Sign Changes Of Coefficients Research Articles

On the Simultaneous Sign Changes of Coefficients of Rankin–Selberg L-Functions over a Certain Integral Binary Quadratic Form

In this paper, we consider the simultaneous sign changes of coefficients of Rankin–Selberg L-functions associated to two distinct Hecke eigenforms supported at positive integers represented by some certain primitive reduced integral binary quadratic form with negative discriminant D. We provide a quantitative result for the number of sign changes of such sequence in the interval (x, 2x] for sufficiently large x.

Sign changes of coefficients of half-integral weight Hecke eigenforms

Let f \mathfrak {f} be a cusp form of half-weight k + 1 / 2 k+1/2 and at most quadratic nebentype character whose Fourier coefficients are denoted as a f ( n ) \mathfrak {a}_\mathfrak {f}(n) . We study sign changes of the family { a f ( t p 2 ) } p ∈ P \{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}} where t t is a square-free number and p p runs through the prime numbers. By taking use of the relationship between the cusp form f \mathfrak {f} of half-integral weight and the Shimura lift f t f_t , we establish that under certain conditions, the Hecke eigenforms of half-integral weight could be determined by sign changes of the sequence { a f ( t p 2 ) } p ∈ P \{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}} .

Sign changes of coefficients of powers of the infinite Borwein product

We denote by ct(m)(n) the coefficient of qn in the series expansion of (q;q)∞m(qt;qt)∞−m, which is the m-th power of the infinite Borwein product. Let t and m be positive integers with m(t−1)≤24. We provide asymptotic formula for ct(m)(n), and give characterizations of n for which ct(m)(n) is positive, negative or zero. We show that ct(m)(n) is ultimately periodic in sign and conjecture that this is still true for other positive integer values of t and m. Furthermore, we confirm this conjecture in the cases (t,m)=(2,m),(p,1),(p,3) for arbitrary positive integer m and prime p.

Sign changes of coefficients and sums of coefficients of L-functions

We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty [24] to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an L-function. As immediate applications, we prove that there are sign changes in intervals within sequences of coefficients of GL(2) holomorphic cusp forms, GL(2) Maass forms, and GL(3) Maass forms. Building on [13,14], we prove that there are sign changes in intervals within sequences of partial sums of coefficients of GL(2) holomorphic cusp forms and Maass forms.

Sign changes of coefficients of certain Dirichlet series

In this paper, we give criteria for infinitely many sign changes of the coefficients of any Dirichlet series if the coefficients are real numbers. We also provide examples where our criteria are applicable.

On the sign changes of coefficients of general Dirichlet series

Under what conditions do the (possibly complex) coefficients of a general Dirichlet series exhibit oscillatory behavior? In this work we invoke Laguerre’s Rule of Signs and Landau’s Theorem to provide a rather simple answer to this question. Furthermore, we explain how our result easily applies to a multitude of functions.