Displaying similar documents to “A note on average behaviour of the Fourier coefficients of j th symmetric power L -function over certain sparse sequence of positive integers”

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

Anubhav Sharma, Ayyadurai Sankaranarayanan (2023)

Czechoslovak Mathematical Journal

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We investigate the average behavior of the n th normalized Fourier coefficients of the j th ( j 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 , ) over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum S j * : = 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 ( 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 ) : = n = 1 λ sym j f ( n ) n s . When j = 2 , the error term which we obtain improves the earlier known result.

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.

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.

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Generalized versions of Ilmanen lemma: Insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove that for a normed linear space X , if f 1 : X is continuous and semiconvex with modulus ω , f 2 : X is continuous and semiconcave with modulus ω and f 1 f 2 , then there exists f C 1 , ω ( X ) such that f 1 f f 2 . Using this result we prove a generalization of Ilmanen lemma (which deals with the case ω ( t ) = t ) to the case of an arbitrary nontrivial modulus ω . This generalization (where a C l o c 1 , ω function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

Multifractal analysis of the divergence of Fourier series

Frédéric Bayart, Yanick Heurteaux (2012)

Annales scientifiques de l'École Normale Supérieure

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A famous theorem of Carleson says that, given any function f L p ( 𝕋 ) , p ( 1 , + ) , its Fourier series ( S n f ( x ) ) converges for almost every x 𝕋 . Beside this property, the series may diverge at some point, without exceeding O ( n 1 / p ) . We define the divergence index at  x as the infimum of the positive real numbers β such that S n f ( x ) = O ( n β ) and we are interested in the size of the exceptional sets E β , namely the sets of  x 𝕋 with divergence index equal to  β . We show that quasi-all functions in  L p ( 𝕋 ) have a multifractal behavior with respect to...

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...

The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

Soufiane Mezroui (2020)

Czechoslovak Mathematical Journal

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Let f = n = 1 a ( n ) q n S k + 1 / 2 ( N , χ 0 ) be a nonzero cuspidal Hecke eigenform of weight k + 1 2 and the trivial nebentypus χ 0 , where the Fourier coefficients a ( n ) are real. Bruinier and Kohnen conjectured that the signs of a ( n ) are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies { a ( t n 2 ) } n , where t is a squarefree integer such that a ( t ) 0 . Let q and d be natural numbers such that ( d , q ) = 1 . In this work, we show that { a ( t n 2 ) } n is equidistributed over any arithmetic progression n d mod q .

Theoretical analysis for 1 - 2 minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

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We investigate the recovery of k -sparse signals using the 1 - 2 minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume k -sparse signals 𝐱 with the prior support T which is composed of g true indices and b wrong...

A new characterization of symmetric group by NSE

Azam Babai, Zeinab Akhlaghi (2017)

Czechoslovak Mathematical Journal

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Let G be a group and ω ( G ) be the set of element orders of G . Let k ω ( G ) and m k ( G ) be the number of elements of order k in G . Let nse ( G ) = { m k ( G ) : k ω ( G ) } . Assume r is a prime number and let G be a group such that nse ( G ) = nse ( S r ) , where S r is the symmetric group of degree r . In this paper we prove that G S r , if r divides the order of G and r 2 does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.