Fourier Coefficients Of Cusp Forms Research Articles

On calculations of the Fourier coefficients of cusp forms of half-integral weight given by the Shintani lift

On calculations of the Fourier coefficients of cusp forms of half-integral weight given by the Shintani lift

On squares of Fourier coefficients twist exponential functions with applications

Let f be a Hecke–Maass cusp form of weight zero for [Formula: see text] and [Formula: see text] be the nth Fourier coefficient. For almost all [Formula: see text] we have [Formula: see text] which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than [Formula: see text] For all [Formula: see text] of type [Formula: see text] we also show that [Formula: see text] where [Formula: see text] This result relies heavily on a generalized double sum (see Theorem 1.3).

On the first simultaneous sign change and non-vanishing of Fourier coefficients of cusp forms

On the first simultaneous sign change and non-vanishing of Fourier coefficients of cusp forms

Estimates for the shifted convolution sum involving Fourier coefficients of cusp forms of half-integral weight

Estimates for the shifted convolution sum involving Fourier coefficients of cusp forms of half-integral weight

Sums of k-th powers and Fourier coefficients of cusp forms

Sums of k-th powers and Fourier coefficients of cusp forms

Weight aspect exponential sums for Fourier coefficients of cusp forms

Weight aspect exponential sums for Fourier coefficients of cusp forms

A short note on sign changes and non-vanishing of Fourier coefficients of half-integral weight cusp forms

We study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight.

Uniform Titchmarsh divisor problems

Asymptotic formulae for Titchmarsh-type divisor sums are obtained with strong error terms that are uniform in the shift parameter. This applies to more general arithmetic functions such as sums of two squares, improving the error term in the representation of the number as a sum of a prime and two squares, and to Fourier coefficients of cusp forms, generalizing a result of Pitt.

Sign Changes of Fourier Coefficients of Cusp Forms of Half-Integral Weight Over Split and Inert Primes in Quadratic Number Fields

In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class $$t\mathbb {Z}^2$$ , we investigate the sign changes in the $$tp^2$$ -th coefficient as p runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. We show that infinitely many sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discriminant also divides the level.

Some remarks on the Fourier coefficients of cusp forms

In this paper, we consider the angular changes of Fourier coefficients of half integral weight cusp forms and sign changes of [Formula: see text]-exponents of generalized modular functions.

Exponential sums of squares of Fourier coefficients of cusp forms

We prove nontrivial estimates for linear sums of squares of Fourier coefficients of holomorphic and Maass cusp forms twisted by additive characters. For holomorphic forms f, we show that if $$|\alpha -a/q| \le 1/q^2$$ with $$(a,q)=1$$ , then for any $$\varepsilon >0$$ , $$\begin{aligned} \qquad \qquad \sum _{n\leqslant X}{\lambda _f(n)}^2 e(n\alpha ) \ll _{f, \varepsilon } X^{{\frac{4}{5}}+\varepsilon } \quad \text {for} \ X^{{\frac{1}{5}}} \ll q \ll X^{{\frac{4}{5}}}. \end{aligned}$$ Moreover, for any $$\varepsilon > 0,$$ there exists a set $$S \subset (0, 1)$$ with $$\mu (S)=1$$ such that for every $$\alpha \in S$$ , there exists $$X_0=X_0(\alpha )$$ such that the above inequality holds true for any $$\alpha \in S$$ and $$X \geqslant X_0(\alpha ).$$ A weaker bound for Maass cusp forms is also established.

Signs of Fourier coefficients of cusp form at sum of two squares

In this article, we investigate the sign changes of the sequence of coefficients at sum of two squares where the coefficients are the Fourier coefficients of the normalized Hecke eigen cusp form for the full modular group. We provide the quantitative result for the number of sign changes of the sequence in a small interval.

