2024/03/23 by Artur Kawalec, Kawalec, Artur
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2403.15741
openalex publication_date 2024/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In article, we explore the secondary zeta function Z(s), which is defined as a generalized zeta type of series over imaginary parts of non-trivial zeros of the Riemann zeta function ζ(s). This function has been analytically continued as a meromorphic function in ℂ with one double pole and an infinity of simple poles. The secondary zeta function is of interest because it can naturally represent an analytical formula for non-trivial zeros of the Riemann zeta function that we will explore, and we show that the non-trivial zeros can be generated directly from primes by introducing a new form of an explicit formula written in terms of the prime zeta function. Additionally, we will also give several new series expansions for Z(s) and numerically compute these coefficients to high precision, and also develop several new methods to analytically extend Z(s) to larger domains and develop algorithms to compute them.