2022/01/25 by Masanori Katsurada, Katsurada, Masanori, Takumi Noda +1
Mathematics · #11F11 #11M35 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Primary 11M36 #Secondary 11E45
paper · pdf · doi:10.48550/arxiv.2201.10124
openalex publication_date 2022/01/25 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
For a class of generalized holomorphic Eisenstein series, we establish complete asymptotic expansions (Theorems~1~and~2), which together with the explicit expression of the latter remainder (Theorem~3), naturally transfer to several new variants of the celebrated formulae of Euler and of Ramanujan for specific values of the Riemann zeta-function (Theorem~4 and Corollaries~4.1--4.5), and to various modular type relations for the classical Eisenstein series of any even integer weight (Corollary~4.6) as well as for Weierstraß' elliptic and allied functions (Corollaries~4.7--4.9). Crucial rôles in the proofs are played by certain Mellin-Barnes type integrals, which are manipulated with several properties of confluent hypergeometric functions.