Abstract
We examine an arithmetical function defined by recursion relations on the sequence {f(pk)}k∈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.
Similar content being viewed by others
References
T. M. Apostol: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics. Springer, New York, 1976.
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.
S. Gun, W. Kohnen, P. Rath: Simultaneous sign change of Fourier-coefficients of two cusp forms. Arch. Math. 105 (2015), 413–424.
M. Knopp, W. Kohnen, W. Pribitkin: On the signs of Fourier coefficients of cusp forms. Ramanujan J. 7 (2003), 269–277.
N. Koblitz: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics 97. Springer, New York, 1993.
W. Kohnen, Y. Martin: Sign changes of Fourier coefficients of cusp forms supported on prime power indices. Int. J. Number Theory 10 (2014), 1921–1927.
J. Meher, M. R. Murty: Sign changes of Fourier coefficients of half-integral weight cusp forms. Int. J. Number Theory 10 (2014), 905–914.
J. Meher, K. D. Shankhadhar, G. K. Viswanadham: A short note on sign changes. Proc. Indian Acad. Sci., Math. Sci. 123 (2013), 315–320.
M. R. Murty: Oscillations of Fourier coefficients of modular forms. Math. Ann. 262 (1983), 431–446.
Acknowledgements
The authors are indebted to Professor B. Ramakrishnan for his encouragement and for many fruitful suggestions. The authors are grateful to anonymous referee for his/her valuable suggestions and remarks which improved the exposition of the paper. The authors acknowledge Harish-Chandra Research Institute for fantastic facilities and for the serene ambience that it facilitates.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Agnihotri, R., Chakraborty, K. Sign changes of certain arithmetical function at prime powers. Czech Math J 71, 1221–1228 (2021). https://doi.org/10.21136/CMJ.2021.0398-20
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/CMJ.2021.0398-20