2021/08/17 by Su Hu, Min-Soo Kim, Hu, Su +1 · 1 citation
Mathematics · #11M35 #33B15 #40A20 #40A30 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Inequalities and Applications #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2108.07677
openalex publication_date 2021/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classical Lerch's formula states the following normalized product: ∏n=0^∞(x+n)=(√(2π))/(Γ(x)), \textrmRe(x)gt;0, where Γ(x) is the Euler gamma function. In this note, by using Barnes' multiple zeta function and its alternating form, we obtain two kinds of generalizations of Lerch's formula, which imply the product ∏n=1^∞ n=√(2π) (in the sense of zeta regularization) and the product (2⋅2)/(1⋅ 3)(4⋅4)/(3⋅ 5)(6⋅6)/(5⋅ 7)⋯=\fracπ2 (Wallis' formula in 1656), respectively.