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Revisiting the average number of divisors of a quadratic polynomial

Abstract

In this work, we employ a well known result due to Zagier to derive an asymptotic formula for the average number of divisors of a quadratic polynomial of the form \(x^{2}-bx+c\) with \(b,\,c\) integers. As simple consequences, we obtain formulas for computing the narrow class number of a quadratic field and relations between the narrow class number and a weighted class number for binary quadratic forms considered by McKee. In the end, we employ the class number relation to compute several examples of the weighted class number.

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Acknowledgements

The author thanks Wei-Lun Tsai for fruitful discussion and many helpful comments on the manuscript, and he would also like to thank the anonymous referee for his/her useful comments, suggestions and corrections.

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Correspondence to Dongxi Ye.

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The author is supported by the Natural Science Foundation of China (Grant No. 11901586), the Natural Science Foundation of Guangdong Province (Grant No. 2019A1515011323) and the Sun Yat-sen University Research Grant for Youth Scholars (Grant No. 19lgpy244).

Communicated by Adrian Constantin.

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Ye, D. Revisiting the average number of divisors of a quadratic polynomial. Monatsh Math (2020). https://doi.org/10.1007/s00605-020-01384-w

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Keywords

  • Class number
  • Number of divisors
  • Quadratic polynomial

Mathematics Subject Classification

  • 11N37
  • 11N56
  • 11D09
  • 11E41