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Displaying similar documents to “On the 4-norm of an automorphic form”

Proof of a conjectured three-valued family of Weil sums of binomials

Daniel J. Katz, Philippe Langevin (2015)

Acta Arithmetica

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We consider Weil sums of binomials of the form W F , d ( a ) = x F ψ ( x d - a x ) , where F is a finite field, ψ: F → ℂ is the canonical additive character, g c d ( d , | F × | ) = 1 , and a F × . If we fix F and d, and examine the values of W F , d ( a ) as a runs through F × , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo | F × | ). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...

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

Guodong Hua (2022)

Czechoslovak Mathematical Journal

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Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ = SL ( 2 , ) . Denote by λ f ( n ) the n th normalized Fourier coefficient of f . We are interested in the average behaviour of the sum a 2 + b 2 x λ f j ( a 2 + b 2 ) for x 1 , where a , b and j 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.

Generalized divisor problem for new forms of higher level

Krishnarjun Krishnamoorthy (2022)

Czechoslovak Mathematical Journal

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Suppose that f is a primitive Hecke eigenform or a Mass cusp form for Γ 0 ( N ) with normalized eigenvalues λ f ( n ) and let X > 1 be a real number. We consider the sum 𝒮 k ( X ) : = n < X n = n 1 , n 2 , ... , n k λ f ( n 1 ) λ f ( n 2 ) ... λ f ( n k ) and show that 𝒮 k ( X ) f , ϵ X 1 - 3 / ( 2 ( k + 3 ) ) + ϵ for every k 1 and ϵ > 0 . The same problem was considered for the case N = 1 , that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for k 5 . Since the result is valid...

On certain G L ( 6 ) form and its Rankin-Selberg convolution

Amrinder Kaur, Ayyadurai Sankaranarayanan (2024)

Czechoslovak Mathematical Journal

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We consider L G ( s ) to be the L -function attached to a particular automorphic form G on G L ( 6 ) . We establish an upper bound for the mean square estimate on the critical line of Rankin-Selberg L -function L G × G ( s ) . As an application of this result, we give an asymptotic formula for the discrete sum of coefficients of L G × G ( s ) .

On higher moments of Hecke eigenvalues attached to cusp forms

Guodong Hua (2022)

Czechoslovak Mathematical Journal

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Let f , g and h be three distinct primitive holomorphic cusp forms of even integral weights k 1 , k 2 and k 3 for the full modular group Γ = SL ( 2 , ) , respectively, and let λ f ( n ) , λ g ( n ) and λ h ( n ) denote the n th normalized Fourier coefficients of f , g and h , respectively. We consider the cancellations of sums related to arithmetic functions λ g ( n ) , λ h ( n ) twisted by λ f ( n ) and establish the following results: n x λ f ( n ) λ g ( n ) i λ h ( n ) j f , g , h , ε x 1 - 1 / 2 i + j + ε for any ε > 0 , where 1 i 2 , j 5 are any fixed positive integers.

Hybrid sup-norm bounds for Hecke–Maass cusp forms

Nicolas Templier (2015)

Journal of the European Mathematical Society

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Let f be a Hecke–Maass cusp form of eigenvalue λ and square-free level N . Normalize the hyperbolic measure such that vol ( Y 0 ( N ) ) = 1 and the form f such that f 2 = 1 . It is shown that f ϵ λ 5 24 + ϵ N 1 3 + ϵ for all ϵ > 0 . This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.

Waring's number for large subgroups of ℤ*ₚ*

Todd Cochrane, Derrick Hart, Christopher Pinner, Craig Spencer (2014)

Acta Arithmetica

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Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain t = p 22 / 39 + ϵ , t = p 15 / 29 + ϵ and for s ≥ 6, t = p ( 9 s + 45 ) / ( 29 s + 33 ) + ϵ . For s ≥ 24 further improvements are made, such as t 32 = p 5 / 16 + ϵ and t 128 = p 1 / 4 .

A note on average behaviour of the Fourier coefficients of j th symmetric power L -function over certain sparse sequence of positive integers

Youjun Wang (2024)

Czechoslovak Mathematical Journal

Similarity:

Let j 2 be a given integer. Let H k * be the set of all normalized primitive holomorphic cusp forms of even integral weight k 2 for the full modulo group SL ( 2 , ) . For f H k * , denote by λ sym j f ( n ) the n th normalized Fourier coefficient of j th symmetric power L -function ( L ( s , sym j f ) ) attached to f . We are interested in the average behaviour of the sum n = a 1 2 + a 2 2 + a 3 2 + a 4 2 + a 5 2 + a 6 2 x ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) 6 λ sym j f 2 ( n ) , where x is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).

On the k -polygonal numbers and the mean value of Dedekind sums

Jing Guo, Xiaoxue Li (2016)

Czechoslovak Mathematical Journal

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For any positive integer k 3 , it is easy to prove that the k -polygonal numbers are a n ( k ) = ( 2 n + n ( n - 1 ) ( k - 2 ) ) / 2 . The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet L -functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums S ( a n ( k ) a ¯ m ( k ) , p ) for k -polygonal numbers with 1 m , n p - 1 , and give an interesting computational formula for it.

Some new sums related to D. H. Lehmer problem

Han Zhang, Wenpeng Zhang (2015)

Czechoslovak Mathematical Journal

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About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let p be a prime, and let N ( k ; p ) denote the number of all 1 a i p - 1 such that a 1 a 2 a k 1 mod p and 2 a i + a ¯ i + 1 , i = 1 , 2 , , k . The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function N ( k ; p ) , and give an interesting asymptotic formula...

On the topology of polynomials with bounded integer coefficients

De-Jun Feng (2016)

Journal of the European Mathematical Society

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For a real number q > 1 and a positive integer m , let Y m ( q ) : = i = 0 n ϵ i q i : ϵ i 0 , ± 1 , ... , ± m , n = 0 , 1 , ... . In this paper, we show that Y m ( q ) is dense in if and only if q < m + 1 and q is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik [8].