Averaging effects on irregularities in the distribution of primes in arithmetic progressions
Authors: Hudson Richard H. 
Published: June 2010

Let

t
t

be an integer taking on values between 1 and

x
x

(

x
x

real), let

π

b
,
c

(
t
)

{\pi _{b,c}}(t)

denote the number of positive primes

⩽
t

\leqslant t

which are

≡
c

\equiv c

(
mod

b
)

\pmod b

, and let

li
⁡
t

\operatorname {li} t

denote the usual integral logarithm of

t
t

. Further, let the ratio of quadratic nonresidues of

b
>
2

b > 2

to quadratic residues of

b
b

be

γ
(
b
)

\gamma (b)

to 1, and let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript b Baseline left-parenthesis x right-parenthesis equals left-parenthesis 1 slash gamma left-parenthesis b right-parenthesis right-parenthesis StartFraction 1 Over x EndFraction left-parenthesis sigma-summation Underscript StartLayout 1st Row t equals 1 2nd Row 1 less-than-or-slanted-equals c less-than-or-slanted-equals b minus 1 EndLayout Overscript x Endscripts pi Subscript b comma c Baseline left-parenthesis t right-parenthesis minus gamma left-parenthesis b right-parenthesis sigma-summation Underscript StartLayout 1st Row t equals 1 2nd Row 1 less-than-or-slanted-equals c prime less-than-or-slanted-equals b minus 1 EndLayout Overscript x Endscripts pi Subscript b comma c Sub Superscript prime Subscript Baseline left-parenthesis t right-parenthesis right-parenthesis">

A
b

(
x
)
=
(
1

/

γ
(
b
)
)

1
x

(

∑

t
=
1

1
⩽
c
⩽
b
−
1

x

π

b
,
c

(
t
)
−
γ
(
b
)

∑

t
=
1

1
⩽

c
′

⩽
b
−
1

x

π

b
,

c
′

(
t
)
)

A_b(x) = (1/\gamma (b)) \frac {1}{x} \left ( \sum _{\substack {t = 1\\ 1 \leqslant c \leqslant b - 1}}^x \pi _{b,c}(t) - \gamma (b) \sum _{\substack {t = 1\\1 \leqslant c’\leqslant b - 1}}^x \pi _{b,c’}(t) \right )

\]
where

c
c

runs over quadratic nonresidues and

c
′

c’

runs over quadratic residues of

b
b

. Nearly periodic oscillations of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A 6 left-parenthesis x right-parenthesis equals left-parenthesis 1 slash x right-parenthesis normal upper Sigma Subscript t equals 1 Superscript x Baseline left-parenthesis pi Subscript 6 comma 5 Baseline left-parenthesis t right-parenthesis minus pi Subscript 6 comma 1 Baseline left-parenthesis t right-parenthesis right-parenthesis">

A
6

(
x
)
=
(
1

/

x
)

Σ

t
=
1

x

(

π

6
,
5

(
t
)
−

π

6
,
1

(
t
)
)

A_6(x) = (1/x)\Sigma _{t = 1}^x(\pi _{6,5}(t) - \pi _{6,1}(t))

about

h
(
x
)
=
(
1

/

x
)

Σ

t
=
1

x

li
⁡
(

t

1

/

2

)

/

2

h(x) = (1/x)\Sigma _{t = 1}^x \operatorname {li}({t^{1/2}})/2

are depicted in Figures 2, 3, 4 over the range of integers less than

2.5
×

10

11

2.5 \times {10^{11}}

. Over this range,

h
(
x
)

h(x)

is a far better "axis of symmetry" for these oscillations than

s
(
x
)
=
(
1

/

x
)

Σ

t
=
1

x

t

1

/

2

/

log
⁡
t

s(x) = (1/x)\Sigma _{t = 1}^x{t^{1/2}}/\log t

(suggested by Shanks [29]). On the other hand, recent work of W. J. Ellison [9], three letters from Andrzej Schinzel to the author, and my own considerations (see Section 4) lead to the following. In contradiction to a conjecture of Shanks [29],
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over x EndFraction left-parenthesis sigma-summation Underscript t equals 1 Overscript x Endscripts left-parenthesis pi Subscript 6 comma 5 Baseline left-parenthesis t right-parenthesis minus pi Subscript 6 comma 1 Baseline left-parenthesis t right-parenthesis right-parenthesis slash left-parenthesis t Superscript 1 slash 2 Baseline slash log t right-parenthesis right-parenthesis right-arrow with stroke 1 as x right-arrow normal infinity period">

1
x

(

∑

t
=
1

x

(

π

6
,
5

(
t
)
−

π

6
,
1

(
t
)
)

/

(

t

1

/

2

/

log
⁡
t
)
)

↛
1

as 

x
→
∞

.

