_2014, 57, 4_

511.325
..

.
( - .. 15.01.2014 .)
, .
: - - - - .
Z (t)
Z(t) = e(t)({1 + A e'(t) = + f
rf I+1
I 4 2 ,
-1
/ Z ) () , .
- () . [1]. 1914 . , ^(1 / 2 + ) . [2] 1921 . , (, + ) > 4+ ^(1 / 2 + ) . [3] 1976 . , > 12 . 1981 . .. [4] - > 53212 .
[5] (, + ) , -, , : (,I) - , (1/2,1/2),
1
1 ^
2
2-&-1(k; l)
l
0(k; l) = - 1 -- , 01(k; l) = , (1)
0.5 - k
: . , ., . , 299/4, . E-mail: shamsullo@rambler.ru
(, + ), > 0 > 0, > (')12 - .
, ( ; I) 1 (; I) .. > 5/32 12 (1)
^=(14#7=18(3)-
(, | = = 0.15625. ^14 14) 32
[6] 6{\) V - (, I) , (1 / 2,1 / 2) :
1(;) = 11 +1,
(,1)
R = 0.8290213568591335924092397772831120... - , , -, .
.. [4] Z ^) Z (^) - Z(1 )(^), 1 > 1. , 1 , Z(^(1), :
1. ] - , > 0(1) > 0. (, + )
> 16 (1 )11, = ( 1) > 0
Z(1)(^).
[7] (, + ) , Z() (1 > 1) , , :
2. (, I) - , ]- ,
1 (; I) = 1
'1- 1 ^
2 --1 (; I)
(; I) = 1 + 1
0.5 - +1
ͻ'(]+1, >0) > 0 (, + ) Z(1 ).

..
, 011,21 = 1 , .. 6 3 / 6 + 6
2 (, I) = ,21. , = 1
, 1,21 = -1. (2)
1 ^63 ) 12
, [8] [6,9,10] .
3. - (,1)
I)= 1+1
- +1.5


inf /) = $f,J = 1.
(./M 1V 11 106 106 J 146'
13 75 ^ = aba2ba2 '1 1
06 106) \2 2,
1. vx - (k,l),
inf (;/) = fil .-71V il = -1
(./M 1V 1 ^ 106 106 J 432 12 2592
, (2), 1, ...
15.01.2014 .

1. Hardy G.H. Sur les zeros de la fonction (s) de Riemann. - Compt.Rend. Acad.Sci-1914.-v.158.-P.1012 - 1014.
2. Hardy G.H., Littlewood J.E. The zeros of Riemann's zeta-function on the critical line. Math.Z.-1921.-Bd 10.-S.283 - 317.
3. . - . Acta arith., 1976, 31, S. 31 - 43.
4. .. - , . . 1981.-. 157..49 - 63.
5. .., .. - , . , 2006, . 49, 5, . 393 - 400.
6. .., .. - , . , 2009, . 52, 5, . 331 - 337.
7. .. Z(t), j >1. , 2006, . 49, 9, . 803 - 809.
8. Graham S.W. Kolesnik G. Vander Corput's Method of Exponential sums. Cambridge university press. 1991, Cambridge, New Vork, Port Chester, Melbourne, Sydney.
9. .. - . , 1994, . 49, 2, . 161 - 162.
10. .. - . - . 2006. . 7. . 1. . 263 - 279.
..
^ ^^

. ^ ^
, , , ,, , .
^ : - ^ - - - .
Sh.A.Khayrulloev
ON THE DISTANCE BETWEEN CONSECUTIVE ZEROS OF THE FIRST ORDER
DERIVATIVE OF THE HARDY FUNCTION
A.Juraev Institute of Mathematics, Academy of Sciences of the Republic of Tajikistan
The lower bound of the gap length in the critical line, which includes the odd-order zero of the first derivative of Hardy function, is found by the method of optimization of exponential pairs. Key words: Hardy function - exponential pair - the Riemann zeta function - critical line.