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We prove that for even integer k, almost all of zeros of the period polynomial associated to a cusp form of weight k on Γ0(N) are on the circle |z| = 1/N under some conditions. [ABSTRACT FROM AUTHOR]

We prove that for even integer k\geq k_0, almost all of zeros of the even period polynomial associated to a newform of weight k on \Gamma _0(N) are on the circle |z|=1/\sqrt {N}. [ABSTRACT FROM AUTHOR]

We develop some asymptotics for a kernel function introduced by Kohnen and use them to estimate the number of normalised Hecke eigenforms in $S_k(\Gamma _0(1))$ whose L-values are simultaneously nonvanishing at a given pair of points each of which lies inside the critical strip. [ABSTRACT FROM AUTHOR]

Iwaniec, Luo, and Sarnak proposed Hypothesis S and its generalization which predicts non-trivial bounds for a smooth sum of the product of an arithmetic sequence { a n } and a fractional exponential function. When a n is the Fourier coefficient λ f (n) of a fixed holomorphic cusp form f , however, a resonance phenomenon prohibits any improvement of the bound beyond a barrier. It is believed that this resonance barrier could be overcome when the weight k of f tends to infinity. The present paper is a first step toward this goal by proving non-trivial bounds for this sum when k and the summation length X both tend to infinity. No such non-trivial bounds are previously known if the form f is allowed to move. Similar bounds are also proved for linear phases and for Maass forms. The main technology is improved large sieve inequalities over a short interval. [ABSTRACT FROM AUTHOR]

In this article, we have considered the problem of first sign change of Fourier coefficients of a primitive cusp form at a sparse sequence, where elements of the sparse sequence can be written as a sum of two squares. Following the methods of Kohnen–Sengupta, we give an effective upper bound on elements of such a sparse sequence to have first sign change. [ABSTRACT FROM AUTHOR]

In this article, we have considered the problem of first sign change of Fourier coefficients of a primitive cusp form at a sparse sequence, where elements of the sparse sequence can be written as a sum of two squares. Following the methods of Kohnen–Sengupta, we give an effective upper bound on elements of such a sparse sequence to have first sign change. [ABSTRACT FROM AUTHOR]

Let f be a holomorphic Hecke cusp form with even integral weight k ⩾ 2 for the full modular group, and let χ be a primitive Dirichlet character modulo q. Let Lf (s, χ) be the automorphic L-function attached to f and χ. We study the mean-square estimate of Lf (s, χ) and establish an asymptotic formula. [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 Hecke eigenvalue or normalized Fourier coefficient by λ f (n) . Let g(z) be another distinct primitive holomorphic cusp form of even integral weight ℓ for the full modular group. In this paper, we establish that the set { p | λ f (p j) < λ g (p j) } with 1 ≤ j ≤ 8 has analytic density at least 1 16 [ j + 1 2 ] 2 , and the set { p | λ f (p j) 2 < λ g (p j) 2 } with 1 ≤ j ≤ 4 has analytic density at least 1 4 j (j + 1) 2 . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let G / H {G/H} be a reductive symmetric space of split rank one and let K be a maximal compact subgroup of G. In a previous article the first two authors introduced a notion of cusp forms for G / H {G/H}. We show that the space of cusp forms coincides with the closure of the space of K-finite generalized matrix coefficients of discrete series representations if and only if there exist no K-spherical discrete series representations. Moreover, we prove that every K-spherical discrete series representation occurs with multiplicity one in the Plancherel decomposition of G / H {G/H}. [ABSTRACT FROM AUTHOR]

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]

Structure theorems for spaces of cusp forms with quadratic characters are presented. It is proved that such spaces of levels N ≠ 3, 17, 19 admit exact cutting if and only if the cutting function is a multiplicative η-product. The cases of the levels N = 3, 17, 19 are also studied. [ABSTRACT FROM AUTHOR]