Angular changes of Fourier coefficients at primes

We study the angle changes of Fourier coefficients of cusp forms and q-exponents of generalized modular functions at primes. More precisely, we prove that both these subsequences, under certain conditions, fall infinitely often outside any given wedge \(\mathcal {W}(\theta _1, \theta _2):=\{re^{i\theta }: r>0, \theta \in [\theta _1,\theta _2]\}\) with \(0\le \theta _2-\theta _1< \pi \).

Poincaré and Eisenstein series for Jacobi forms of lattice index

Poincaré and Eisenstein series are building blocks for every type of automorphic forms. We define Poincaré series for Jacobi forms of lattice index and show that they reproduce Fourier coefficients of cusp forms under the Petersson scalar product. We compute the Fourier expansions of Poincaré and Eisenstein series and give an explicit formula for the Fourier coefficients of the trivial Eisenstein series in terms of values of Dirichlet L-functions at negative integers. For even weight and fixed index, finite linear combinations of Fourier coefficients of non-trivial Eisenstein series are equal to finite linear combinations of Fourier coefficients of the trivial one. This result is used to obtain formulas for the Fourier coefficients of Eisenstein series associated with isotropic elements of small order.

On triple correlations of Fourier coefficients of cusp forms. II

The triple correlation [Formula: see text] for arbitrary coefficients [Formula: see text] is estimated on average over [Formula: see text] in some short intervals. By introducing a device of short intervals, we are able to reduce this problem to uniform oscillations of one of the three coefficients, say [Formula: see text], against additive characters of [Formula: see text] over short intervals. The argument is simple, but refines previous arguments in certain cases. More precise estimates are also obtained by taking [Formula: see text] to be Fourier coefficients of cusp forms and Möbius functions, which substantially improve previous results.

On the Ramanujan–Petersson conjecture for modular forms of half-integral weight

Abstract We investigate the (still unknown) Ramanujan–Petersson conjecture about the growth of the Fourier coefficients of cusp forms of half-integral weight and prove that it is optimal, at least for newforms in the plus space.

Oscillatory behavior and equidistribution of signs of Fourier coefficients of cusp forms

In this paper, we discuss questions related to the oscillatory behavior and the equidistribution of signs for certain subfamilies of Fourier coefficients of integral weight newforms with a non-trivial nebentypus as well as Fourier coefficients of eigenforms of half-integral weight reachable by the Shimura correspondence.

On the coefficients of symmetric power L-functions

We study the signs of the Fourier coefficients of a newform. Let [Formula: see text] be a normalized newform of weight [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the [Formula: see text]th Fourier coefficient of [Formula: see text]. For any fixed positive integer [Formula: see text], we study the distribution of the signs of [Formula: see text], where [Formula: see text] runs over all prime numbers. We also find out the abscissas of absolute convergence of two Dirichlet series with coefficients involving the Fourier coefficients of cusp forms and the coefficients of symmetric power [Formula: see text]-functions.

On the first sign change of Fourier coefficients of cusp forms

Let f be a nonzero cusp form of even integral weight k⩾2 on the Hecke congruence subgroup Γ0(N) with N squarefree. Suppose that the normalized Fourier coefficients λf(n) of f are real. We prove that the first sign change of λf(n) occurs in the range n≪(kN)2+ε. This improves upon the earlier result of Choie and Kohnen [2].

On the fourth power moment of Fourier coefficients of cusp form

Let $a(n)$ be the Fourier coefficients of a holomorphic cusp form of weight $\kappa=2n\geqslant12$ for the full modular group and $A(x)=\sum\limits_{n\leqslant x}a(n)$. In this paper, we establish an asymptotic formula of the fourth power moment of $A(x)$ and prove that \begin{equation*} \int_1^TA^4(x)\mathrm{d}x=\frac{3}{64\kappa\pi^4}s_{4;2}(\tilde{a}) T^{2\kappa}+O\big(T^{2\kappa-\delta_4+\varepsilon}\big) \end{equation*} with $\delta_4=1/8$, which improves the previous result.