Ramanujan's function on small primes

2025/12/02 by Barry Brent, Brent, Barry
Mathematics · Physics and Astronomy · #Analytic Number Theory Research #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications

paper · pdf · doi:10.48550/arxiv.2512.02345

Abstract

We denote functions mapping n to the Fourier coefficient of qⁿ in the expansion of a cusp form as Ramanujan functions. We empirically study the eigenvalues of determinants that represent values of these Ramanujan functions. In some cases, considered as point sets in the complex plane, they appear to oscillate as n increases. We look for regularities in this phenomenon and discuss the possibility of exploiting it to attack Lehmer's question about the existence of zeros of Ramanujan's tau function.

Citations

• Pointwise ergodic theorem along primes of the form x² + ny²
• Polynomial interpolation of modular forms for Hecke groups
• Matrix methods for arithmetic functions
• Modular Elliptic Curves and Fermat's Last Theorem

Related

• Small Primes And The Tau Function
• On the prime factors of the iterates of the Ramanujan τ--function
• On some claims in Ramanujan's `unpublished' manuscript on the partition and tau functions
• Prime values of Ramanujan's tau function
• Ramanujan's congruence primes