Search

Let f be a nonzero cusp form of even integral weight k ⩾ 2 on the Hecke congruence subgroup Γ 0 ( N ) with N squarefree. Suppose that the normalized Fourier coefficients λ f ( n ) of f are real. We prove that the first sign change of λ f ( n ) occurs in the range n ≪ ( k N ) 2 + ε . This improves upon the earlier result of Choie and Kohnen [2] . [ABSTRACT FROM AUTHOR]

Let a(n) be the Fourier coefficients of a holomorphic cusp form of weight κ = 2n ≥ 12 for the full modular group and A(x) = Ʃn≤xa(n). In this paper, we establish an asymptotic formula of the fourth power moment of A(x) and prove that ∫1TA4(x)dx=364κπ4s4;2(a˜)T2κ+O(T2κ−δ4+ε) with δ4 = 1/8, which improves the previous result. [ABSTRACT FROM AUTHOR]

Let λf (n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∊ Sk(Г). In this paper, we study the asymptotics for double sum involving Fourier coefficient λi f (n) of cusp forms, where i = 2, 4, 6, 8. [ABSTRACT FROM AUTHOR]

In this paper we consider the mean value of hybrid arithmetic function of the form λ6f (n)σb(n)Øc(n), where λf (n) is the n-th Fourier coefficients of the holomorphic cusp form f, σ(n) is the sum-of-divisors function, Ø(n) is the Euler's totient function. In detail, we prove that for any e > 0, ∑ n⩽x λ6f (n)σb(n)Øc(n) = x(b+c+1)P4(log x) + O(xb+c+631/652+ε); where b; c ∊ ℝ. [ABSTRACT FROM AUTHOR]

Let f be a Hecke eigencusp form of even integral weight k or Maass cusp form for the full modular group S L 2 ( Z ) . Denote by λ sym m f ( n ) the n th normalized coefficient of the Dirichlet expansion of the m th symmetric power L -function associated to f . In this paper, we establish some bounds for ∑ n ⩽ x λ sym m f ( n ) , ∑ n ⩽ x λ f ( n m ) , which improve the corresponding results of Lau & Lü [8] . [ABSTRACT FROM AUTHOR]

Let f( z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth normalized Fourier coefficient (Hecke eigenvalue) by λ( n). Let d( n) be the Dirichlet divisor function. In this paper, we establish that where $${0 < \delta_{j} < 1}$$ , $${0 < \delta_{i,j}\leqq 1}$$ ( $${\delta_{i,j} = 1}$$ if and only if i = j), and . [ABSTRACT FROM AUTHOR]

From a result of Waldspurger [W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math.64 (1981) 175-198], it is known that the normalized Fourier coefficients a(m) of a half-integral weight holomorphic cusp eigenform 픣 are, up to a finite set of factors, one of $\pm \sqrt{L(\frac{1}{2}, f, \chi_m)}$ when m is square-free and f is the integral weight cusp form related to 픣 by the Shimura correspondence [G. Shimura, On modular forms of half-integral weight, Ann. of Math.97 (1973) 440-481]. In this paper we address a question posed by Kohnen: which square root is a(m)? In particular, if we look at the set of a(m) with m square-free, do these Fourier coefficients change sign infinitely often? By partially analytically continuing a related Dirichlet series, we are able to show that this is so. [ABSTRACT FROM AUTHOR]