Zagier's conjecture on the asymptotic expansion of the Lambert series where is the Ramanujan's tau function, was proved by Hafner and Stopple. Recently, Chakraborty, Kanemitsu and Maji have extended this result to any cusp forms over the full modular group. The goal of this paper is to extend the asymptotic behavior to cusp forms over any congruence subgroup of the full modular group. [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 Maass cusp form for Γ( N) with Fourier coefficients λ( n) and Laplace eigenvalue $$\frac{1} {4} + k^2 $$ . For real α ≠ 0 and β > 0, consider the sum S ( f; α, β) = ∑ λ( n) e( αn ) ϕ( n/ X), where ϕ is a smooth function of compact support. We prove bounds for the second spectral moment of S ( f; α, β), with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X. This implies that if f has its eigenvalue beyond $$X^{\tfrac{1} {2} + \varepsilon } $$ , the standard resonance main term for S ( f; $$ \pm 2\sqrt q $$ , 1/2), q ∈ ℤ, cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL(2) × GL(2). It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of K ⩽ L ⩽ K . The same bounds can be proved in a similar way for holomorphic cusp forms. [ABSTRACT FROM AUTHOR]

Let g be a holomorphic or Maass Hecke newform of level D and nebentypus χD, and let a ( n) be its n-th Fourier coefficient. We consider the sum $${S_1} = \sum {_{X < n \leqslant 2X}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)}$$ and prove that S has an asymptotic formula when β = 1/2 and α is close to $$\pm 2\sqrt {q/D}$$ for positive integer q ≤ X/4 and X sufficiently large. And when 0 < β < 1 and α, β fail to meet the above condition, we obtain upper bounds of S . We also consider the sum $${S_2} = \sum {_{n > 0}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)\phi \left( {n/X} \right)}$$ with ø( x) ∈ C (0,+∞) and prove that S has better upper bounds than S at some special α and β. [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]

In this paper, we study the exponential sums involving Fourier coefficients of cusp forms for SL ( 2 , Z ) over primes. We are able to establish some new results. In addition, similar results have been obtained for the coefficients of symmetric-square lifts. [ABSTRACT FROM AUTHOR]

We obtain a discrete universality theorem on the approximation of analytic functions by shifts of L-functions of normalized Hecke-eigen cusp forms with respect to congruence subgroups. Such a result is already proved in the authors' previous paper [A. Lauriňcikas, K. Matsumoto, and J. Steuding, Discrete universality of L-functions for new forms, Math. Notes, 78(4):551-558, 2005] under a certain arithmetical condition, which we remove in this paper. Also, we present discrete analogues of results of [A. Lauriňcikas, K. Matsumoto, and J. Steuding, Universality of some functions related to zeta-functions of certain cusp forms, Osaka J. Math., 50(4):1021-1037, 2013]. [ABSTRACT FROM AUTHOR]

We study the zero-density estimates for automorphic L-functions $${L(s, \pi)}$$ for GL when $${\sigma}$$ is near 1. In particular, we get a range of $${\sigma}$$ for which the density hypothesis holds. The proofs use a zero detecting argument, the Halász-Montgomery inequality and a bound for an integral power moment of $${L(1/2+it, \pi)}$$ . [ABSTRACT FROM AUTHOR]

Let f be a primitive holomorphic or Maass cusp form for the group SL ( 2 , Z ) , and a f ( n ) its n th normalized Fourier coefficient. It is proved that, for any α , β ∈ R , there exists an effective positive constant c such that ∑ n ≤ N Λ ( n ) a f ( n ) e ( α n 2 + β n ) ≪ f N exp ⁡ ( − c log ⁡ N ) . [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]

Abstract: We prove, in respect of an arbitrary Hecke congruence subgroup , some new upper bounds for sums involving Fourier coefficients of Γ-automorphic cusp forms on . The Fourier coefficients in question may arise from the Fourier expansion at any given cusp of Γ (our results are not limited to the case ). Our proof utilises an extension, to arbitrary cusps, of a spectral-Kloosterman summation formula for that was obtained by Lokvenec-Guleska (in her doctoral thesis). [Copyright &y& Elsevier]

