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Венский доклад о количестве нулей дзета-функции Римана на коротких промежутках критической прямой.

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??????????????????????
??? 511
??????? ??????: ? ??????????
????? ?????-??????? ?????? ??
???????? ???????????
??????????? ??????1
?. ?. ???????? (?. ??????)2
?????????
??????, ????????? ?? ???????? ?. ?. ??????? ??????? ??????????????? ??????? ??????????????? ?????????? ???????????? ????????????
???? 13 ???? 1994 ?.
???????? ?????: ?????-??????? ??????, ???????? ??????, ????????
?????????.
????????????: 13 ????????.
UDK 511
Vienna Talk: On the Number of Zeros of the
Riemann Zeta Function in Short Intervals of the
Critical Line3
A. A. Karatsuba4
Abstract
The invited talk presented at the seminar of Prof. P.M. Gruber at the
Chair of Mathematical Analysis of the Department of Mathematics of Vienna
University of Technology at the June 13, 1994.
Keywords: Riemann Zeta Function, the Riemann hypothesis (RH), the
Selberg hypothesis (SH).
Bibliography: 13 titles.
1
?????? ??????????? ? ?????? ?. ?. ?????????.
???????? ?????????? ???????? (31.01.1937 ? 28.09.2008) ? ?????? ??????-?????????????? ????, ?????????, ?????????? ??????? ?????? ????? ??????????????? ?????????
??. ?. ?. ???????? ???, ????????? ??????????? ???????????????? ???????????? ????? ?.
?. ??????????.
3
The report was prepared for publication E. A. Karatsuba.
4
Anatoly Alekseevich Karatsuba (01.31.1937 ? 28.09.2008) ? Doctor of Physical and Mathematical Sciences, Professor, Head of Department of Number Theory Mathematics Institute. Russian
Academy of Sciences, professor of the Moscow State University named after M. V. Lomonosov.
2
20
?. ?. ????????
??????? ??????: ? ?????????? ????? ?????-???????. . . 21
� ??? ?????? ????? ? ????? ??????? ?????? ?(s), ??????? ?? ???????????
??????. ????? ???? s = ? + it, i2 = ?1, ?, t ? ???????????? ?????.
??????? ?(s) ??? ? = Res > 1 ???????????? ????? ???????:
?
?
1
?(s) =
.
ns
n=1
???? ? ?????? ??????? ??? ?(s) :
)?1
?(
1
1? s
?(s) =
,
p
p
(1)
p ? ???????.
(2)
??????? (2) ?????????? ???????? ?????? ??? ????????? ?????????????.
??? Res > 1 ??????????? ?????????:
? ?(
(s)
)
s+1
s?2
1
? 2s
? ?
?(s) =
+
x? 2 + x 2 ?1 (x)dx,
(3)
2
s(s ? 1)
1
??? ? (s) ? ?????-??????? ??????,
?1 (x) =
?
?
e??xn .
2
n=1
?????? ????? (3) ?????????? ??? ????? s. ????????????? ??????? (3) ?????????? ?(s) ?? ??? s -?????????.
?? (3) ??????? ??????????? ?????????????? ????????? ?(s) :
(s)
s
g(s) = g(1 ? s), g(s) = ? ? 2 ?
?(s).
(4)
2
� ????????, ??? ??? ??????????? ???? ?(s) ????? ? ?????? 0 ? Res ? 1
(????????????? ??????). ???????? ???????? (RH) ??????????, ??? ??? ??? ?????
?? ????? ?????? Res = 12 (????????????? ??????).
????? T ? 2, N (T ) ? ?????????? ????? ?(s) ? ?????????????? 0 ? Res ? 1,
0 < Ims ? T, N0 (T ) ? ?????????? ????? ?(s) ?? ??????? Res = 21 , 0 < Ims ? T.
????, ???
N0 (T ) ? N (T ).
(5)
RH ??????????, ???
N0 (T ) = N (T ).
??? N (T ) ??????????? ??????? ??????-??????????, ??????????? ?????? ?
???????? N (T ) ??? T ? +? :
N (T ) =
T
T
T
log
?
