Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Variations on a theorem of Davenport
concerning abundant numbers
Abundant,
perfect,
deficient
A statistical
approach
Emily Jennings, Paul Pollack & Lola Thompson
A
generalization
of
Davenport’s
theorem
University of Georgia & Oberlin College
Representing
sums of 2
squares
October 27, 2013
1 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Abundant, perfect, deficient – oh my!
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
Let σ(n) =
P
d|n d.
We say that n is:
abundant if σ(n) > 2n,
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
2 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Abundant, perfect, deficient – oh my!
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
Let σ(n) =
P
d|n d.
We say that n is:
abundant if σ(n) > 2n, eg. 12, 18, 20, 24, 30, 36, 40,
42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96,...
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
3 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Abundant, perfect, deficient – oh my!
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Let σ(n) =
P
d|n d.
We say that n is:
abundant if σ(n) > 2n, eg. 12, 18, 20, 24, 30, 36, 40,
42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96,...
perfect if σ(n) = 2n,
Representing
sums of 2
squares
4 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Abundant, perfect, deficient – oh my!
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Let σ(n) =
P
d|n d.
We say that n is:
abundant if σ(n) > 2n, eg. 12, 18, 20, 24, 30, 36, 40,
42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96,...
perfect if σ(n) = 2n, eg. 6, 28, 496, 8128, 33550336,
8589869056, 137438691328, 2305843008139952128,...
Representing
sums of 2
squares
5 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Abundant, perfect, deficient – oh my!
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
Let σ(n) =
P
d|n d.
We say that n is:
abundant if σ(n) > 2n, eg. 12, 18, 20, 24, 30, 36, 40,
42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96,...
perfect if σ(n) = 2n, eg. 6, 28, 496, 8128, 33550336,
8589869056, 137438691328, 2305843008139952128,...
deficient if σ(n) < 2n,
6 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Abundant, perfect, deficient – oh my!
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
Let σ(n) =
P
d|n d.
We say that n is:
abundant if σ(n) > 2n, eg. 12, 18, 20, 24, 30, 36, 40,
42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96,...
perfect if σ(n) = 2n, eg. 6, 28, 496, 8128, 33550336,
8589869056, 137438691328, 2305843008139952128,...
deficient if σ(n) < 2n, eg. 1, 2, 3, 4, 5, 7, 8, 9, 10, 11,
13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29,...
7 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Perfect vs. the rest
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
“Perfect numbers, like perfect men, are very rare.” – Descartes
8 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Perfect vs. the rest
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
“Perfect numbers, like perfect men, are very rare.” – Descartes
Representing
sums of 2
squares
9 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Perfect vs. the rest
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
“Just as...ugly and vile things abound, so [abundant] and
deficient numbers are plentiful...” –Nicomachus
10 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A statistical approach
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
What do we mean by “rare” versus “plentiful?”
Definition
If A is a subset of N = {1, 2, 3, ...}, we define the density of A
to be
#A ∩ [1, x]
.
lim
x→∞
x
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
11 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A statistical approach
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
What do we mean by “rare” versus “plentiful?”
Definition
If A is a subset of N = {1, 2, 3, ...}, we define the density of A
to be
#A ∩ [1, x]
.
lim
x→∞
x
A statistical
approach
A
generalization
of
Davenport’s
theorem
Example The prime numbers have density
Representing
sums of 2
squares
12 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A statistical approach
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
What do we mean by “rare” versus “plentiful?”
Definition
If A is a subset of N = {1, 2, 3, ...}, we define the density of A
to be
#A ∩ [1, x]
.
lim
x→∞
x
A statistical
approach
A
generalization
of
Davenport’s
theorem
Example The prime numbers have density 0.
Representing
sums of 2
squares
13 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A statistical approach
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
What do we mean by “rare” versus “plentiful?”
Definition
If A is a subset of N = {1, 2, 3, ...}, we define the density of A
to be
#A ∩ [1, x]
.
lim
x→∞
x
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
Example The prime numbers have density 0.
Example The set of natural numbers with 0 as the units digit
has density
14 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A statistical approach
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
What do we mean by “rare” versus “plentiful?”
Definition
If A is a subset of N = {1, 2, 3, ...}, we define the density of A
to be
#A ∩ [1, x]
.
lim
x→∞
x
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
Example The prime numbers have density 0.
Example The set of natural numbers with 0 as the units digit
1
has density 10
.
15 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A statistical approach
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
What do we mean by “rare” versus “plentiful?”
Definition
If A is a subset of N = {1, 2, 3, ...}, we define the density of A
to be
#A ∩ [1, x]
.
lim
x→∞
x
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
16 / 52
Example The prime numbers have density 0.
Example The set of natural numbers with 0 as the units digit
1
has density 10
.
Example The set of natural numbers with 1 as the leading
digit
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A statistical approach
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
What do we mean by “rare” versus “plentiful?”
Definition
If A is a subset of N = {1, 2, 3, ...}, we define the density of A
to be
#A ∩ [1, x]
.
lim
x→∞
x
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
17 / 52
Example The prime numbers have density 0.
