Displaying similar documents to “A note on Riemann integrability.”

Weaker forms of continuity and vector-valued Riemann integration

M. A. Sofi (2012)

Colloquium Mathematicae

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It was proved by Kadets that a weak*-continuous function on [0,1] taking values in the dual of a Banach space X is Riemann-integrable precisely when X is finite-dimensional. In this note, we prove a Fréchet-space analogue of this result by showing that the Riemann integrability holds exactly when the underlying Fréchet space is Montel.

The Basic Existence Theorem of Riemann-Stieltjes Integral

Kazuhisa Nakasho, Keiko Narita, Yasunari Shidama (2016)

Formalized Mathematics

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In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are...

A family of deformations of the Riemann xi-function

Masatoshi Suzuki (2013)

Acta Arithmetica

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We introduce a family of deformations of the Riemann xi-function endowed with two continuous parameters. We show that it has rich analytic structure and that its conjectural (mild) zero-free region for some fixed parameter is a sufficient condition for the Riemann hypothesis to hold for the Riemann zeta function.