On the Ternary Additive Divisor Problem and the Sixth Moment of the Zeta-Function

The sixth moment of the family of Γ1(q)-automorphic L-functions

“…They are all called formulas of Voronoi type, and we refer to [8] for a general survey of recent developments. The summation formula for τ k is obtained by Ivić in [3], and we will quote here some of his results that we will need.…”

“…The required summation formula is derived in [3] using analytic properties of the corresponding generating Dirichlet series defined for (c, d) = 1 and (s) > 1 as: Its only singularity is the pole of order 3 at s = 1 (for the precise functional equation we refer to Lemma 1 in [3]). For any smooth and compactly supported ψ ∈ C ∞ 0 (0, ∞) letψ(s) = ∞ 0 ψ(x)x s dx x be its Mellin transform-an entire function of rapid decay.…”

“…For any smooth and compactly supported ψ ∈ C ∞ 0 (0, ∞) letψ(s) = ∞ 0 ψ(x)x s dx x be its Mellin transform-an entire function of rapid decay. Then, for any (c, d) = 1 we have the following summation formula ( [3], Theorem 2):…”

Bounds for $\rGL(3) \times \rGL(2)$ $\rL$-functions and $\rGL(3)$ $\rL$-functions

“…Note that a special case of the above lemma (when α = β = γ = 0 ) was given by Ivic (see [Ivi97]). Now let f be a self-dual Hecke-Maass form of type (ν, ν) for SL(3, Z), normalized to have the first Fourier coefficient A(1, 1) equal to 1.…”

Shifted convolution and the Titchmarsh divisor problem over 𝔽q[t]

“…But even a conjectural description of the polynomials P 2(k−1) (u; h) is difficult to obtain (see §7, [5,6]). A variant of the problem about the autocorrelation of the divisor function is to determine an asymptotic for the more general sum given by (1.7)…”