Applied Mathematics E-Notes, 12(2012), 1-4 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
On A Sum Related By Non-trivial Zeros Of The
Riemann Zeta Function
Mehdi Hassaniy
Received 27 September 2010
Abstract
P
In this note we obtain explicit upper and lower bounds for the sum 0<
where is the imaginary part of the non-trivial zero = + i of (s).
1
1
T
,
Introduction
P1
The Riemann zeta function is de…ned for s 2 C with <(s) > 1 by (s) = n=1 n s ,
and extended by analytic continuation to the whole complex plan with a simple pole
with residues 1 at s = 1. A symmetric functional equation for (s) is
s
2
s
2
(s) =
1
s
2
1
s
2
(1
s);
R1
where (s) = 0 e t ts 1 dt is the Euler’s gamma function. (s) is meromorphic with
simple poles at s = 0; 1; 2; ::: (see [3]). By using these facts, we may see that
trivial zeros (zeros on real line R) of (s) are s = 2; 4; 6; ::: . Furthermore, we get
symmetry of non-trivial zeros (other zeros = + i with properties 0
1 and
6= 0) with respect to the vertical line <(s) = 1=2. Our intention in writing this paper
is to approximate the function
X
1
:
A(T ) =
0<
T
( +i )=0
More precisely, we show the following.
THEOREM 1. Let 1 = minf > 0 : ( + i ) = 0g
T > 1 we have
3
< A(T )
50
1
log2 T
4
log(2 )
log T
2
14:13472514. Then, for any
<
109
:
250
(1)
Our strategy to prove this result is to consider the zero counting function N (T )
de…ned by
X
N (T ) =
1;
0<
T
( +i )=0
Mathematics Subject Classi…cations: 11M06, 11M26.
of Mathematics, University of Zanjan, University Blvd., 45371-38111 Zanjan, Iran.
y Department
1
2
Estimates for Roots of the Riemann Zeta Function
and then translate known approximations about it to get desired approximations for
A(T ). The key for doing such translation is using Stieljes integration and integrating
by parts. Indeed, if we assume that 1 < U V and (t) 2 C 1 (U; V ) is a non-negative
function, then we have
Z V
Z V
X
N (t) 0 (t)dt + N (V ) (V ) N (U ) (U ):
(t)dN (t) =
( )=
U<
U
U
V
Among his various conjectures about the function (s) and its non-trivial zeros, B.
Riemann [5] claimed that
T
T
log
2
2
N (T ) =
T
+ O(log T ):
2
(2)
This conjecture of Riemann is proved by H. von Mangoldt more than 30 years later
[1, 2]. An immediate consequence of (2), which follows by partial summation, is the
asymptotic approximation
A(T ) = O(log2 T ):
To obtain a more precise approximation, we use the relation (2) by replacing ( ) = 1 ,
and putting 0 < U < 1 and V = T . We have
A(T ) =
Z
T
dN (t)
=
t
U
Z
T
U
N (t)
N (T )
dt +
:
2
t
T
(3)
We substitute N (T ) from (2) to obtain
1
A(T ) =
2
Z
T
t
2
log
U
t
1
dt
2
Z
T
U
T
dt 1
+
log
t 2
2
1
+O
2
Z
T
U
!
log T
log(t)
dt +O
:
2
t
T
Then, we simplify the right hand side of this relation, and we let U ! 1 . Therefore,
we get
log(2 )
1
log2 T
log T + O(1):
A(T ) =
4
2
Now, we are very close to the truth of Theorem 1. Our remaining duty is to estimate
the constant in error term O(1) in the last relation.
2
Proof of Theorem
The working engine of our paper is a result due to J. B. Rosser (Theorem 19 of [6]),
which asserts that
jN (T )
F (T )j
R(T )
(for T
2);
(4)
where
F (T ) =
T
T
log
2
2
T
7
+ ;
2
8
and
R(T ) =
137
433
397
log T +
log log T +
:
1000
1000
250
M. Hassani
3
This is indeed an explicit version of (2), and it allows us to obtain some explicit approximations of A(T ). In fact, by considering (4) and by using (3) with 2 U < 1 ,
for every T > 1 we obtain
Z
T
F (T ) R(T )
R(t)
dt +
2
t
T
U
A(T )
Z
T
U
F (t)
dt
t2
Z
T
R(t)
F (T ) + R(T )
dt +
:
2
t
T
U
A simple calculation shows
d
F (t)
=
2
t
dt
1 + log(2 )
log2 (2 ) 2 log(2 )
log t +
2
4
1
log2 t
4
7
8t
:
Also, we have
d
R(t)
433 log log t
137 log t
69
433
=
E(t) ;
2
t
dt
1000
t
1000 t
40t 1000
R 1 ds
1
where E(t) = 1 st
s . This integral converges for t > 1; in fact E(t)
t log t as t ! 1.
d
1
Moreover, by using the relation dt
E(t) = t2 log
t , we obtain
1
1
< E(t) <
t log t
t log2 t
1
t log t
Therefore, by letting U !
A(T ) <
1
log2 T
4
31
95t log2 t
(for t
we get the following explicit upper bound
1
log(2 )
137 log2 T + 433 log T
log T +cau
2
1000T log2 T
where cau = 0:43596427
T 2:222, we obtain
A(T ) <
2):
<
1
log2 T
4
109
250 .
433
(for T >
1 );
Since 137 log2 T + 433 log T > 433 is valid for
log(2 )
109
log T +
2
250
(for T >
1 ):
This completes the proof of the right hand side of (1). To prove the left hand side of
(1), we follow same steps to get
A(T ) >
1
log(2 )
log2 T
log T
4
2
3
274 log T + 866(log log T ) log2 T + 3313 log2 T + 433 log T
+ cal +
1000T log2 T
3
. We note that for T
for T > 1 , where cal = 0:06058187
> 50
in the above inequality is strictly positive. Thus, we obtain
A(T ) >
1
log2 T
4
log(2 )
3
log T +
2
50
(for T >
433
;
2 the last fraction
1 ):
4
Estimates for Roots of the Riemann Zeta Function
References
[1] H. Davenport, Multiplicative Number Theory (Second Edition), Springer-Verlag,
1980.
[2] A. Ivić, The Riemann Zeta Function, John Wiley & sons, 1985.
[3] N. N. Lebedev, Special Functions and their Applications, Translated and edited by
Richard A. Silverman, Dover Publications, New York, 1972.
[4] A. Odlyzko,
Tables of zeros of the Riemann
http://www.dtc.umn.edu/~odlyzko/zeta tables/
zeta
function:
[5] B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse (On
the Number of Prime Numbers less than a Given Quantity), Monatsberichte der
Berliner Akademie, November 1859.
[6] J. B. Rosser, Explicit bounds for some functions of prime numbers, Amer. J. Math.,
63(1941), 211–232.