February 22, 2024

Ramanujan type congruences for quotients of Klein Forms

In this work, Ramanujan type congruences modulo powers of primes p greater than 5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and eta quotients and are generating functions for certain important arithmetic functions known to satisfy Ramanujan type congruences for p = 5, 7, 11. The vectors of exponents corresponding to these products that are modular forms for some Gamma function of p are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for some Gamma function of p, of weights k for primes 5 less than p less than 19 and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. Here, we provide a comprehensive characterization of these products modulo powers of 5 for the level 5 products, and a characterization modulo 7 for the level 7 products.

About the speaker

Jeffery Opoku is a dedicated scholar and educator with a passion for mathematics and its applications. Currently pursuing a Doctor of Philosophy (Ph.D.) in Mathematics and Statistics at The University of Texas, Rio Grande Valley (UTRGV), Jeffery has a strong academic background. He earned his Master of Philosophy (M.Phil.) in Applied Mathematics from Kwame Nkrumah University of Science and Technology, Ghana, and a Bachelor of Science in Pure Mathematics from the University of Ghana. Jeffery’s research interests span various mathematical domains, Analytic Number Theory (with a focus on Ramanujan Mathematics), Computational Number Theory, Combinatorics, and Special Functions. In addition to his research pursuits, Jeffery has a passion for teaching and has served as a Graduate Teaching Assistant at UTRGV. He’s been involved in mentoring and tutoring mathematics students, both at the university and school levels. With a strong foundation in mathematics, dedication to education, and a commitment to excellence, Jeffery Opoku is poised to make a meaningful impact in the world of mathematics and beyond.