Riemann zeta function and arithmetic progression of higher order

Hamilton Brito, Éder Furtado and Fernando Matos
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 1, Pages 1–7
DOI: 10.7546/nntdm.2020.26.1.1-7
Full paper (PDF, 567 Kb)

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Authors and affiliations

Hamilton Brito
Instituto Federal de Educação, Ciência e Tecnologia do Pará
Avenida Almirante Barroso, 1155, Brasil

Éder Furtado
Instituto Federal de Educação, Ciência e Tecnologia do Pará
Avenida Almirante Barroso, 1155, Brasil

Fernando Matos
Instituto Federal de Educação, Ciência e Tecnologia do Pará
Avenida Almirante Barroso, 1155, Brasil

Abstract

Riemann zeta function has a great importance in number theory, constituting one of the most studied functions. The zeta function, being a series, has a close relationship with the arithmetic progressions (AP). AP of higher order allows the understanding of several probabilities involving sequences. In this paper, we will approach Riemann zeta function with an AP of higher order. We will deduce a formula from the progression that will allow to express of the zeta function for a natural number greater than or equal to 2. In this way, we will show that the study of an AP of higher order can be very useful in the study of Riemann zeta function, and it may open other possibilities for studying the value of this function for odd numbers.

Keywords

• Riemann zeta function
• Arithmetic progression of higher order
• Sequence

2010 Mathematics Subject Classification

• 11M06
• 11M26

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