+ O (log T ) .
2?
2? 2?
22
?. ?. ????????
� ? 1942 ?. ?. ???????? [1] ??????? ??????? A:
??????? 1. (??????? A.)
N0 (T + H) ? N0 (T ) ? c (N (T + H) ? N (T )) ,
??? H = T a ,
1
2
(6)
< a ? 1, c ? 10?6 .
?? (6) ??????? ??????? B:
??????? 2. (??????? B.)
N0 (T ) ? cN (T ).
(7)
? [1] ???????? ????? ??????? C:
??????? 3. (??????? C.) ?If ?(t) is a function of t which is positive and
increases
(
) to infinity with t, then for ?almost all? t > 0, there is at least one zero of
? 21 + it between t and t + ?(t)
; that is: the measure of those t, 0 < t ? T, for which
log t (
)
there is no zero in the interval t, t + ?(t)
, is o(T ).?
log t
?? ??????? C, ? ?????????, ???????:
N0 (T ) >
N (T )
.
?(T )
(8)
????, ??? (8) ?????? (7).
? ????? ? ???????? A ???????? ????? ? [1] (?. 5):
?The result of the present paper do not pretend to be best which can be obtained
by these and similar methods. In fact, several things seem to suggest, that the
condition a > 12 of Theorem A and Theorem D, if we use still more sophisticated
arguments, may be replaced by a > ? where ? < 12 .?
??????????? (6) ? H = T a , a > ?, ? < 12 , ???????? ??? ????????? ??????????
(SH).
� ? 1984 ?. ????? ??????? (??. [2]) ??????? D:
??????? 4. (??????? D.) ??????????? (6) ??????????? ???
H = T a,
a>?=
27
1
1
= ?
.
82
3 246
????? ???????, ??????? D ???? SH ? ? =
????? ?? ????? ??????? ??????? E ([3]):
27
.
82
??????? 5. (??????? E.) ??? ????? ???? T ??????????? (6) ???????????
??? H = T a , 0 < a ? 1, a ? ?????.
??????? ??????: ? ?????????? ????? ?????-???????. . . 23
??????? ???????, ??? ????? ???? T SH ??????????? ??? ? = ? > 0, ? ?
????? ?????? ????.
???????? ???????? ?????? C ? E. ??????, ??????? E ? ???????????? ?????? ??????? ??????? C, ??? ??? ?? ??? ?????????? (7), ? ?? ????? ??? ??
??????? C ??????? ?????? (8).
� ??? ?????????????? SH ? ????????? ???????? ????????????? ???? ?????????????????? ????. ?????? ?????? ????? ????, ??????? ?????? ??????????,
? ???? ??????? ???????? ?, ? = 27
, ? ???? ???? ???????, ???? ?????????
82
??????????? ?????? ????????-??????-?????????-??????.
? ??????? ??? ?????????????????? ????? ????? ???????? ??????, ?? ????
????????? ?????????. ??? ???? ??????????? ??? ????? ???? T ????????
? = ? > 0, ? ? ????? ?????? ????, ??????? ???????, ???????? ??????? E.
� ??? ?????????????? ??????? E ? ??????????? ????????? ???, ??? H = T a ,
a > 0, a ? ?????????????.
? ???? ??????? ??????-?????????? ?? RH ???????, ??? ?????? ???????)
(
??? (T, T +H), ??? H ? c1 > 0, ???????? ?? ?????? ??? cH log T ????? ? 12 + it .
??????? ??????????? ??????? ??????? ???????? ??????????? ??????? E ???
H ??????????? ???????, ??? T a .
? 1992 ?. ? ??????? ??????? F ([4]):
??????? 6. (??????? F.) ??? ????? ???? T ? ??? H ? ????????
(
)
?
?
3
exp exp log log T ? H ? T
(9)
??????????? ???????????
(
N0 (T + H) ? N0 (T ) ? cH log H exp ?a1
?
log T
log
log H
)
.
(10)
????????? ??????? F:
????????? 1. ??? ????? ? > 0, ?1 > 0, ? H ? ????????
exp (log? T ) ? H ?
?
