Research Statement
Brandon Fodden
The central focus of my research is the study of L-functions. A combination of
powerful results and fundamental open questions makes this an exciting area in which to
do research. The study of L-functions also brings together many different areas of number
theory, intersecting with the study of the distribution of prime numbers, the structure
of number fields, and the study of elliptic curves and modular forms, for example. To
this point, my research has focused on the relation between the generalized Riemann
hypothesis and the existence of solutions to Diophantine equations, and the study of the
growth of fractional moments of L-functions. I look forward to continuing along these
lines of study in the future, as well as broadening the scope of my studies and considering
other problems associated with L-functions and with number theory in general.
At the International Congress of Mathematicians in 1900, Hilbert gave a famous list
of problems which he felt were fundamental to the progress of mathematics in the 20th
century. The tenth problem on his list asks for a computing algorithm which will tell
whether or not an arbitrary Diophantine equation with integer coefficients has a solution
in the integers. The work of Martin Davis, Yuri Matiyasevic̆, Hilary Putnam and Julia
Robinson showed that no such algorithm exists. However, several new and interesting
ideas came out of the work. My research uses a result which stems from this work.
If we have an algorithm to tell in a finite number of steps whether or not an arbitrary
positive integer n has a particular property P (written P (n)), then we say that the
property is a decidable property of n. One may prove the following theorem:
Theorem 1 (Davis, Matiyasevic̆, Putnam, Robinson) A statement of the form
(∀n)(P (n)) where P (n) is a decidable property of n is equivalent to the unsolvability
in the integers of a particular Diophantine equation.
If a mathematical statement can be shown to be equivalent to a statement of the
form (∀n)(P (n)) for P a decidable property of n, we say that it is Diophantine. The
Riemann hypothesis, denoted by RH, states that the non-trivial zeroes of the Riemann
zeta function ζ(s) lie on the line Re(s) = 21 . The following result was previously known:
Theorem 2 (Davis, Matiyasevic̆, Robinson, Shapiro)
2
X 1 x2
RH ⇔ 
−  ≤ 36x3 for x = 1, 2, 3...
k
2

k≤δ(x)
where the function δ(x) (which can be given explicitly) may be computed in finitely many
steps for any positive integer x.
We see that the property on the right side of the above equivalence is a decidable
property, and so the Riemann hypothesis is Diophantine. That is, there is a specific
Diophantine equation which has no solutions in the integers if and only if the Riemann
hypothesis is true.
Given an algebraic number field K, one may define the Dedekind zeta function (denoted ζK (s)) analogous to ζ(s). A hypothesis about the zeroes of ζK (s) similar to the
1
Riemann hypothesis may be made. This is denoted by GRHK . I have extended Theorem
2 to hold for GRHK . Working over a number field many complications arise which do
not appear in the classical case. However, I have proved the following theorem:
Theorem 3 (Fodden) Let K = Q(θ) for θ an algebraic integer with minimal polynomial f . Let n be the degree of f and let R(f, f 0 ) be the resultant of f and f 0 . Then
2

X 1 x2
−  ≤ (5n|R(f, f 0 )| + 13n + 10n2 )2 x3 for x = 1, 2, 3...
GRHK ⇔ 
k
2
k≤δf (x)
where the function δf (x) (which can be given explicitly) may be computed in finitely many
steps for any positive integer x.
Given the minimal polynomial f , one may compute R(f, f 0 ), n and δf (x) in finitely
many steps. Thus the property on the right side of the above equivalence is decidable.
We therefore have the following theorem:
Theorem 4 (Fodden) The generalized Riemann hypothesis for an algebraic number
field K with a given minimal polynomial f is Diophantine.
Using this theorem, we may consider the statement that GRHK holds for a collection
of number fields K. I have shown the following:
Theorem 5 (Fodden) Let P ⊆ Z[x] be recursively enumerable (listable). Then the
statement “for all roots θ of all polynomials in P, the generalized Riemann hypothesis
for Q(θ) holds” is Diophantine.
From this, a number of corollaries may be derived. For example, we may show the
following:
Corollary 6 Let ζn be a primitive nth root of unity. Then the statement
(∀n)(GRHQ(ζn ) is true)
is Diophantine.
That is, there is a specific Diophantine equation which has no integer solutions if and
only if GRHQ(ζn ) holds for every integer n.
We may also show:
Corollary 7 The statement
(∀K)(GRHK is true)
is Diophantine.
That is, there is a specific Diophantine equation which has no integer solutions if and
only if GRHK holds for every number field K.
I have also shown that an equivalence of the form given in Theorems 2 and 3 may be
extended to hold for L-functions belonging to the Selberg class. Let GRHF denote the
generalized Riemann hypothesis for the L-function F . I have the following theorem:
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Theorem 8 (Fodden) Let F (s) be a member of the Selberg class. Let m be the order
of the pole of F at s = 1. Then
2

