We use the Weyl bound for Dirichlet L-functions to derive zero-density estimates for L-functions associated to families of fixed-order Dirichlet characters. The results improve on previous bounds given by the author when $\sigma $ is sufficiently distant from the critical line.

A new reciprocity formula for Dirichlet L-functions associated to an arbitrary primitive Dirichlet character of prime modulus q is established. We find an identity relating the fourth moment of individual Dirichlet L-functions in the t-aspect to the cubic moment of central L-values of Hecke–Maaß newforms of level at most $q^{2}$ and primitive central character $\psi ^{2}$ averaged over all primitive nonquadratic characters $\psi $ modulo q. Our formula can be thought of as a reverse version of recent work of Petrow–Young. Direct corollaries involve a variant of Iwaniec’s short interval fourth moment bound and the twelfth moment bound for Dirichlet L-functions, which generalise work of Jutila and Heath-Brown, respectively. This work traverses an intersection of classical analytic number theory and automorphic forms.

The aim of this article is to establish the behaviour of partial Euler products for Dirichlet L-functions under the generalised Riemann hypothesis (GRH) via Ramanujan’s work. To understand the behaviour of Euler products on the critical line, we invoke the deep Riemann hypothesis (DRH). This work clarifies the relation between GRH and DRH.

In Gun and Ramakrishnan [‘On special values of certain Dirichlet L-functions’, Ramanujan J.15 (2008), 275–280], we gave expressions for the special values of certain Dirichlet L-function in terms of finite sums involving Jacobi symbols. In this note we extend our earlier results by giving similar expressions for two more special values of Dirichlet L-functions, namely L(−1,χm) and L(−2,χ−m′), where m,m′ are square-free integers with m≡1 mod 8 and m′≡3 mod 8 and χD is the Kronecker symbol . As a consequence, using the identities of Cohen [‘Sums involving the values at negative integers of L-functions of quadratic characters’, Math. Ann.217 (1975), 271–285], we also express the finite sums with Jacobi symbols in terms of sums involving divisor functions. Finally, we observe that the proof of Theorem 1.2 in Gun and Ramakrishnan (as above) is a direct consequence of Equation (24) in Gun, Manickam and Ramakrishnan [‘A canonical subspace of modular forms of half-integral weight’, Math. Ann.347 (2010), 899–916].