3
T,
??? ????? ???? T ??????????? ??????:
N0 (T + H) ? N0 (T ) ? H (log H)1??1 .
(11)
??????????? (10) ? (11) ??????, ??? (6), ??? ??? ?????? ????? (6) ???? ???????? ??????? H log T. ?? (10) ? (11) ????? ????? ????, ??? ?????????? ??????
??? ????? ????? H.
24
?. ?. ????????
� ??????????? ?? ??????? ??????? ??????, ???????????? ????????
??????? F. ?????? ?????, ?????? ????????? ????????? ?????????.
????
( ?(s),
) ??????? ?? ??????????? ??????, ??? ???????????? ???? ??????? ? 12 + it . ????????? ???????????? ??????? ? (t) ??????????
( 1?s ) ? 1?s
2 ?
?
( s 2) ei?(t) =
.
? 2s
? ? 2 1
s= 2 +it
?????, ? ???? (4), ??????? Z(t),
(
)
1
Z(t) = e ?
+ it ,
2
??? ???????????? t ????????? ???????????? ????????. ?????????????, ???????????? ???? Z(t) ? ??? ???? ?(s), ??????? ?? ??????????? ??????.
?? ??????? ????????? ??? ?(s) ???????:
( )
t
t
t
7
1
?(t) =
log
?
+ +O
,
(12)
2?
2? 2? 8
t
t ? t1 > 0.
???? ??????????? ???????????:
? t+h
? t+h
|Z(u)| du > Z(u)du ,
(13)
i?(t)
t
t
?? ??????? Z(u) ????? ???? ????????? ??????? ?? ????????? (t, t + h) (????
??????????????????????).
� ????? h > 0, h ? ????????? ????????. ????? E ? ???????????? ?????
t ? (T, T + H) ?????, ??? ??? ??? ??????????? (13). ???? �(E) ? ???? E, ??
??? ?????????? ????? Z(t) ?? (T, T + H) ??????????? ??????
1 �(E)
.
(14)
2 h
?????????????, ???? ????? ?????? ????????? ????? �(E) ??? ???????? ??????? h. ???? ?? ??????? ?????? ????? �(E) ??????? ? ?????????. ?? ??????????? ????????? E ???????? ? ??????? ???????????:
)
? T +H (? t+h
I1 =
|Z(u)| du dt =
N0 (T + H) ? N0 (T ) ?
? (?
=
E
T
t
? (?
=
E
t
)
? (?
t+h
|Z(u)| du dt +
t
t+h
E?
|Z(u)| du dt =
t
? ?
|Z(u)| du dt +
)
)
t+h
E?
t
t+h
Z(u)du dt ?
??????? ??????: ? ?????????? ????? ?????-???????. . . 25
? I2 + I3 ,
(15)
???
? (?
t+h
I2 =
E
)
|Z(u)| du dt,
?
T +H
I3 =
t
T
?
t+h
t
Z(u)du dt.
??? ?????, ???????? ? I2 ??????????? ????, ????????
I1 ? I3 ?
?
(16)
�(E)J2 ,
???
?
T +H
(?
)2
t+h
J2 =
Z(u)du
T
dt.
t
????????? ??????????? (16) ???? ?????? ????? ??? �(E). ???? ?????? ?????
????????????? ?????? ?????? ??? ???????? ?????????? I1 ? I3 . ????????? I1
? I3 ????? ??????. ??? ?? ???? ??????????? ?????????? ?? ????????? ??? ????????? ??????? ????????? ??????????????. ??? Z(t) ???????? ??????? ???????
??????:
Z(t) =
( 1
)
? cos (?(t) ? t log n)
?
+ O t? 4 log t .
n
?t
n?
(17)
2?
?? ??????? (12) ??? ?(t) ? (17) ?????, ??? ????????? ????? ? Z(t) ???????????. ? I1 ?????????? ???????? ??????? ?? |Z(u)|, ? ?? ????? ??? ? I3 ? ??
Z(u). ??? ?????????????? ? ????????? ?????????? ??????? I1 ?? I3 . ????? ??
?????, ??? ??? ?????? ???????? ????????? h, ???, ????????, ????? ???????
