Ethan S. Lee

Associate Lecturer at UWE Bristol and the University of Chichester
and
Honorary Research Fellow at the University of Bristol

About Me

I received my PhD in Mathematics from the University of New South Wales in January 2023, where I was supervised by Prof. Timothy Trudgian. I am currently a Heilbronn Research Fellow at the University of Bristol, where I am continuing my research in Analytic Number Theory, with particular focus on:

the distribution of primes (including generalisations),
properties of L-functions, and
explicit results.

Beyond my research, which will be summarised below, I have been:

an Organiser of the Heilbronn Number Theory Seminars at the University of Bristol,
an Editorial Assistant for the Integers journal (since 2021),
a Contributor to the TMT-EMT Project, and
supervising student projects on topics in Additive and Analytic Number Theory.

I also have experience in teaching Mathematics, Computer Science, and Cyber Security to diverse audiences.

Outside of academia, I support Everton FC and enjoy hobbies such as games and exploring the countryside.

See my Abridged CV for a complete summary of my activities, positions, talks, etc.

Contact Me!

If you wish to contact me for any reason, please email me at:

ethan.lee@bristol.ac.uk (institutional)
ethan.s.lee@unswalumni.com (permanent)

My Research

Analytic properties of L-functions

One of the central goals of analytic number theory has been to study the distribution of arithmetic objects through the lens of L-functions, which are complex functions with nice properties. In particular, the distribution of these arithmetic objects can be related to the distribution of the zeros and poles of an L-function through an explicit formula. For example:

The distribution of rational primes (2 ,3, 5, ...) is explicitly linked to the distribution of zeros of the Riemann zeta-function.
The distribution of prime ideals in a number field is explicitly linked to the distribution of zeros of a Dedekind zeta-function, which can be thought of as an algebraic extension of the Riemann zeta-function.

Therefore, to study the distribution of prime ideals in any number field, it is paramount that we study the distribution of the zeros of the Dedekind zeta-function (the key results here are zero-free regions and zero-density bounds) and quantify the residue of the unique simple pole of the Dedekind zeta-function. To this end, I have produced the following research works:

On an explicit zero-free region for the Dedekind zeta-function.
E. S. Lee | J. Number Theory | 2021 | DOI | arXiv:2002.05456
Explicit estimates for Artin L-functions: Duke’s short-sum theorem and applications.
S. R. Garcia and E. S. Lee | J. Number Theory | 2021 | DOI | arXiv:2101.11853
The error term in the truncated Perron formula for the logarithm of an L-function.
S. R. Garcia, J. Lagarias, and E. S. Lee | Can. Math. Bull. | 2023 | DOI | arXiv:2206.01391

Arithmetic structure in number fields

The arithmetic structure of number-theoretic objects in a number field is a central problem in algebraic number theory. The gold standard result in this area is the Chebotarev denisty theorem (CDT), which approximates the number of prime ideals in a number field with prescribed algebraic properties. However, the error in the CDT is hard to quantify, due to technical obstructions (such as the Siegel zero) and limitations in our knowledge on the distribution of the zeros of the Dedekind zeta-function.

Unconditional Results I: One direction of my research in this area focuses on obtaining concrete descriptions for the error in weighted variants of the CDT. My results, which are presented in the following research papers, overcome all of these technical obstructions and can be applied more broadly than the CDT.

Unconditional explicit Mertens’ theorems for number fields and Dedekind zeta residue bounds.
S. R. Garcia and E. S. Lee | Ramanujan J. | 2021 | DOI | arXiv:2007.10313
On the number of integral ideals in a number field.
E. S. Lee | J. Math. Anal. Appl. | 2023 | DOI | arXiv:2203.00389
Explicit Mertens' theorems for number fields.
E. S. Lee | Bull. Aust. Math. Soc. | 2023 | DOI
Explicit Mertens’ theorems for number fields and Dedekind zeta residue bounds with GRH.
S. R. Garcia and E. S. Lee | 2020 | arXiv:2006.03337

Applications: The number of solutions (mod p) to the congruence F(x) = 0 (mod p) for most primes p is equal to the number of prime ideals in a certain number field satisfying a norm condition. Building upon this connection, the following papers apply my explicit results on Mertens' theorems for number fields (from above).

