Rafael Jakimczuk

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 3, Pages 44—64

DOI: 10.7546/nntdm.2019.25.3.44-64

**Download full paper: PDF, 241 Kb**

## Details

### Authors and affiliations

Rafael Jakimczuk

*Division Matematica, Universidad Nacional de Lujan
Buenos Aires, Argentina
*

### Abstract

In this article we study functions related to numbers which have the same kernel. We apply the results obtained to the sums , where is an arbitrary but fixed positive integer and denotes the kernel of . For example, we prove that

where

and the positive coefficients of the series have a strong connection with the prime numbers. We also prove that

where . The methods used are very elementary. The case , namely , was studied, as it is well-known, by N. G. de Bruijn (1962) and W. Schwarz (1965).

### Keywords

- Kernel function
- Numbers with the same kernel

### 2010 Mathematics Subject Classification

- 11A99
- 11B99

### References

- Beukers, F. (1975). The lattice points of n-dimensional tetrahedra, Indag. Math., 37, 365–372.
- De Koninck, J., Diouf, I., & Doyon, N. (2012). On the truncated kernel function, J. Integer Seq., 15, Article 12.3.2.
- Hardy, G. H., & Wright, E. M. (1960). An Introduction to the Theory of Numbers, Oxford.
- Jakimczuk, R. (2008). An observation on the Cipolla’s expansion, Mathematical Sciences. Quarterly Journal, 2, 219–222.
- Jakimczuk, R. (2007). Integers of the form , where are primes fixed, International Journal of Contemporary Mathematical Sciences, 2, 1327–1333.
- Jakimczuk, R. (2017). On the kernel function, International Mathematical Forum, 12, 693–703.
- Rey Pastor, J., Pi Calleja, P., & Trejo, C. A. (1969). Analisis Matematico, Volume 1, Kapelusz.

## Related papers

## Cite this paper

APAJakimczuk, R. (2019). Numbers with the same kernel. Notes on Number Theory and Discrete Mathematics, 25(3), 44-64, doi: 10.7546/nntdm.2019.25.3.44-64.

ChicagoJakimczuk, Rafael. “Numbers with the Same Kernel.” Notes on Number Theory and Discrete Mathematics 25, no. 3 (2019): 44-64, doi: 10.7546/nntdm.2019.25.3.44-64.

MLAJakimczuk, Rafael. “Direct Parametrization of Pythagorean Triples.” Notes on Number Theory and Discrete Mathematics 25.3 (2019): 44-64. Print, doi: 10.7546/nntdm.2019.25.3.44-64.