We compute zeros off the critical line of a Dirichlet series considered by H. Davenport and H. Heilbronn. This computation is accomplished by deforming a Dirichlet series with a set of known zeros into the DavenportHeilbronn series.

This integral trick assumes greater significance when the function f is known to have strong decay properties both at 0 and at ∞, since then the Mellin transform is entire in s. One way to ensure such rapid decay is via eigenfunction properties in the context of automorphic forms. [2] • The archetype Mellin transform: zeta from theta • Abstracting to holomorphic modular forms • Variation: wavef...

For a fixed hyperelliptic curve C given by the equation y 1⁄4 f ðxÞ with f A Z1⁄2x having distinct roots and degree at least 5, we study the variation of rational points on the quadratic twists Cm whose equation is given by my 2 1⁄4 f ðxÞ. More precisely, we study the Dirichlet series Df ðsÞ 1⁄4 P 0 m00aCmðQÞjmj s where the summation is over all non-zero squarefree integers. We show that Df ðsÞ...

We determine the maximum possible size of the qth moment of a Dirichlet series, for 1 6 q 6 2.

1 n : (n, f) = 1 0 : otherwise , the absolute convergence for s > 1 follows from the convergence of ζQ(s) for s > 1. (ii) Given the convergence for s > 1, the proof of the product expansion is identical to the case ζQ(s). We also observe that L(s, χ1) = ζQ(s). As we will see, for non-trivial characters, the L-series behave rather differently. We will analyze these functions further, but first w...

We attach a certain n × n matrix An to the Dirichlet series L(s) = ∑ ∞ k=1 akk . We study the determinant, characteristic polynomial, eigenvalues, and eigenvectors of these matrices. The determinant of An can be understood as a weighted sum of the first n coefficients of the Dirichlet series L(s). We give an interpretation of the partial sum of a Dirichlet series as a product of eigenvalues. In...