?????????? Z(u). ????, ?????:
(
) ( 1 )
? 1
1
? n?iu + O T ? 2 ;
|Z(u)| = ?
+ iu = 1 +
2
n
2?n?T
)
T +H (
1
?
I1 ? h
+ iu du ?
2
T +h
?
(
) T +H
( 1)
?
1
? n?it + O T ? 2
1+
? h
dt ? hH (1 + o(1)) .
T
n
?
(18)
2?n?T
?????, ??? ??????? ???? ????????? ?? ?????????, ?????? 1. ??? ?????????
????????? ?1n n?it , n ? 2, ?????? ??????????? ? ????? ?????????????? ????
1
???????. ????? ????? ?????, ??? H ?????? ???? ????????, ?????? H > T 6 ,
????? ?????? ????????? ????? ?????????????????? ?????????.
???????? I3 ????????? ??????, ???????? ??????????? ????, ?????????????? ??????????????? ?? ?????????? ?????????:
26
?. ?. ????????
?
I3 ?
?
T +H
(?
)2
t+h
H
Z(u)du
T
dt.
t
???? h ? H ?? ????? ????, ?? ??? I3 ??????? ???????? ??????:
1
I3 ? hH.
2
?????? ? ?? (16) ???????:
?
1
(19)
hH ? I1 ? I3 ? �(E)J2 ,
2
???????? J2 ? (19) ??????????? ????? ??????????? ??????????? ????.
?????????? ????? ??????????? ??????????????????????. ?? ??? ??????????? (??. [5]) ????? ? ????????? ? 1921 ?. ???????? ???????????:
N0 (T + H) ? N0 (T ) ? cH,
H = T a,
1
a> .
2
� ???????? ????? ??????????????????????, ?? ????????? ??? ???????????? ???????????? ???? ????:
(?
)2
T +H
F (t)dt
?
T +H
?H
T
F 2 (t)dt.
(20)
T
????????, ??? ??????????? ???? ???????? ??????, ???? F (t) = const. ???????,
??? ????? F (t) ?? (T, T + H) ? ?????????, ??? ?????? (20). ??? ???????????
? ??????? ??????? ???? ???????????? ?. ?????????? ? 1942 ?. ?????? Z(t) ??
?????????? ??????? F (t),
(
)2
1
F (t) = Z(t) ?
+ it ,
2
?????? ?(s) ??????????? ???, ????? F (t) ? ??????? ???? ?????? ? ?????????.
??? ????????? ??? ???????? ??????? A, B, C (??. [1]). ??? ?????? ?(s) ?.
???????? ??????????? ?????????? ???, ??? ?(s) ????? ???????? ????????????.
????? ?????,
(
)
{
log ?
? ?(?)
?(?) 1 ? log
, ?<X
X
?(s) =
, ?(?) =
s
?
0,
? ? X,
??X
? ????? ?(?) ???????????? ??????????:
?
?
?(?)
?=1
?s
)
?(
1 2
=
1? s
,
p
p
1
Res > 1,
??????? ??????: ? ?????????? ????? ?????-???????. . . 27
X ? 1 ? ????????? ????? (???????? ??????).
??? ???????????? ??? ????????, ???? ????????????? ???????, ????????
F (t), ??????????? ???? ? ??????.
?The underlying idea of our methods is to introduce an auxiliary function, which
to a certain extent neutralizes the peculiarities of |?(s)| on the line ? = ?0 ; in this
paper we have always ?0 = 12 . This idea, with ?0 > 12 , was first invented by Bohr
and Landau in their researches concerning N (?, T ) for fixed ? > 21 , and has later
been used by other authors in connection with the same problem? (??. [1], ?.5).
????? ????????, ??? ??? H = T a , a > 12 ,
? T +H
?1
|Z(t)|2 dt ? log T ;
(21)
H
H ?1
?
T
T +H
|F (t)|2 dt ? const.
(22)
T
?
?????????????, |Z(t)| ? ??????? ?? (T, T + H) ???? log T , ? |F (t)| ???? ?????????.
