Ethan S. Lee

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

• ethan.lee@uwe.ac.uk (institutional)

• ethan.s.lee@unswalumni.com (permanent)

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:

1. On an explicit zero-free region for the Dedekind zeta-function.
   E. S. Lee | J. Number Theory | 2021 | DOI | arXiv:2002.05456

2. 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

3. 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.  

4. 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

5. On the number of integral ideals in a number field.
   E. S. Lee | J. Math. Anal. Appl. | 2023 | DOI | arXiv:2203.00389

6. Explicit Mertens' theorems for number fields.
   E. S. Lee | Bull. Aust. Math. Soc. | 2023 | DOI

7. 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).

8. 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

9. 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.

10. Explicit interval estimates for prime numbers.
    M. Cully-Hugill and E. S. Lee | Math. Comp. | 2022 | DOI | arXiv:2103.05986

11. E. S. Lee. Extensions of Bertrand's postulate.
    Lond. Math. Soc. Newsl. | 2023 | Issue 505

12. 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. 

13. The prime number theorem for primes in arithmetic progressions at large values.
    E. S. Lee | Q. J. Math. | 2023 | DOI | arXiv:2301.13457

14. 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

15. 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. 

16. 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.

17. Additive representations of natural numbers.
    F. J. Francis and E. S. Lee | Integers (volume 22, A14) | 2022 | Journal | arXiv:2003.08083

18. 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