An effective analytic formula for the number of distinct irreducible factors of a polynomial.
S. R. Garcia, E. S. Lee, J. Suh and J. Yu | J. Aust. Math. Soc. | 2021 | DOI | arXiv:2012.03102
Explicit upper bounds for the number of primes simultaneously representable by any set of irreducible polynomials.
M. Bordignon and E. S. Lee | 2022 | arXiv:2211.11012

Unconditional Results II: In some special cases, we have extra knowledge about L-functions which we can use to establish stronger descriptions of the error in the CDT in that context. For example, the L-function associated to the rationals (the prototypical number field) is the Riemann zeta-function, and we know the location of millions of the lowest lying complex zeros of the Riemann zeta-function (see the LMFDB). In the following papers, we use this extra knowledge to study problems closely related to the distribution of rational primes. The paper [10] is a critical ingredient that has been used to make progress toward Legendre's conjecture.

Explicit interval estimates for prime numbers.
M. Cully-Hugill and E. S. Lee | Math. Comp. | 2022 | DOI | arXiv:2103.05986
E. S. Lee. Extensions of Bertrand's postulate.
Lond. Math. Soc. Newsl. | 2023 | Issue 505
New explicit bounds for Mertens function and the reciprocal of the Riemann zeta-function.
E. S. Lee and N. Leong | 2024 | arXiv:2208.06141

Distribution of primes assuming the Riemann Hypothesis

The Generalised Riemann Hypothesis (GRH) postulates exact knowledge about the horizontal distribution of the zeros of an L-function. Assuming the GRH, we can establish the strongest descriptions for the the error in the Chebotarev denisty theorem (CDT) with no technical obstructions. The following papers are my contributions to the literature in this area, which focus on two special cases of the CDT, namely the prime number theorem (PNT) and the PNT for primes in arithmetic progressions.

The prime number theorem for primes in arithmetic progressions at large values.
E. S. Lee | Q. J. Math. | 2023 | DOI | arXiv:2301.13457
On the constants in Mertens' theorems for primes in arithmetic progressions.
D. Keliher and E. S. Lee | Proceedings of the Integers Conference 2023 | 2024 | arXiv:2306.09981
Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis.
E. S. Lee and P. Nosal | 2023 | arXiv:2312.05628

One can think of the CDT as a global result, which describes how primes are distributed on average, but famous conjectures (such as the twin prime conjecture) suggest that the distribution of primes is not regular. To quantify the extent and nature of this irregularity, we also study the distribution of primes in short intervals in the following paper.

The error in the prime number theorem in short intervals.
E. S. Lee | 2024 | arXiv:2403.11814

Problems in Additive Number Theory

The goal of additive number theory is to understand how integers (or other arithmetic objects) can be expressed as sums of simpler components, such as primes, powers, or square-free integers. In many cases, we can convert the problems in additive number theory into problems about the distribution of primes in an arithmetic progression, which can be tackled using the tools we have developed in other aspects of my research. In the following paper, we study a problem that is related to the Goldbach conjecture.

Additive representations of natural numbers.
F. J. Francis and E. S. Lee | Integers (volume 22, A14) | 2022 | Journal | arXiv:2003.08083
On the sum of a prime and a square-free number with divisibility conditions.
E. S. Lee and R. O'Clarey | 2025 | arXiv:2506.11814

A Frequently Asked Question

Why do I care about explicit results? I care about these, because a result that is asymptotically "worse" can have unexpected benefits over its asymptotically "superior" counter-part when we consider the size of the implied constants and the range that these bounds hold for. Sure, if you only care about the long term behaviour of an error term, then you shouldn't care about explicit results, but if you also care (like me) about the behaviour of that same error for reasonable x, say x < 10^{10000}, then you definitely should care about results that completely describe the error term (i.e. explicit results), because these will clarify any dependencies and reveal how practical that error is. In some cases, explicit results are so practical that they can be used to inform computations that verify long-standing conjectures up to some height (such as the Goldbach conjecture). On the other hand, some implied constants are large enough that that result is only decent when x is abominably large. In the last case, we need to refine that constant or consider using an asymptotically inferior result that has a better controlled error (in the explicit sense) to obtain the application at hand when x is not abominably large!

An illustrative example: A classic problem is to find the shortest interval that contains at least one prime; but the answer to this problem depends on what you consider shortest to mean. An interval of the form [x, x+o(x)] that contains a prime is called a short interval, because it will be the shortest in the asymptotic sense (i.e. in the long run), whereas an interval of the form [x, x+Cx] for some constant 0<C<1 that contains a prime is called a long interval. Finding the smallest value for C such that the long interval contains a prime for all x>Y is a classic problem for an explicit analytic number theorist. However, are long intervals going to be worse, because we know that we can do better asymptotically? No! In fact, for "small" x, where small can mean x<10^{1000}, the best long intervals are shorter than the best short intervals in the numerical (or physical) sense.

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