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The convergence of certain Diophantine series
- Source :
- Journal of Number Theory. 229:179-198
- Publication Year :
- 2021
- Publisher :
- Elsevier BV, 2021.
-
Abstract
- For x irrational, we study the convergence of series of the form ∑ n − s f ( n x ) where f is a real-valued, 1-periodic function which is continuous, except for singularities at the integers with a potential growth. We show that it is possible to fully characterize the convergence set and to approximate the series in terms of the continued fraction of x. This improves and generalizes recent results by Rivoal who studied the examples f ( t ) = cot ( π t ) and f ( t ) = sin − 2 ( π t ) .
- Subjects :
- Algebra and Number Theory
Series (mathematics)
Diophantine equation
010102 general mathematics
010103 numerical & computational mathematics
Function (mathematics)
01 natural sciences
Combinatorics
Irrational number
Convergence (routing)
Gravitational singularity
Fraction (mathematics)
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 0022314X
- Volume :
- 229
- Database :
- OpenAIRE
- Journal :
- Journal of Number Theory
- Accession number :
- edsair.doi...........4af55ae6f921035201fc42fc89b57660