Akash Singha Roy

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Graduate Student
PhD candidate
Instructor of record

I am currently a final year PhD student in the Department of Mathematics at the University of Georgia (UGA). I am working under the direction of Prof. Paul Pollack, and hope to receive my PhD by April 2025. Hence, I am currently on the job market. I completed my undergraduate studies at the Chennai Mathematical Institute, where I obtained a BSc. Honors in Mathematics and Computer Science.

My primary research interests lie in elementary, analytic and combinatorial number theory. In part of my thesis, I obtain new results on the distribution in arithmetic progressions of values taken by arithmetic functions: This was born out of a series of joint papers with my advisor and his former student Dr. Noah Lebowitz-Lockard, -- where we had set out to investigate analogues of the Siegel-Walfisz theorem for large classes of additive and multiplicative functions as well as for the joint distribution of families consisting of such functions, -- and recently culminated in three solo papers obtaining some of the best possible results in this direction. 

My current research (which constitutes the second part of my thesis) concerns new results on mean values of multiplicative functions as well as various new applications of these results. Besides these, I have coauthored multiple papers with my advisor, and with Profs. Vorrapan (Fai) Chandee, Xiannan Li and Nathan Mc New, on a variety of topics such as Benford's law (studying this phenomenon for various interesting sequences such as Hecke eigenvalues of newforms), distributions of intermediate prime factors and the behavior of the famous "aliquot sum" function s(n). I am also interested in a variety of other questions on the “anatomy of integers”, Erdos-type problems and general statistical questions on distributions of arithmetic functions, including Fourier coefficients of modular forms and the partition function. 

My research blends several areas of mathematics beyond my aforementioned primary interests, such as linear algebra and module theory, probability, commutative algebra, algebraic number theory, as well as arithmetic and algebraic geometry. Moreover, my work relies heavily on character sums and exponential sums, and many of the questions I have worked on and am working on have interesting analogues in the realms of L-functions, modular forms (and more generally automorphic forms), probability and probabilistic number theory. As part of my research, I have used multiple mathematical software packages such as Sage, Pari/GP, Macaulay2, and Magma.

I am the 2024 recipient of the William Armor Wills Memorial Scholarship Award from the Department of Mathematics at UGA. Please see my personal webpage for more details (and recent details) on my research, teaching and service/outreach activities.

 

 

 

Research Areas:
Number Theory and Arithmetic Geometry
Research Interests:

My primary research interests lie in elementary, analytic and combinatorial number theory. In part of my thesis, I obtain new results on the distribution in arithmetic progressions of values taken by arithmetic functions: This was born out of a series of joint papers with my advisor and his former student Dr. Noah Lebowitz-Lockard, -- where we had set out to investigate analogues of the Siegel-Walfisz theorem for large classes of additive and multiplicative functions as well as for the joint distribution of families consisting of such functions, -- and recently culminated in three solo papers obtaining some of the best possible results in this direction. 

My current research (which constitutes the second part of my thesis) concerns new results on mean values of multiplicative functions as well as various new applications of these results. Besides these, I have coauthored multiple papers with my advisor, and with Profs. Vorrapan (Fai) Chandee, Xiannan Li and Nathan Mc New, on a variety of topics such as Benford's law (studying this phenomenon for various interesting sequences such as Hecke eigenvalues of newforms), distributions of intermediate prime factors and the behavior of the famous "aliquot sum" function s(n). I am also interested in a variety of other questions on the “anatomy of integers”, Erdos-type problems and general statistical questions on distributions of arithmetic functions, including Fourier coefficients of modular forms and the partition function. 

My research blends several areas of mathematics beyond my aforementioned primary interests, such as linear algebra and module theory, probability, commutative algebra, algebraic number theory, as well as arithmetic and algebraic geometry. Moreover, my work relies heavily on character sums and exponential sums, and many of the questions I have worked on and am working on have interesting analogues in the realms of L-functions, modular forms (and more generally automorphic forms), probability and probabilistic number theory. As part of my research, I have used multiple mathematical software packages such as Sage, Pari/GP, Macaulay2, and Magma.

I am the 2024 recipient of the William Armor Wills Memorial Scholarship Award from the Department of Mathematics at UGA. Please see my personal webpage for more details (and recent details) on my research.

Degree Completion Date:
Selected Publications:

Published, Accepted and submitted Works

(Oldest first)

1. Steps into analytic number theory: A problem-based introduction (with P. Pollack) 
Springer, Problem Books in Mathematics.  

