Values of Dirichlet series associated with modular forms at the points S=1/2, 1

Let denote a parabolic form of even weights k which is an eigenfunction of all Hecke operators, and let be a real character. It is known that the Dirichlet series satisfies a functional equation of the Riemann type for. A number of asymptotic results on L_f(1/2,), L_f(1,) for d are proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 134, pp. 117–137, 1984.     

Application of spherical functions to a certain problem in the theory of quadratic forms

The problem regarding the number of integral points on multidimensional ellipsoids is investigated with the aid of modular forms. In the paper we consider the simplest special case of the following problem: one considers a multidimensional sphere and as a domain on it one selects a cap. The precise result is formulated in the following manner: let r(n) be the number of the representations of n by a sum of squares, 0n₁, 2 n; the expression for K(A), 4, is given in the paper. It is also shown that one can refine somewhat the results on the distribution of integral points on multidimensional ellipsoids, obtained by A. V. Malyshev by the circular method, remaining within the framework of the same methods.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 38–45, 1985.We express our gratitude to M. A. skopina for a consultatin regarding the theory of multiple fourier series     

Photochemical cyclization of anils of polyfluoroaromatic ketones

Conclusions The UV irradiation of the anils polyfluoroaromatic ketones in strongly acidic media leads to photochemical cyclization to phenanthridines. The insertion of substituents into the anil molecule lowers the phototransformation rate.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 7, pp. 1606–1611, July, 1980.     

Comparative analysis of essential oil compositions from leaves and stems of Rhododendron adamsii, R. aureum, and R. dauricum

Chemical compositions of essential oils (EO) from leaves and stems of Rhododendron adamsii, R. aureum, and R. dauricum were studied by GC/MS. The analysis demonstrated that the EO of these plants contained mainly mono-and diterpene hydrocarbons. A large amount of alkanes was also found in the EO of R. aureum. __________ Translated from Khimiya Prirodnykh Soedinenii, No. 4, pp. 344–347, July–August, 2006.     

Series ∑F(m)qm,where F(m) is the number of odd classes of binary quadratic forms of determinant −m

Consideration of the analytic continuation of the Eisenstein series of weight 3/2 for the group ₀(4) leads to a new proof of Mordell's formula connecting the values X()= _m=1 F(m)e^im, Im > 0, and (–1/). The behavior of the function () for ₀(4) is examined by the same method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 69–79, 1976.     

Some properties related to torsion of a two-dimensional surface

We consider a two-dimensional surface F² of class C³ in Euclidean spaceE ⁿ ,n4. We introduce the concepts of the normal curvature vectork _N (x; t) and the Euler curvature vectorv _N (x; t) of the normalp _N (x;t) and the Euler _N (x;t) torsion of surface F² at the point x in the tangent directiont. We show that these magnitudes are characteristics of surface F² at the point x in the directiont, and derive formulas for their calculation. We establish necessary and sufficient conditions under which the directions of vectorsk _N (x; t) andp _N (x; t) are parallel displaced in the normal fiber bundle on F² from the pointx ¯F ² in the directiont. In particular, the following assertion holds: the direction of the Euler curvature vectorp _N (x; t) is parallel displaced in the normal fiber bundle on F² along any geodesic on F² if and only ifv _N (x; t) 0, x F ². t.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 45–52, 1990.     

On quadratic congruences

It is proved that under appropriate assumptions, the solutions f of quadratic congruences Q(f) ≡ 0 (mod n), n ≤ x, with (f|n) = ±1, are distributed asymptotically equally as x → ∞. Bibliography: 5 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 195–200.     

Direct measurement of the 7Be solar neutrino flux with 192 days of borexino data

We report the direct measurement of the 7Be solar neutrino signal rate performed with the Borexino detector at the Laboratori Nazionali del Gran Sasso. The interaction rate of the 0.862 MeV 7Be neutrinos is 49+/-3stat+/-4syst counts/(day.100 ton). The hypothesis of no oscillation for 7Be solar neutrinos is inconsistent with our measurement at the 4sigma C.L. Our result is the first direct measurement of the survival probability for solar nu(e) in the transition region between matter-enhanced and vacuum-driven oscillations. The measurement improves the experimental determination of the flux of 7Be, pp, and CNO solar nu(e), and the limit on the effective neutrino magnetic moment using solar neutrinos.     

Distribution of Lattice Points over the Four-Dimensional Sphere

Let r_l(n) be the number of representations of n by a sum of l squares of integers and let 0 < A < 1 be a constant. It is proved that if (n,2)=1, then . Previously, the author obtained this asymptotics with a weaker error term O( . Bibliography: 12 titles.     

Order of the Epstein Zeta-Function in the Critical Strip

Let Q(x₁,...,x_k) be a positive quadratic form of k2 variables and let (s,Q) be the Epstein zeta-function of the form Q. The growth rate of (s,Q) on the line Re s = (k–1)/2 is investigated. For k4 and for an integral form Q, the problem is reduced to a similar problem on the behavior of the Dirichlet L-series on the line Re s = 1/2. In the case k=3, the diagonal form over is investigated by the van der Corput method. For k=2, the known result due to Titchmarsh is re-proved by using a variant of the van der Corput method given by Heath-Brown. Bibliography: 9 titles.