X 1
− mx2  ≤ CF x3 for x = 1, 2, 3...
GRHF ⇔ 
k
k≤δF (x)
where CF is a constant that may be given explicitly in terms of the constants appearing
in the functional equation for F , and δF (x) may be given explicitly.
The difficulty in using Theorem 8 to show that the corresponding generalized Riemann hypothesis is Diophantine is in showing that the property on the right side of the
equivalence is decidable. Essentially this is possible if one has an algorithm to compute
the Dirichlet coefficients of F to an arbitrary degree of accuracy. I am currently working
on this problem, and on applications of the resulting theorem.
I believe that it is important to know when the generalized Riemann hypothesis
for a given L-function is Diophantine. By exploring statements that are equivalent to
the generalized Riemann hypothesis, we gain a better understand of exactly what the
hypothesis entails. Furthermore, knowing that we may write the generalized Riemann
hypothesis in the form (∀n)(P (n)), where P is a decidable property of n, allows us
to give it the classification Π01 in the arithmetical hierarchy, which measures the logical
complexity of a statement. This leads to a better understanding of the logical complexity
of the generalized Riemann hypothesis.
Recently, I have been studying the growth
of fractional moments of L-functions. Given
2k
RT 1
an L-function g(s), let Ik (g, T ) = 1 g 2 + it denote the kth moment of g(s). In
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the case of the Riemann zeta function, it is expected that Ik (ζ, T ) ∼ T (log T )k , but this
has only been shown for k = 0, 1 and 2. A weaker result for which to aim is to show that
2
2
T (log T )k Ik (g, T ) T (log T )k .
My research has focused on establishing the lower bound for a certain class of L-functions.
In particular, working jointly with Dr. Amir Akbary, we have shown the following theorem:
P
g(n)
Theorem 9 (Akbary, Fodden) Suppose g(s) = ∞
n=1 ns is absolutely convergent for
σ > 1. Suppose that except for a pole of degree m at s = 1 (with m possibly equal to 0),
g(s) extends to an entire function for Re (s) ≥ 12 and
|g(σ + it)| |t|A
for a fixed A > 0, σ ≥
1
2
and |t| ≥ 1. Suppose g(s) has an Euler product of the form
−1
d YY
αg (p, j)
1−
g(s) =
ps
p j=1
for Re (s) > 1, where αg (p, j) ∈ C and
1
|αg (p, j)| (2p)γ for a fixed 0 ≥ γ < .
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3
Suppose also that we have
X
|ag (p)|2 = β
p≤x
x
(1 + o(1))
log x
as x → ∞, and
X |ag (p)|2
p≤x
p
= β log log x + O(1)
for some fixed constant β. Then for any rational k ≥ 0 we have
2
Ik (g, T ) T (log T )βk .
The proof of this theorem extends a method of Heath-Brown and makes essential use
of a theorem of Levin and Fainleib on the sums of multiplicative functions. By focusing
on as general a set of L-functions as reasonably possible we have allowed the most salient
aspects of the method used to shine through. Thus previously murky connections between
results on moments of L-functions and the prime number theorem on the absolute value
of the squares of the Dirichlet coefficients have been made clearer. Using Theorem 9, we
have the following corollaries:
Corollary 10 Let L(π, s) be a principal automorphic L-function. Suppose that for the
local parameters at unramified primes we have
|απ (p, j)| ≤ pγ , for a fixed 0 ≤ γ < 1/4.
Then
Ik (π, T ) T (log T )k
2
for rational k ≥ 0.
Corollary 11 Let L(f, s) be the L-function associated to a newform of weight `, level
N , and nebentypus ψ. Then for any rational k ≥ 0 we have
2
Ik (f, T ) T (log T )k .
Corollary 12 Let L(f, s) be a Maass cusp newform of weight zero and level N with
nebentypus ψ. Then for any rational k ≥ 0 we have
2
Ik (f, T ) T (log T )k .
Corollary 13 Assume Artin’s conjecture for the Artin L-function L(K/Q, ρ, s). Then
for any rational k ≥ 0, we have
2
Ik (K/Q, ρ, T ) T (log T )hϕ,ϕik ,
where Ik (K/Q, ρ, T ) denotes the k-th moment of L(K/Q, ρ, s). Here ϕ is the character
associated to ρ and
1 X
|ϕ(g)|2
hϕ, ϕi =
|G| g∈G
where G = Gal(K/Q).
Corollary 14 Let K be a degree n Galois extension of Q, and let ζK (s) be the Dedekind
zeta function of K. Then for any rational k ≥ 0 we have
2
Ik (ζK , T ) T (log T )nk .
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