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We investigate the average behavior of the $n$ th normalized Fourier coefficients of the $j$ th ( $j \geq 2$ be any fixed integer) symmetric power $L$ -function (i.e., $L (s, {sym}^{j} f)$ ), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $S L (2, ℤ)$ over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum $S_{j}^{*} : = \sum_{a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} + a_{5}^{2} + a_{6}^{2} \leq x (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}) \in ℤ^{6}} λ_{{sym}^{j} f}^{2} (a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} + a_{5}^{2} + a_{6}^{2}),$ where $x$ is sufficiently large, and $L (s, {sym}^{j} f) : = \sum_{n = 1}^{\infty} \frac{λ_{{sym}^{j} f} (n)}{n^{s}} .$ When $j = 2$ , the error term which we obtain improves the earlier known result.

On the coefficients of Drinfeld modular forms.

Ernst-Ulrich Gekeler (1988)

Inventiones mathematicae

On the cuspidal divisor class group of a Drinfeld modular curve.

Gekeler, Ernst-Ulrich (1997)

Documenta Mathematica

On the existence of cusp forms over function fields.

Isaac Efrat (1989)

Journal für die reine und angewandte Mathematik

On the Fourier coefficients of modular forms

Douglas L. Ulmer (1995)

Annales scientifiques de l'École Normale Supérieure

On the Fourier coefficients of small positive powers of 0 ... .

M.L Knopp (1986)

Inventiones mathematicae

On the functional properties of Bessel zeta-functions

Takumi Noda (2015)

Acta Arithmetica

Zeta-functions associated with modified Bessel functions are introduced as ordinary Dirichlet series whose coefficients are J-Bessel and K-Bessel functions. Integral representations, transformation formulas, a power series expansion involving the Riemann zeta-function and a recurrence formula are given. The inverse Laplace transform of Weber's first exponential integral is the basic tool to derive the integral representations. As an application, we give a new proof of the Fourier series expansion...

On the Generating Functions of Certain Eisenstein Series.

Michael Ackerman (1979)

Mathematische Annalen

On the higher power moments of cusp form coefficients over sums of two squares

Guodong Hua (2022)

Czechoslovak Mathematical Journal

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $Γ = SL (2, ℤ)$ . Denote by $λ_{f} (n)$ the $n$ th normalized Fourier coefficient of $f$ . We are interested in the average behaviour of the sum $\sum_{a^{2} + b^{2} \leq x} λ_{f}^{j} (a^{2} + b^{2})$ for $x \geq 1$ , where $a, b \in ℤ$ and $j \geq 9$ is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power $L$ -functions and Rankin-Selberg $L$ -functions.

On the image of $Λ$ -adic Galois representations

Ami Fischman (2002)

Annales de l’institut Fourier

We explore the question of how big the image of a Galois representation attached to a $Λ$ -adic modular form with no complex multiplication is and show that for a “generic” set of $Λ$ -adic modular forms (normalized, ordinary eigenforms with no complex multiplication), all have a large image.

On the number of representations of positive integers by a direct sum of binary quadratic forms with discriminant $- 23$ .

Lomadze, G. (1997)

Georgian Mathematical Journal

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Displaying 21 – 40 of 63

On Ramanujan's cubic continued fraction

On some entire modular forms of an integral weight for the congruence subgroup $Γ_{0} (4 N)$ .

On some entire modular forms of weights 5 and 6 for the congruence group $Γ_{0} (4 N)$ .

On the Absoluate Value of Ramanuja n's ...-Function.

On the asymptotics of the number of p̅-core partitions of integers

On the average behavior of the Fourier coefficients of $j$ th symmetric power $L$ -function over certain sequences of positive integers