A generalization of a theorem of Erdős and Niven

Jun Qiu; Hongguang Wu; Jun Qiu; Hongguang Wu

doi:10.3934/math.2026766

AIMS Mathematics

2026, Volume 11, Issue 6: 18835-18844. doi: 10.3934/math.2026766

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A generalization of a theorem of Erdős and Niven

Jun Qiu ^{1
,
,},
Hongguang Wu ²

1.
School of Big Data and Statistics, Tongling University, Tongling 244000, China
2.
Department of Mathematics, Changzhou University, Changzhou 213164, China

Received: 28 April 2026 Revised: 04 June 2026 Accepted: 11 June 2026 Published: 26 June 2026
MSC : 11B75, 11B83, 11Y70

In 1946, Erdős and Niven proved that no two distinct partial sums of the harmonic series can be equal. In this paper, we extend their result by establishing the analogous property for a generalized harmonic series of the form $ \sum\frac{1}{a+ib} $, where $ a $ and $ b $ are coprime positive integers with $ a > b\ge 1 $.
- harmonic series,
- generalized harmonic series,
- partial sums
Citation: Jun Qiu, Hongguang Wu. A generalization of a theorem of Erdős and Niven[J]. AIMS Mathematics, 2026, 11(6): 18835-18844. doi: 10.3934/math.2026766

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Abstract

In 1946, Erdős and Niven proved that no two distinct partial sums of the harmonic series can be equal. In this paper, we extend their result by establishing the analogous property for a generalized harmonic series of the form $ \sum\frac{1}{a+ib} $, where $ a $ and $ b $ are coprime positive integers with $ a > b\ge 1 $.

References

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