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We estimate short exponential sums weighted by the Fourier coefficients of a Maass form. This requires working out a certain transformation formula for non-linear exponential sums, which is of independent interest. We also discuss how the results depend on the growth of the Fourier coefficients in question. As a byproduct of these considerations, we can slightly extend the range of validity of a short exponential sum estimate for holomorphic cusp forms. The short estimates allow us to reduce smoothing errors. In particular, we prove an analogue of an approximate functional equation previously proven for holomorphic cusp form coefficients. As an application of these, we remove the logarithm from the classical upper bound for long linear sums weighted by Fourier coefficients of Maass forms, the resulting estimate being the best possible. This also involves improving the upper bounds for long linear sums with rational additive twists, the gains again allowed by the estimates for the short sums. Finally, we shall use the approximate functional equation to bound somewhat longer short exponential sums., Comment: 58 pp

In this paper, we study two-dimensional divisor problems of the Fourier coefficients of some automorphic product L-functions attached to the primitive holomorphic cusp form f(z) with weight k for the full modular group SL2(Z). Additionally, we establish the upper bound and the asymptotic formula for these divisor problems on average, respectively.

In this paper, we show that half of non-zero coefficients of the spinor zeta function of a Siegel cusp form of genus 2 are positive and half are negative., Comment: 19 pages

Let a ( n ) be the normalized Fourier coefficient of a holomorphic cusp form of weight k or a Maass cusp form with Laplacian eigenvalue 1 4 + r 2 for SL 2 ( Z ) . We consider exponential sum of the form ∑ n ⩽ X a ( n ) e ( α n 2 + β n ) and obtain the upper bound X 7 8 + e which is uniformly in α , β ∈ R .

Lehmer's conjecture asserts that τ ( p ) ≠ 0 where τ is the Ramanujan τ-function. This is equivalent to the assertion that τ ( n ) ≠ 0 for any n. A related problem is to find the distribution of primes p for which τ ( p ) ≡ 0 ( mod p ) . These are open problems. We show that the variant of estimating the number of integers n for which n and τ ( n ) do not have a non-trivial common factor is more amenable to study. In particular, we show that the number of such n ⩽ x is ≪ x / log log log x . 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.

In this paper, we establish a Voronoi formula for the spinor zeta function of a Siegel cusp form of genus 2. We deduce from this formula quantitative results on the number of its positive (resp. negative) coefficients in some short intervals., Comment: 12 pages

Let f be a cusp form of weight k+1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato-Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp^2)}_p where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn^2)}_n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato-Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind-Dirichlet density., 8 pages; typos corrected, final version, accepted for publication in Archiv der Mathematik

The purpose of the article is to estimate the mean square of a squareroot length exponential sum of Fourier coefficients of a holomorphic cusp form., Comment: fixed an error at the end of the paper, and corrected some typos

Simple homomorphisms to elliptic modular forms are defined on the ring of Siegel modular forms and linear relations on the Fourier coefficients of Siegel modular forms are implied by the codomains of these homomorphisms. We use the linear relations provided by these homomorphisms to compute the Siegel cusp forms of degree n and weight k in some new cases: ( n , k ) = ( 4 , 14 ) , ( 4 , 16 ) , ( 5 , 8 ) , ( 5 , 10 ) , ( 6 , 8 ) . We also compute enough Fourier coefficients using this method to determine the Hecke eigenforms in the nontrivial cases. We also put the open question of whether our technique always succeeds in a precise form. As a partial converse we prove that the Fourier series of Siegel modular forms are characterized among all formal series by the codomain spaces of these homomorphisms and a certain boundedness condition.