2. Distribution mod p of Euler's totient and the sum of proper divisors (with N. Lebowitz-Lockard and P. Pollack)
Michigan Math. J. 74 (2024), 143–166.
Links:     Journal     arXiV

3. Joint distribution in residue classes of polynomial-like multiplicative functions (with P. Pollack) Acta Arith. 202 (2022), 89–104.
Link:     Journal     arXiV

4. Powerfree sums of proper divisors (with P. Pollack)
Colloq. Math 168 (2022), 287–295.
Link:     Journal     arXiV

5. Dirichlet, Sierpinski, and Benford (with P. Pollack)
J. Number Theory 239 (2022), 352–364.
Link:     Journal

6. On Benford's Law for multiplicative functions (with V. Chandee, X. Li and  P. Pollack)
Proc. Amer. Math. Soc. 151 (2023), 4607–4619.
Link:     Journal     arXiV

7. Benford behavior and distribution in residue classes of large prime factors (with P. Pollack)
Canad. Math. Bull. 66 (2023), 626–642.
Link:     Journal

8. Distribution in coprime residue classes of polynomially-defined multiplicative functions (with P. Pollack)
Math. Z. 303 (2023), no. 4, paper 93, 20 pages.
Link:     Journal (older)     arXiV

9. Intermediate prime factors in specified subsets (with N. McNew and P. Pollack)
Monatsh. Math. 202 (2023), 837–855.
Link:     Journal

10. The distribution of intermediate prime factors (with N. McNew and P. Pollack)
Illinois J. Math. 68 (2024), no. 3, 537-576. 

Link:     Journal     arXiV

11. Mean values of multiplicative functions and applications to residue-class distribution (with P. Pollack)
Proc. Edinb. Math. Soc., accepted for publication.
Link:     Most recent version

12. Joint distribution in residue classes of families of polynomially-defined multiplicative functions I, 53 pages
Submitted to J. London Math. Soc.
Link:     Most recent version

13. Joint distribution in residue classes of families of polynomially-defined multiplicative functions II, 31 pages
Submitted to Acta. Arith.
Link:     Most recent version

14. Joint distribution in residue classes of families of polynomially-defined additive functions, 34 pages
Submitted to Math Z.
Link:     Most recent version

15. Anatomical mean value bounds on  multiplicative functions and the distribution of the sum of divisors, 34 pages 
Submitted to Michigan Math. J.
Link:     Most recent version

Manuscripts under preparation 

16. The Landau-Selberg-Delange method for products of Dirichlet L-functions and applications.

17. Distribution in residue classes of hybrid families of polynomially-defined additive and multiplicative functions. 

18. Weighted equidistribution and mean values of multiplicative functions in twisted progressions.

recent talks and slides

1. PAlmetto Number Theory Series (PANTS)  XXXVII: December 2023
Distribution in coprime residue classes of Euler’s totient and the sum of divisors
Link to slides

2. University of Georgia Number Theory Seminar: April 2024
Joint distribution in residue classes of families of ``polynomially-defined” multiplicative functions
Link to slides 

3. Dartmouth College Algebra and Number Theory Seminar: November 2024
Distribution and mean values of families of multiplicative functions in arithmetic progressions
Link to slides

4. AMS Spring Eastern Sectional Meeting 2025: "Counting and Asymptotics in Number Theory" (upcoming, April 2025)
 

Of note:

Recipient of:

  1. William Armor Wills Memorial Scholarship Award 2024.
  2. UGA Graduate School Dean's Award 2023.
  3. UGA Graduate School Research Assistantship: August 2022 to May 2023.
  4. Exemplary Counselor Award at Ross/Asia Mathematics Program 2019.

Please see CV for list of other awards.  

Courses Regularly Taught:
MATH2250
MATH1113
Teaching:

Refereeing

Have refereed for

  1. Women in Numbers Europe 4 – Research Directions in Number Theory, Springer, Association for Women in Mathematics Series.
  2. Rose-Hulman Undergraduate Mathematics Journal.

teaching and service at UGA

Fall 2024

  1. Instructor of MATH 2250 (Calculus I)
    Flipped/hybrid classroom structure.
  2. Math Study Hall tutor.
  3. Committee for UGA High School Math Tournament 2024
    Contributed several questions and was involved in the design of the contest.

Spring 2024

  1. Instructor of MATH 2250 (Calculus I)
    Flipped/hybrid classroom structure.
  2. Committee for design of MATH 2250 final exam.
  3. MATH 2250 Active Learning Working Group.
  4. Math Study Hall tutor.

Fall 2023

  1. Instructor for MATH 1113 (Precalculus)
    Flipped classroom structure.
  2. Grader for MATH 3100 (Sequences and series)
    Instructed by Prof. Paul Pollack.

Teaching and service prior to UGA

  1. Counselor in the Ross/Asia Mathematics program 2019.
    The Ross Program is a residential summer math camp for high school students, primarily focused on algebra and number theory, where students are immersed in the process of mathematical discovery for six weeks. As a Counselor, my responsibility was to mentor the students by guiding their thinking and providing detailed feedback on their work; in addition, I also discussed several interesting mathematical problems with students and gave several informal lectures.

    Received Exemplary Counselor Award “in recognition of outstanding work at the 2019 Ross/Asia Mathematics Program”.

  2. Served on committee for evaluating applications to the Ross Mathematics Program: 2020-2021.
  3. Teaching assistant in the courses Algebra III and Algebra IV at the Chennai Mathematical Institute: 2020-2021.
  4. Contributed questions to and served as grader for the Scholastic Test for Excellence in Mathematical Sciences (STEMS) conducted by the Chennai Mathematical Institute: 2019.
  5. Junior Counselor in the Ross Mathematics Program at the Ohio State University: 2018.

Contact Information

Akash.SinghaRoy@uga.edu
Office: 427J Boyd GSRC

Other Information

Curriculum Vitae: AkashSRoy_CVDec29.pdf (115.74 KB)
Professional Web site: Personal webpage

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