Results of physical modeling of stopes in soft ores on substitutes are presented. Stope of cusp form is found to be more stable as compared with square chambers. [ABSTRACT FROM AUTHOR]

It is known that if the Fourier coefficients a(n)(n ≥ 1) of an elliptic modular form of even integral weight k ≥ 2 on the Hecke congruence subgroup Γ0(N)(N ∈ N) satisfy the bound a(n) ≪f nc for all n ≥ 1, where c > 0 is any number strictly less than k - 1, then f must be cuspidal. Here we investigate the case of half-integral weight modular forms. The main objective of this note is to show that to deduce that f is a cusp form, it is sufficient to impose a suitable growth condition only on the Fourier coefficients a(|D|) where D is a fundamental discriminant with (-1)kD > 0. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, zero density estimates for automorphic L-functions for are deduced from a bound for an integral power moment of on the critical line . In particular for the Riemann zeta function, classical zero density estimates are extended to short vertical strips. For g being a holomorphic or Maass eigenform for , bounds for zero density for in short strips are proved, which extend Ivićʼs results on long strips. For a self-dual Hecke Maass eigenform f for , estimates of zero density for in short and long strips are also proved. The proofs use a zero detecting argument, a large sieve inequality, a bound for an integral power moment of , the Rankin–Selberg theory, and the Halász–Montgomery–Jutila method. [Copyright &y& Elsevier]

Abstract: Let be the space of all normalized holomorphic Hecke-eigen cusp forms of even integral weight k for the full modular group , and denote by the corresponding symmetric square L-function associated to . In this paper, the lower bound of the higher moment of is established, i.e., for any even number , holds for , where . [Copyright &y& Elsevier]

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]

In this work we consider the representations of positive integers by quadratic forms Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. and Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. of discriminant -31 and we obtain some results concerning the modular forms ℘ (τ ; F, φrs). Moreover we construct a basis for the cusp form space S4(Γ0(31), 1), and then we give some formulas for the number of representations of positive integer n by positive definite quadratic forms. [ABSTRACT FROM AUTHOR]

We obtain a limit theorem for L-functions of elliptic curves twisted by Dirichlet characters with increasing modulus. [ABSTRACT FROM AUTHOR]

This paper is devoted to upper and lower bounds of fractional moments of L-functions attached to certain cusp forms. The upper bound is proved under the analogue of the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

In this paper we prove zero-density estimates of the large sieve type for the automorphic L-function L( s, f ⊗ χ), where f is a holomorphic cusp form and χ(mod q) is a primitive character. [ABSTRACT FROM AUTHOR]

Abstract: Recently Blomer showed that if denote the normalized Fourier coefficients of any holomorphic cusp form f with integral weight, then holds uniformly in . By an elementary argument we show that independent of q, where could be the normalized Fourier coefficients of any reasonable cusp forms, including Maass cusp forms, holomorphic cusp forms with half-integral or integral weights. [Copyright &y& Elsevier]

Abstract: Lehmer''s conjecture asserts that where τ is the Ramanujan τ-function. This is equivalent to the assertion that for any n. A related problem is to find the distribution of primes p for which . These are open problems. We show that the variant of estimating the number of integers n for which n and do not have a non-trivial common factor is more amenable to study. In particular, we show that the number of such is . We prove a similar result for more general cusp forms. This may be seen as a modular analogue of an old result of Erdős on the Euler ϕ function. [Copyright &y& Elsevier]

Suppose that the zeta-function f( s , F ) attached to a normalized eigenform F ( z ) of weight ? does not have zeros for Res>?/2. Let , where ? T >0, ? T ?8 and log ? T =o(log log T ), as T ?8, and let . It is proved the estimate as T ?8. [ABSTRACT FROM AUTHOR]