??? ??? ?????????, ??? ????? ?. ????????? ???????? ?????? ? ????????,
??????? ???????? ????????????.
�. ????????? ??????????, ?????????? ? ?????? ?????????????? ?????
???????, ?? ??????? ???????? ????????????, ?? ??????? ???????? ?????????, ? ????? ????? ?????, ??????????? ????????????????? ????????? ?????
?????????? ????? ????? ????? ?? ???????? ??????????? ??????. ??????? ????
??????? ? ??????? ??????? F.
?????? ???????? ?????? ?? ????? ??????? ??????? (??. [12], [13]).
????? ?? ????? ??????? ?????????????? ????? ??????? ???????? ??????? ??????????-??????????? f (s),
f (s) =
?
?
r(n)
n=1
ns
,
Res > 0,
(23)
??? r(1)
? =? 1, r(2) = ?, r(3) = ??, r(4) = ?1, r(5) = 0, r(n + 5) = r(n),
10?2 5?2
?
?=
.
5?1
??????? f (s) ????????????? ????????? ???????? ????:
( ? )? 2s ( s + 1 )
?
g(s) = g(1 ? s), g(s) =
f (s).
(24)
5
2
??? f (s) RH ?? ????? ?????. ?. ????????? ? ?. ?????????? (??. [6]) ????????,
??? f (s) ????? ?? ??????, ??? cT, c > 0, ????? ? ?????? Res > 1, 0 < Ims ? T.
?.?. ??????? ???????, ??? ? ?????? ?????????????? ???? 21 < ? < Res ? ? <
1, 0 < Ims ? T, ??????? f (s) ???????? ?? ??????, ??? cT ?????, c = c(?, ?) > 0
(??. [7]).
28
?. ?. ????????
??? ?? ?????, ??????????? ?????? Res = 12 ???????? ?????? ??????????
??? ????? f (s). ??? ??????? ? 1980 ?. ?.?. ??????? (??. [8]), ?? ??? ?????????? ????????? ????? ????? f (s). ????? ?????, ??? N0 (T ), ?????????? f (s),
??????????? ??????:
(
)
1?
(25)
N0 (T ) ? T exp
log log log log T .
20
??????? ????????? ????????? ???????????? ?. ??????? (??. [9]) ????? f (s) ?
?????-??????? ????????. ?. ?????? ? [9] ????????, ??? ??????????? (25) ???
???????? ?. ???????? ??? ? 1942 ?. ????? ????, ?. ???????? ? ?. ?????? (??.
[10]) ??????? ???????? ??????? ? ???, ??? ????? ??? ???? f (s) ????? ?? ??????????? ??????. ??? ???????????? ???????????? ??????????? RH ? ??? ????
???????? ? ???????????? ????? L -??????? ???????. ???????, ?. ????????
? [11] ???????, ???
? ??? ??????????? ?????? ? ?????? 0 < Ims ? T ????? ??
??????, ??? cT log log T ????? f (s).
??? ????? ???????????? ??????? f (s) !
? 1989 ?. ? ??????? ????? ??????? ?????? (25) ????? ??????????? (??. [12] ?
[13]):
( ?
)
?
N0 (T ) ? T log T exp ?c log log T .
(26)
???????, ??? (26) ????? ??????? (25). ??? ?? ?????????? (26)?
??????? f (s) ????? ??????????? ??? ???:
1 ? i?
1 + i?
L(s, ?) +
L(s, ??),
(27)
2
2
??? ? = ?(n) ? ???????? ??????? ?? ?????? 5 ?????, ??? ?(2) = i.
???? f (s) ?? ????? ???????? ????????????. ?? ?????? ????????? ? (27) ????? ???????? ????????????, ??? ???
f (s) =
)?1
?(
?(p)
L(s, ?) =
1? s
.
p
p
????????, ??? L(s, ?) ? L(s, ??) ??????? ?? ????? ????? ????? ????????? ????????????, ??????, ?? h(s),
?
h(s) =
p?1
(mod 5)
(
)?1
1
.
1? s
p
?????????????, ?? h(s) ??? Res > 1 ??????? f (s). ??????? ??? f (s) ?????
