The sign changes of Fourier coefficients of self-adjoint forms is an important area of research in number theory. Professor Yingnan Wang and his collaborators have made progress on two projects in this field.
1.Paper Title：On the first sign change of Fourier coefficients of cusp forms
Professor Yingnan Wang, in collaboration with Associate Professor Guohua Chen of North China University of Water Resources and Electric Power and Professor Xujin Liu of the University of Hong Kong, has proposed a new method addressing the problem of the first sign change of Fourier coefficients for general holomorphic cusp forms of weight k and level N over GL(2). This method first constructs the Rankin-Selberg L-function of a general holomorphic cusp form using the form of the Fourier coefficients. Then, by leveraging the relationship between Eisenstein series of level N and Eisenstein series on the total modular group, it proves the analytic extension and functional equations of the Rankin-Selberg L-function constructed from the aforementioned form, thereby deriving relevant results regarding the first sign change of the Fourier coefficients. This method is equally applicable to the sign problem of Fourier coefficients for semi-integer-weighted holomorphic cusp forms. These research findings were recently published in the American Journal of Mathematics.
Full text link:
https://muse.jhu.edu/article/986597
2.Paper Title：On Signs of Fourier Coefficients on GL(n)
Professor Yingnan Wang, in collaboration with Associate Professor Didier Lesesvre of the University of Lille, France, and Dr. Minghao Wu of The Chinese University of Hong Kong, proved that for any n > 2, the proportion of Hecke-Maass cusp forms on GL(n) that satisfy the condition of having a positive density of sign changes in the sequence of real Fourier coefficients is at least 1-a, where a is an arbitrarily small positive real number; Furthermore, by using the vertical version of the Sato-Tate distribution formula with a residual term to replace the generalized Ramanujan conjecture, they unconditionally proved that the proportion of Hecke-Maass cusp forms on GL(n) with real Fourier coefficients and a sequence where the ratio of positive to negative numbers is constant is at least 1-b, where b is also an arbitrarily small positive real number. These research findings were published in “International Mathematics Research Notices.”
Full text link:
https://doi.org/10.1093/imrn/rnaf381
Biography:
Yingnan Wang is a professor and doctoral advisor at the School of Mathematical Sciences, Shenzhen University. He received his bachelor’s and master’s degrees from Shandong University and his Ph.D. from the University of Hong Kong. His research focuses on analytic number theory, specifically on automorphic forms and automorphic L-functions. His research findings have been published in journals such as the American Journal of Mathematics, Advances in Mathematics, and International Mathematics Research Notices.
