Hardy function $Z(t)$ takes real values for real values of $t$ and real zeros of $Z(t)$ are the zeros of $\zeta(s)$, that are on the critical line.
The first result of the zeros of the Riemann zeta function on the critical line is the G.Hardy theorem. In 1914 he proved that $\zeta(1/2+it)$ has infinitely many real zeros. Then Hardy and Littlewood in 1921 proved that the interval $(T, T+H)$ when $H\ge T^{1/4+\varepsilon}$ contains odd order zero of function $\zeta(1/2+it)$. Jan Moser in 1976 proved that this assertion holds for $H\ge T^{1/6}\ln^2T$. In 1981 A.A.Karatsuba proved Hardy-Litllvud theorem already for $H\ge T^{5/32}\ln^2T$.
In 2006 Z.Kh.Rahmonov, Sh.A.Khayrulloev have reduced a problem of the magnitude of the interval $(T, T+H)$ of the critical line, which contains an odd order zero of the zeta function to the problem of finding an exponential pairs for estimating the special trigonometric sums.
In 2009 Z.H.Rahmonov, Sh.A.Khayrulloev find the lower bound of $\theta_1(k, l)$ on $\mathcal{P}$ – the set of all exponential pairs $(k, l)$ of different from $(1/2, 1/2)$ and having the form: $$ \mathop {\inf }\limits_{(k,l) \in \mathcal{P} } \theta _1 (k;l) = R + 1, $$ where $R = 0.8290213568591335924092397772831120\ldots $ – Rankin constant.
In 1981 A.A.Karatsuba has studied the problem of neighboring zeros of the function $Z(t)$ together with the problem of the neighboring extremum points or of inflection points of the function $Z(t)$ or in a more general substitution – of the neighboring zeros of functions $Z^{(j)}(t)$, $j\ge 1$.
The main result of our work is to reduce the problem of the magnitude of the interval $(T, T+H)$ of the critical line, which is known to be a odd order zero of the function $Z^{(j)}(t)$, ($j\ge 1$) to the problem of finding the exponential pairs for estimating the special trigonometric sum and to improve the A.A.Karatsuby theorem for $j=1$.
