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A PROOF OF MERCA’S CONJECTURES ON SUMS OF ODD DIVISOR FUNCTIONS

Part of: Additive number theory; partitions Elementary number theory

Published online by Cambridge University Press:  10 September 2021

KAYA LAKEIN  and
ANNE LARSEN
KAYA LAKEIN*
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA94305, USA
ANNE LARSEN
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA02138, USA e-mail: larsen@college.harvard.edu
*
e-mail: epi2@stanford.edu
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Abstract

Merca [‘Congruence identities involving sums of odd divisors function’, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci.22(2) (2021), 119–125] posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalised m-gonal numbers. Extending Merca’s work, we complete the proof of these conjectures.

Keywords

divisor functionpolygonal numberpartition

MSC classification

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 11P81: Elementary theory of partitions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 2 , April 2022 , pp. 197 - 201
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The authors are grateful for the generous support of the National Science Foundation (DMS 2002265 and 205118), the National Security Agency (H98230-21-1-0059), the Thomas Jefferson Fund at the University of Virginia and the Templeton World Charity Foundation.

References

Ballantine, C. M. and Merca, M., ‘Parity of sums of partition numbers and squares in arithmetic progressions’, Ramanujan J. 44 (2017), 617–630.CrossRefGoogle Scholar
Hong, L.-T. and Zhang, S.-T., ‘Proof of the Ballantine–Merca conjecture and theta function identities modulo 2’, Proc. Amer. Math. Soc., to appear.Google Scholar
Merca, M., ‘Congruence identities involving sums of odd divisors function’, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 22(2) (2021), 119–125.Google Scholar