[Paper Review] Note on Malmstèn's paper De Integralibus quibusdam definitis seriebusque infinitis

Alexander Aycock|arXiv (Cornell University)|Jun 16, 2013

History and Theory of Mathematics1 references3 citations

TL;DR

This paper presents a previously overlooked proof of the functional equation for the Dirichlet eta-function, derived directly from formulas in Malmstön's 1846 paper De integralibus quibusdam definitis seriebusque infinitis. By applying limit processes and series expansions to Malmstön's integral identity, the author reconstructs the functional equation for η(s) using only tools available to Malmstön, demonstrating that he could have derived it with minimal additional work.

ABSTRACT

We present a proof of the functional equation of the Riemann zeta-function or more precisely the Dirichlet eta-function, which proof seems to be new but follows almost immediately from Malmstèn's paper ``De integralibus quibusdam definitis seriebusque infinitis``

Motivation & Objective

• To demonstrate that Malmstön's 1846 paper contains the essential ingredients for deriving the functional equation of the Dirichlet eta-function, despite its omission.
• To address the historical oversight of Malmstön's work by showing that his methods could have yielded the functional equation with only minor extensions.
• To highlight the deep mathematical insight in Malmstön's work, which predates Riemann's 1859 functional equation proof by 13 years.
• To advocate for renewed scholarly attention to Malmstön's Latin-language paper, which remains underappreciated despite its foundational contributions.

Proposed method

• Starts from Malmstön's formula 30, an integral involving hyperbolic functions and a parameter a, relating it to a logarithmic integral over [0,1].
• Takes the limit as a → 0 to eliminate the sin(a) factor, transforming the left-hand side into an expression involving u^{1-s}/(e^{πu} - e^{-πu}).
• Expands both sides using geometric series: 1/(1 - e^{-2πu}) on the left and 1/(1+y)^2 on the right, leading to series representations.
• Interchanges summation and integration, then applies the gamma function identity ∫₀^∞ e^{-y} y^{1-s} dy = Γ(2-s) to evaluate the integrals.
• Uses the known integral ∫₀¹ ln^{s-1}(1/y) y^{n-1} dy = Γ(s)/n^s to express the right-hand side in terms of η(s-1).
• Substitutes the relation λ(2-s) = (2^{2-s}-1)/(2^{2-s}-2) ⋅ η(2-s) and rewrites the equation in terms of η(1-s), yielding the final functional equation.

Experimental results

Research questions

• RQ1 Could Malmstön have derived the functional equation for the Dirichlet eta-function using only the tools and formulas present in his 1846 paper?
• RQ2 Why was Malmstön's paper, despite its depth, overlooked for over a century and not cited by contemporaries?
• RQ3 What is the precise connection between Malmstön's integral identity and the functional equation of the eta-function?
• RQ4 How does the functional equation derived from Malmstön's work compare to Riemann's 1859 proof of the zeta-function's functional equation?
• RQ5 To what extent does Malmstön's method anticipate later developments in analytic number theory, such as the use of Mellin transforms and analytic continuation?

Key findings

• The functional equation for the Dirichlet eta-function, η(1-s) = (2^s - 1)/(1 - 2^{s-1}) ⋅ π^{-s} ⋅ cos(πs/2) ⋅ Γ(s) ⋅ η(s), is derivable directly from Malmstön's formula 30 via limit and series manipulation.
• The derivation relies only on standard integral identities and series expansions known to Malmstön, suggesting the functional equation was within his reach.
• Malmstön's formula 57 for x=0 is equivalent to the key integral identity used in the proof, confirming its presence in his work.
• The relation λ(s) = (2^s - 1)/(2^s - 2) ⋅ η(s) connects the Dirichlet beta-function to the eta-function, enabling the final transformation.
• The paper argues that Malmstön's failure to derive the eta-function equation was not due to lack of tools, but likely due to the absence of explicit analytic continuation concepts.
• The author concludes that Malmstön's paper contains deep, underappreciated results, including a derivation of the Fourier series for lnΓ(x), which predates Kummer's 1847 result.