????? ????? ?(s), ??? ??????? F (t),
) (
)2
(
1
1
i?(t)
+ it ?
+ it ,
F (t) = e f
2
2
??????? ??????: ? ?????????? ????? ?????-???????. . . 29
?
???? ? ?? ????? ? ??????? ??????????, ?? ????? ????? ??????, ??? log t (?????? ???? ?
?. ?????????). ????? ?????, ????????? F (t) ? ??????? ???? ????????
??????? 4 log t. ??????, ????? ??? ?? ??????????, ????? ???????? (26). ?????? ???????????? ???????????? ???????? ????? ?????????? ??????? ???????
??????????, ?? ???? ?????? ??????????? ?????-?????????-?????? (13) ????????????? ????????? ???????????:
?
(? t+h
)? ? t+h
|F (u)|du
> F (u)du ,
t
t
??? ? ? 0 ??? t ? ?.
?? ???? ???? ???? ??????????? ?? ? ???????????? ??????? ??????????????
?????????? ???????????? ??????????? ???? ? ???????? (26). ???? ?? ?????
??? ???????? ???? ? ?????????????? ??????? F.
?????? ???????????? ??????????
1. Selberg, A. On the zeros of Riemann?s zeta-function // Skr. Norske Vid. Akad.
Oslo, 1942, nr. 10, pp. 1?59.
2. ???????? ?. ?. ? ????? ??????? ?(s) ?? ???????? ??????????? ??????????? ?????? // ???. ?? ????. ???. ?????., 1984, ?. 48, ? 3, ?. 569?584.
(
)
3. ???????? ?. ?. ????????????? ????? ??????? ? 21 + it // ???. ?? ????.
???. ?????., 1984, ?. 48, ? 6, ?. 1214?1224.
4. ???????? ?. ?. ? ?????????? ????? ?????-??????? ??????, ??????? ??
????? ???? ???????? ??????????? ??????????? ?????? // ???. ?? ????.
???. ?????., 1992, ?. 56, ? 2, ?. 372?397.
5. Hardy, G. H., Littlewood, J. E., The zeros of Riemann?s zeta-function on the
critical line // Mathematische Zeitschrift, 1921, Vol. 10, Issue 3?4 , pp. 283?317.
6. Davenport, H. and Heilbronn, H., On the zeros of certain Dirichlet series // J.
London Math. Soc. 11, 1936, pp. 181?185 and pp. 307?312.
7. ??????? ?. ?., ? ????? ?????-??????? ???????????? ???? // ?????? ?????, ?????????????? ?????? ? ?? ??????????, ??????? ??????. ??????????? ????????? ????? ?????????? ??????????? ? ??? ?????????????????????, ????? ???? ????, ?. 142, 1976, ?. 135?147.
8. ??????? ?. ?., ? ????? ????????? ????? ???????, ??????? ?? ??????????? ?????? // ???. ?? ????. ???. ?????., 1980, ??? 44, ? 1, ?. 63?91.
9. Hejhal, D.A. Zeros of Epstein zeta functions and supercomputers // Proceedings
of the International Congress of Mathematicians 1986 (Berkeley), pp.1362?1384.
30
?. ?. ????????
10. Bombieri, E. and Hejhal, D.A. Sur les ze?ros des fonctions ze?ta d?Epstein //
Comptes Rendus Acad. Sci. Paris 304, 1987, pp. 213?217.
11. Selberg, A. Old and new conjectures and results about a class of Dirichlet series
// Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori,
1989), Univ. Salerno, Salerno, 1992, pp. 367?385.
12. ???????? ?. ?. ? ????? ??????? ??????????-???????????, ??????? ??
??????????? ?????? // ???. ?? ????. ???. ?????., 1990, ?. 54, ? 2, ?.
303?315.
13. ???????? ?. ?. ? ????? ?????????????? ????? ???????, ?? ??????? ???????? ???????????? // ???. ???. ???. ?????., 1993, ?. 57, ? 5, ?. 3?14.
REFERENCES
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)
(
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??????? ??????: ? ?????????? ????? ?????-???????. . . 31
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????????? 10.02.2015
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