\frac {1}{x} \left ( \sum \limits _{t = 1}^x (\pi _{6,5}(t) - \pi _{6,1}(t)) / (t^{1/2}/\log t) \right ) \nrightarrow 1\quad \text {as $x \to \infty $} .

\]
Moreover, I prove in Theorem 4.1 that

A
6

(
x
)

/

h
(
x
)
↛
1

{A_6}(x)/h(x) \nrightarrow 1

as

x
→
∞

x \to \infty

, and Schinzel has provided a heuristic argument that no amount of averaging of

A
6

(
x
)

{A_6}(x)

will provide an asymptotic relationship of this sort. However, let

h

(
1
)

(
x
)
=
h
(
x
)

{h^{(1)}}(x) = h(x)

,

A
6

(
1
)

(
x
)
=

A
6

(
x
)

A_6^{(1)}(x) = {A_6}(x)

, and for

k
>
1

k > 1

let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Superscript left-parenthesis k plus 1 right-parenthesis Baseline left-parenthesis x right-parenthesis equals StartFraction 1 Over x EndFraction sigma-summation Underscript t equals 1 Overscript x Endscripts h Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis t right-parenthesis comma upper A 6 Superscript left-parenthesis k plus 1 right-parenthesis Baseline left-parenthesis x right-parenthesis equals StartFraction 1 Over x EndFraction sigma-summation Underscript t equals 1 Overscript x Endscripts upper A 6 Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis t right-parenthesis period">

h

(
k
+
1
)

(
x
)
=

1
x

∑

t
=
1

x

h

(
k
)

(
t
)
,

A
6

(
k
+
1
)

(
x
)
=

1
x

∑

t
=
1

x

A
6

(
k
)

(
t
)
.

{h^{(k + 1)}}(x) = \frac {1}{x}\sum \limits _{t = 1}^x {{h^{(k)}}(t),\quad A_6^{(k + 1)}(x) = \frac {1}{x}} \sum \limits _{t = 1}^x {A_6^{(k)}(t).}

\]
Assuming the truth of the generalized Riemann hypothesis for

L
(
s
,
χ
)

L(s,\chi )

,

χ
\chi

the nonprincipal character

mod

6

\bmod \,6

, we prove
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript k right-arrow normal infinity Endscripts limit Underscript x right-arrow normal infinity Endscripts Overscript bar Endscripts StartFraction upper A 6 Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis x right-parenthesis Over h Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis x right-parenthesis EndFraction equals 1 equals limit Underscript k right-arrow normal infinity Endscripts limit Underscript bar Endscripts Underscript x right-arrow normal infinity Endscripts StartFraction upper A 6 Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis x right-parenthesis Over h Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis x right-parenthesis EndFraction period">

lim

k
→
∞

lim

x
→
∞

¯

A
6

(
k
)

(
x
)

h

(
k
)

(
x
)

=
1
=

lim

k
→
∞

lim
_

x
→
∞

A
6

(
k
)

(
x
)

h

(
k
)

(
x
)

.

\lim \limits _{k \to \infty } \;\overline {\lim \limits _{x \to \infty } } \;\frac {{A_6^{(k)}(x)}}{{{h^{(k)}}(x)}} = 1 = \lim \limits _{k \to \infty } \;\underline {\lim } \limits _{x \to \infty } \;\frac {{A_6^{(k)}(x)}}{{{h^{(k)}}(x)}}.

\]
The behavior of

A
6

(
x
)

{A_6}(x)

is a special case of a far more general phenomenon. In Section 3, reasons are given why

A
b

(
x
)

{A_b}(x)

can be expected to oscillate more or less symmetrically about

h
(
x
)

h(x)

for every modulus

b
>
2

b > 2

.