Example The set of natural numbers with 0 as the units digit
1
has density 10
.
Example The set of natural numbers with 1 as the leading
digit does not have a density.
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Some natural questions
Variations on
a theorem of
Davenport
Does the set of abundant numbers have a density? What
about the perfect and deficient numbers?
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
18 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Some natural questions
Variations on
a theorem of
Davenport
Does the set of abundant numbers have a density? What
about the perfect and deficient numbers?
Emily
Jennings,
Paul Pollack
& Lola
Thompson
It will be useful to re-formulate the conditions for abundant vs.
perfect vs. deficient as follows:
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
19 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Some natural questions
Variations on
a theorem of
Davenport
Does the set of abundant numbers have a density? What
about the perfect and deficient numbers?
Emily
Jennings,
Paul Pollack
& Lola
Thompson
It will be useful to re-formulate the conditions for abundant vs.
perfect vs. deficient as follows:
Abundant,
perfect,
deficient
n is abundant if
n
σ(n)
<
1
2
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
20 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Some natural questions
Variations on
a theorem of
Davenport
Does the set of abundant numbers have a density? What
about the perfect and deficient numbers?
Emily
Jennings,
Paul Pollack
& Lola
Thompson
It will be useful to re-formulate the conditions for abundant vs.
perfect vs. deficient as follows:
Abundant,
perfect,
deficient
n is abundant if
A statistical
approach
n is perfect if
n
σ(n)
n
σ(n)
=
<
1
2
1
2
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
21 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Some natural questions
Variations on
a theorem of
Davenport
Does the set of abundant numbers have a density? What
about the perfect and deficient numbers?
Emily
Jennings,
Paul Pollack
& Lola
Thompson
It will be useful to re-formulate the conditions for abundant vs.
perfect vs. deficient as follows:
Abundant,
perfect,
deficient
n is abundant if
A statistical
approach
n is perfect if
A
generalization
of
Davenport’s
theorem
n is deficient
n
σ(n)
<
1
2
n
1
σ(n) = 2
n
if σ(n)
> 21
Representing
sums of 2
squares
22 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Some natural questions
Variations on
a theorem of
Davenport
Does the set of abundant numbers have a density? What
about the perfect and deficient numbers?
Emily
Jennings,
Paul Pollack
& Lola
Thompson
It will be useful to re-formulate the conditions for abundant vs.
perfect vs. deficient as follows:
Abundant,
perfect,
deficient
n is abundant if
A statistical
approach
n is perfect if
A
generalization
of
Davenport’s
theorem
n is deficient
n
σ(n)
1
2
n
1
σ(n) = 2
n
if σ(n)
> 21
So,
X
Representing
sums of 2
squares
23 / 52
<
1
n≤x
n/σ(n)<1/2
represents the count of abundant numbers up to x.
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Davenport’s theorem
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
Theorem (Davenport, 1933)
A
generalization
of
Davenport’s
theorem
Let
Representing
sums of 2
squares
24 / 52
1
x→∞ x
D(u) := lim
X
1.
n≤x
n/σ(n)≤u
Then D(u) exists for all u ∈ [0, 1] and varies continuously with
u.
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Density of abundant numbers
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
Theorem (Kobayashi, 2010)
We have
0.24761 < D(1/2) < 0.24765.
25 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A generalization of Davenport’s theorem
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
Co-authors: Lola Thompson, Giant Sloth, Emily Jennings and Paul Pollack
26 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A generalization of Davenport’s theorem
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
Theorem (Jennings, Pollack, T., 2013)
Let f be a multiplicative function that is bounded in mean
square. Suppose that for every nonnegative integer k, the
function n 7→ f (n)(n/σ(n))k possesses a mean value. Then for
every real u ∈ [0, 1], the limit
1
x→∞ x
Df (u) := lim
X
f (n)
n≤x
n/σ(n)≤u
exists. Moreover, Df (u) is continuous as a function of u.
27 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A useful corollary
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Corollary (Jennings, Pollack, T., 2013)
Let f be a multiplicative function bounded in mean square.
Then the aforementioned theorem holds if
X |f (p) − 1|
p
p
< ∞ and
X X |f (pj )|
p
j≥2
pj
< ∞.
(1)
If |f (n)| ≤ 1 for all n ∈ N, then (1) can be replaced with the
weaker assumption that the series
X f (p) − 1
p
Representing
sums of 2
squares
p
converges (possibly conditionally).
28 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Examples where our theorem is useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
Example f (n) = 1 identically.
Example The function
1 if n is squaref ree
f (n) =
0 otherwise
satisfies the hypotheses of our corollary.
Example To obtain a result for f (n) = ϕ(n) or σ(n), we can
apply our corollary to ϕ(n)/n or σ(n)/n and remove the
weight of 1/n by partial summation.
29 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Let τ (n) :=
P
d|n 1.
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
30 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
Let τ (n) :=
P
d|n 1.
Elementary Fact: The mean value of τ on [1, x] is asymptotic
to log x as x → ∞.
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
31 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
Let τ (n) :=
P
d|n 1.
Elementary Fact: The mean value of τ on [1, x] is asymptotic
to log x as x → ∞.
So, our theorem is not the right tool for this job.
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
32 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
Let τ (n) :=
P
d|n 1.
Elementary Fact: The mean value of τ on [1, x] is asymptotic
to log x as x → ∞.
So, our theorem is not the right tool for this job.
To obtain the ‘correct’ generalization of Davenport’s theorem,
we should be dividing by something proportional to x log x
(rather than dividing by x).
33 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
The representation function for sums of two squares:
1
r(n) := #{(x, y) ∈ Z2 : x2 + y 2 = n}.
4
Example: What is r(5)?












