ser. Mat. vol. 40, no. 1, ??. 209?220.
6. Arkhipov, G. I., Karatsuba A. A. & C?ubarikov, V. N., 1987, ?Theory of multiple
trigonometric sums?, Moscow: Nauka, 368 pp.
7. Arkhipov G. I., C?ubarikov V. N. & Karatsuba A. A. 2004, ?Trigonometric Sums
in Number Theory and Analysis 39?, De Gruyter expositions in mathematics,
554 pp.
8. Franel J. 1924, ?Les suites de Farey et le probleme des nombres premiers?,
Go?ttinger Nachrichten, ??. 198?201.
9. Landau E. 1927, ?Vorlesungen u?ber Zahlentheorie?, Leipzig, vol. 2, ?. 240.
10. Romanov, N. P. 2013, ?Teorija chisel i funkcional?nyj analiz: sbornik trudov?,
[Number theory and functional analysis: collected papers], Tom. Univ., Tomsk,
478 pp. (Russian)
11. Greaves G. R. H., Hall R. R., Huxley M. N. & Wilson J. C. 1993, ?Multiple Franel
Integrals? Mathematika, vol. 40, pp. 51?70.
318
?. ?. ?????????
12. C?ubarikov V. N. 1976, ?On a multiple trigonometric integral? Dokl. AN SSSR,
vol. 227, no. 6, pp. 1308?1310.
13. C?ubarikov V. N. 1978, ?Multiple rational trigonometric sums and multiple
integrals?, Mat. Zametki vol. 20, no. 1, ??. 61?68.
14. C?ubarikov V. N. 2015, ?Convergence exponent of singular integral in a multidimensional additive problem? Dokl. RAN, vol. 463, no. 5, p. 530.
15. C?ubarikov V. N. 2015, ?The arithmetic sum and gaussian multiplication theorem? Chebyshevskii Sb., vol. 16, no. 2(54), ??. 231?253.
16. C?ubarikov V. N. 2015, ?Elementary of the complete rational arithmetical sums?
Chebyshevskii Sb., vol. 16, no. 3(55), ??. 452?461.
17. C?ubarikov V. N. 2016, ?Full rational arithmetic sums? Vestnik Moskov. Univ.
Ser. Mat., Mech., vol. 1, ??. 60?61.
18. C?ubarikov V. N. 2016, ?The arithmetic sum of the values of polynomials? Dokl.
RAN, vol. 466, no. 2, ??. 1?2.
19. Shihsadilov, M. Sh. ?A class of oscillatory integrals? Vestnik Moskov. Univ. Ser.
Mat., Mech., vol. 5, ??. 61?63.
????????-?????????????? ????????? ??????????? ???????????????? ???????????? ??. ?. ?. ??????????.
????????? 01.12.2015.
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