Arithmetic equivalent of essential simplicity of zeta zeros
Authors: Mueller Julia 
Published: June 2010

Let

R
(
x
)

R(x)

and

S
(
t
)

S(t)

be the remainder terms in the prime number theorem and the Riemann-von Mangoldt formula respectively, that is

ψ
(
x
)
=
x
+
R
(
x
)

\psi (x) = x + R(x)

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis t right-parenthesis equals left-parenthesis 1 slash 2 pi right-parenthesis integral Subscript 0 Superscript t Baseline log left-parenthesis tau slash 2 pi right-parenthesis d tau plus upper S left-parenthesis t right-parenthesis plus 7 slash 8 plus upper O left-parenthesis 1 slash t right-parenthesis">

N
(
t
)
=
(
1

/

2
π
)

∫
0
t

log
⁡
(
τ

/

2
π
)

d
τ
+
S
(
t
)
+
7

/

8
+
O
(
1

/

t
)

N(t) = (1/2\pi )\int _0^t {\log (\tau /2\pi )\,d\tau + S(t) + 7/8 + O(1/t)}

. We are interested in the following integrals:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J left-parenthesis upper T comma beta right-parenthesis equals integral Subscript 1 Superscript upper T Superscript beta Baseline Baseline left-parenthesis upper R left-parenthesis x plus x slash upper T right-parenthesis minus upper R left-parenthesis x right-parenthesis right-parenthesis squared d x slash x squared">

J
(
T
,
β
)
=

∫
1

T
β

(
R
(
x
+
x

/

T
)
−
R
(
x
)
)

2

d
x

/

x
2

J(T,\beta ) = \int _1^{{T^\beta }} {{{(R(x + x/T) - R(x))}^2}dx/{x^2}}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis upper T comma alpha right-parenthesis equals integral Subscript 1 Superscript upper T Baseline left-parenthesis upper S left-parenthesis t plus alpha slash upper L right-parenthesis minus upper S left-parenthesis t right-parenthesis right-parenthesis squared d t">

I
(
T
,
α
)
=

∫
1
T

(
S
(
t
+
α

/

L
)
−
S
(
t
)
)

2

d
t

I(T,\alpha ) = \int _1^T {{{(S(t + \alpha /L) - S(t))}^2}dt}

, where

L
=

(
2
π

)

−
1

log

T

L = {(2\pi )^{ - 1}}\log \,T

. Furthermore, denote by

N
(
T
,
α
)
(

N

∗

(
T
)
)

N(T,\alpha )(N^{\ast }(T))

the number of pairs of zeros

1
2

+
i
Υ
,

1
2

+
i

Υ
′

\frac {1} {2} + i\Upsilon ,\frac {1} {2} + i\Upsilon ’

with

0
>
Υ
⩽
T

0 > \Upsilon \leqslant T

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than left-parenthesis normal upper Upsilon prime minus normal upper Upsilon right-parenthesis upper L less-than-or-slanted-equals alpha left-parenthesis left-parenthesis normal upper Upsilon prime minus normal upper Upsilon right-parenthesis upper L equals 0 right-parenthesis">

0
>
(

Υ
′

−
Υ
)
L
⩽
α

(
(

Υ
′

−
Υ
)
L
=
0
)

0 > (\Upsilon ’ - \Upsilon )L \leqslant \alpha \,((\Upsilon ’ - \Upsilon )L = 0)

—i.e., off-diagonal and diagonal pairs. Theorem. Assume the Riemann hypothesis. The following three hypotheses (A), (B) and

(

C

1

,

C

2

)

({{\text {C}}_1},{{\text {C}}_2})

are equivalent: for

β
→
∞

\beta \to \infty

and

α
→
0

\alpha \to 0

as

T
→
∞

T \to \infty

we have (A)

J
(
T
,
β
)
∼
β

T

−
1

log
2

T

J(T,\beta ) \sim \beta {T^{ - 1}}{\log ^2}T

, (B)

I
(
T
,
α
)
∼
α
T

I(T,\alpha ) \sim \alpha T

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper C Subscript 1 Baseline right-parenthesis upper N Superscript asterisk Baseline left-parenthesis upper T right-parenthesis tilde upper T upper L comma left-parenthesis upper C Subscript 2 Baseline right-parenthesis upper N left-parenthesis upper T comma alpha right-parenthesis equals o left-parenthesis upper T upper L right-parenthesis">

(

C

1

)

N

∗

(
T
)
∼
T
L
,
(

C

2

)
N
(
T
,
α
)
=
o
(
T
L
)

({{\text {C}}_1})\;N^{\ast }(T) \sim TL,({{\text {C}}_2})N(T,\alpha ) = o(TL)

. Hypothesis

(

C

1

,

C

2

)

({{\text {C}}_1},{{\text {C}}_2})

is called the essential simplicity hypothesis.