Sign changes of certain arithmetical function at prime powers. (English) Zbl 07442487

Summary: We examine an arithmetical function defined by recursion relations on the sequence \(\{f(p^k)\}_{k\in\mathbb{N}}\) and obtain sufficient condition(s) for the sequence to change sign infinitely often. As an application we give criteria for infinitely many sign changes of Chebyshev polynomials and that of sequence formed by the Fourier coefficients of a cusp form.


11A25 Arithmetic functions; related numbers; inversion formulas
11B83 Special sequences and polynomials
11F30 Fourier coefficients of automorphic forms
Full Text: DOI


[1] Apostol, T. M., Introduction to Analytic Number Theory (1976), New York: Springer, New York · Zbl 0335.10001 · doi:10.1007/978-1-4757-5579-4
[2] S. Banerjee: A note on signs of Fourier coefficients of two cusp forms. Proc. Indian Acad. Sci., Math. Sci. 128 (2018), Article ID 43, 6 pages. · Zbl 1448.11087
[3] Gun, S.; Kohnen, W.; Rath, P., Simultaneous sign change of Fourier-coefficients of two cusp forms, Arch. Math., 105, 413-424 (2015) · Zbl 1339.11055 · doi:10.1007/s00013-015-0829-3
[4] Knopp, M.; Kohnen, W.; Pribitkin, W., On the signs of Fourier coefficients of cusp forms, Ramanujan J., 7, 269-277 (2003) · Zbl 1045.11027 · doi:10.1023/A:1026207515396
[5] Koblitz, N., Introduction to Elliptic Curves and Modular Forms (1993), New York: Springer, New York · Zbl 0804.11039 · doi:10.1007/978-1-4612-0909-6
[6] Kohnen, W.; Martin, Y., Sign changes of Fourier coefficients of cusp forms supported on prime power indices, Int. J. Number Theory, 10, 1921-1927 (2014) · Zbl 1304.11022 · doi:10.1142/S1793042114500626
[7] Meher, J.; Murty, M. R., Sign changes of Fourier coefficients of half-integral weight cusp forms, Int. J. Number Theory, 10, 905-914 (2014) · Zbl 1304.11029 · doi:10.1142/S1793042114500067
[8] Meher, J.; Shankhadhar, K. D.; Viswanadham, G. K., A short note on sign changes, Proc. Indian Acad. Sci., Math. Sci., 123, 315-320 (2013) · Zbl 1281.11043 · doi:10.1007/s12044-013-0139-2
[9] Murty, M. R., Oscillations of Fourier coefficients of modular forms, Math. Ann., 262, 431-446 (1983) · Zbl 0489.10020 · doi:10.1007/BF01456059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.