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On the small values of the Riemann zeta-function on the critical line M.A.Korolev

Moscow Steklov Institute e -mail:

hardy_ramanujan@mail.ru, korolevma@mi.ras.ru

)

Let tn ,

n = 1, 2, 3, . . .,

b e the sequence of Gram p oints. It is known that

tn n
where

2 n , ln n

n +, (s)
lying

n

denotes

n

th ordinate of complex zeros of the Riemann zeta-function

in the upp er half -plane, and, moreover, that the values connected with the b ehavior of



(s)

1 at the sequence 2

+ it + itn

1 2

n

are real. The problems

b elongs to the ѕdiscrete

theory of the Riemann zeta-functionї. The recent result due to J. Kalp okas, J. Steuding and the sp eaker shows that there exist some p ositive constants c1 and c2 such that the following relations hold:

max
n N

1 2

+ it

n

c1 ln N
1 2

5/4

,

n

min
N

1 2

+ it

n

- c1 ln N

5/4

,

max
n N

+ it

n

c2 exp

1 2

-

ln N ln ln N

,

as

N

grows unb oundedly. This theorem means that the p oints of the intersection of the

curve

t

1 2

+ it

with the real axis can ѕrunї very far to the right or to the left from

the origin. such that At present, nothing is known ab out the existence or non-existence of the p oints tn 1 + itn = 0. The main goal of the talk is to describe some recent results of 2

the sp eaker concerning the existence of ѕvery smallї values In particular, the following result will b e discussed:



1 2

+ itn

.

lim inf
n+

1 2

+ itn e

(n)

= 0,

where

(n) = (ln ln n)

1-

.

1