Theta-series associated with indefinite ternary quadratic forms and the related Dirichlet series

One obtains theoretic-numerical results connected with the investigation of the Fourier coefficients of theta-series, associated with indefinite ternary quadratic forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 41–50, 1981.     

Dirichlet series associated with square of class numbers of binary quadratic forms

We give meromorphic continuations and functional equations of Mellin transform and Rankin–Selberg convolution associated with a real analytic Eisenstein series of half-integral weight. Applications to Dirichlet series whose coefficients are the class numbers of binary quadratic forms are given.     

Theta series of inhomogeneous quadratic forms

Ohne Zusammenfassung     

Weyl group multiple Dirichlet series constructed from quadratic characters

We construct multiple Dirichlet series in several complex variables whose coefficients involve quadratic residue symbols. The series are shown to have an analytic continuation and satisfy a certain group of functional equations. These are the first examples of an infinite collection of unstable Weyl group multiple Dirichlet series in greater than two variables having the properties predicted in [2].     

The mean square of Dirichlet series associated with automorphic forms

Letf be a non-holomorphic automorphic form of real weight and eigenvalue λ=1/4?ρ ², ?ρ≥0, which is defined with respect to a Fuchsian group of the first kind. Assume that ∞ is a cusp of this group and denote bya _∞,n,a _∞,n ,n ∈ ?, the Fourier coefficients off at ∞. Following Hecke and Maas we prove that under suitable assumptions the associated Dirichlet seriesL ⁺ (f, s) = ∑ _{n > 0} a _∞,n (n + μ_221E;)^?s andL ^? (f, s) = ∑ _{n < 0} a _∞,n |n + μ_221E;|^?s have meromorphic continuation in the entire complex plane and statisfy a certain functional equation (μ_∞ denotes the cusp parameter of the cusp ∞). We are interested in mean square estimates of these functions. Iff is not a cusp form we prove $$\int_0^T {|L^ \pm (f,\Re _\rho + it)|^2 dt = T(\log T)^a (B^ \pm + o(1)),}$$ wherea is either 1, 2 or 4, andB ^± is a constant. A similar result is true iff is a cusp form. In case of a congruence group the termo(1) can be replaced byO ((logT)^?1).     

The correspondence between theta series of ternary and quaternary quadratic forms

This article uses orthogonal groups and quaternion algebras to describe the Shimura lifting for theta series of general three-dimensional lattices with spherical coefficients.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniye Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 196, pp. 61–82, 1991.     

Linear relations of theta series of genera of quadratic forms

In this paper we study linear relations among theta series of genera of positive definite n-ary quadratic forms with given level D,2D,4D and 8D for square free D. We obtain a basis for the space generated by genus theta series. This forms a basis of Eisenstein space.     

Eisenstein series of weight-3/2 and singular Hardy-Littlewood series for ternary quadratic forms

By means of analytic continuation Eisenstein series of weight-3/2 are constructed with respect to a main congruence subgroup (N). This makes it possible to define the singular series of an arbitrary integral ternary quadratic form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 50, pp. 156–168, 1975.     

A series of a discrete group and its application to the Dirichlet series associated with automorphic forms

Starting from the Poincaré-Selberg series, another series is defined for a discrete group; the relationship is established between this series and the Dirichlet series for which the coefficients are products of the Fourier coefficients of the automorphic forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 63, pp. 3–7, 1976.