A statistical
approach
A
generalization
of
Davenport’s
theorem











Representing
sums of 2
squares
34 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
The representation function for sums of two squares:
1
r(n) := #{(x, y) ∈ Z2 : x2 + y 2 = n}.
4
Example: What is r(5)?

5 = (−2)2 + (−1)2 





















Representing
sums of 2
squares
35 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
The representation function for sums of two squares:
1
r(n) := #{(x, y) ∈ Z2 : x2 + y 2 = n}.
4
Example: What is r(5)?
5 = (−2)2 + (−1)2
= (−2)2 + 12
A
generalization
of
Davenport’s
theorem























Representing
sums of 2
squares
36 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
The representation function for sums of two squares:
1
r(n) := #{(x, y) ∈ Z2 : x2 + y 2 = n}.
4
Example: What is r(5)?
5 = (−2)2 + (−1)2
= (−2)2 + 12
= 22 + (−1)2























Representing
sums of 2
squares
37 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
The representation function for sums of two squares:
1
r(n) := #{(x, y) ∈ Z2 : x2 + y 2 = n}.
4
Example: What is r(5)?
5 = (−2)2 + (−1)2
= (−2)2 + 12
= 22 + (−1)2
= 22 + 12























Representing
sums of 2
squares
38 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
The representation function for sums of two squares:
1
r(n) := #{(x, y) ∈ Z2 : x2 + y 2 = n}.
4
Example: What is r(5)?
5 = (−2)2 + (−1)2
= (−2)2 + 12
= 22 + (−1)2
= 22 + 12
= (−1)2 + (−2)2
Representing
sums of 2
squares
39 / 52
Emily Jennings, Paul Pollack & Lola Thompson























Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
The representation function for sums of two squares:
1
r(n) := #{(x, y) ∈ Z2 : x2 + y 2 = n}.
4
Example: What is r(5)?
5 = (−2)2 + (−1)2
= (−2)2 + 12
= 22 + (−1)2
= 22 + 12
= (−1)2 + (−2)2
= (−1)2 + 22
40 / 52
Emily Jennings, Paul Pollack & Lola Thompson























Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
The representation function for sums of two squares:
1
r(n) := #{(x, y) ∈ Z2 : x2 + y 2 = n}.
4
Example: What is r(5)?
5 = (−2)2 + (−1)2
= (−2)2 + 12
= 22 + (−1)2
= 22 + 12
= (−1)2 + (−2)2
= (−1)2 + 22
= 12 + (−2)2
41 / 52
Emily Jennings, Paul Pollack & Lola Thompson























Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
42 / 52
The representation function for sums of two squares:
1
r(n) := #{(x, y) ∈ Z2 : x2 + y 2 = n}.
4
Example: What is r(5)?
5 = (−2)2 + (−1)2
= (−2)2 + 12
= 22 + (−1)2
= 22 + 12
= (−1)2 + (−2)2
= (−1)2 + 22
= 12 + (−2)2
= 12 + 22
Emily Jennings, Paul Pollack & Lola Thompson























Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
43 / 52
The representation function for sums of two squares:
1
r(n) := #{(x, y) ∈ Z2 : x2 + y 2 = n}.
4
Example: What is r(5)?
5 = (−2)2 + (−1)2
= (−2)2 + 12
= 22 + (−1)2
= 22 + 12
= (−1)2 + (−2)2
= (−1)2 + 22
= 12 + (−2)2
= 12 + 22
Emily Jennings, Paul Pollack & Lola Thompson























∴ r(5) =
1
·8=2
4
Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Our theorem is not so useful in dealing with
1
r(n) := #{(x, y) ∈ Z2 : x2 + y 2 = n}.
4
Abundant,
perfect,
deficient
A statistical
approach
What fails?
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
44 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Examples where our theorem is not so useful
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Our theorem is not so useful in dealing with
1
r(n) := #{(x, y) ∈ Z2 : x2 + y 2 = n}.
4
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
What fails?
r(n) is not bounded in mean square.
Representing
sums of 2
squares
45 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Another generalization of Davenport’s theorem
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
To address the problem raised in the τ (n) example, normalize
by
X
S(f ; x) :=
f (n).
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
n≤x
Define:
1
x→∞ S(f ; x)
D̃f (u) = lim
X
f (n),
n≤x
n/σ(n)≤u
whenever the limit exists.
46 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Davenport generalization, v 2.0
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Theorem (Jennings, Pollack, T., 2013)
Suppose that f is a nonnegative multiplicative function with
the property that as x → ∞,
X
f (p)
p≤x
log p
∼ κ log x
p
for some κ > 0. Suppose also that f (p) is bounded for primes p
and that f is tame on prime powers. Then D̃f (u) exists for all
u ∈ [0, 1] and is both continuous and strictly increasing.
Representing
sums of 2
squares
47 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Our Davenport generalization in action!
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
Example When f = τ , the hypotheses hold with κ = 2.
Example
When f = r, the hypotheses hold with κ = 1. Since
P
r(n)
∼ π4 x, then
n≤x
1
#{(x, y) ∈ Z2 : 0 < x2 + y 2 ≤ R
R→∞ πR
x2 + y 2
and
≤ u}.
σ(x2 + y 2 )
D̃r (u) = lim
Our theorem tells us that D̃r (u) exists and is both continuous
and strictly increasing.
48 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A different sum of two squares example
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Example Let S = {n ∈ Z : ∃ a, b ∈ Z st. n = a2 + b2 }.
0 = 02 + 02
1 = 12 + 02
2 = 12 + 12
4 = 22 + 02
5 = 22 + 12
8 = 22 + 22
9 = 32 + 02
10 = 32 + 12
13 = 32 + 22



























Representing
sums of 2
squares
49 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A different sum of two squares example
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Example Let S = {n ∈ Z : ∃ a, b ∈ Z st. n = a2 + b2 }.
0 = 02 + 02
1 = 12 + 02
2 = 12 + 12
4 = 22 + 02
5 = 22 + 12
8 = 22 + 22
9 = 32 + 02
10 = 32 + 12
13 = 32 + 22














∴ S = {0, 1, 2, 4, 5, 8, 9, 10, 13...}













Representing
sums of 2
squares
50 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
A different sum of two squares example
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
51 / 52
Example Let S = {n ∈ Z : ∃ a, b ∈ Z st. n = a2 + b2 }.
0 = 02 + 02
1 = 12 + 02
2 = 12 + 12
4 = 22 + 02
5 = 22 + 12
8 = 22 + 22
9 = 32 + 02
10 = 32 + 12
13 = 32 + 22














∴ S = {0, 1, 2, 4, 5, 8, 9, 10, 13...}













Let f = 1S . We can apply our theorem with κ = 1/2; this
gives a different two-squares analog of Davenport’s theorem.
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport
Variations on
a theorem of
Davenport
Emily
Jennings,
Paul Pollack
& Lola
Thompson
Abundant,
perfect,
deficient
A statistical
approach
Thank you!
A
generalization
of
Davenport’s
theorem
Representing
sums of 2
squares
52 / 52
Emily Jennings, Paul Pollack & Lola Thompson
Variations on a theorem of Davenport