Fourier coefficients of Siegel cusp forms of genus n

Let F(Z) be a cusp form of integral weight k relative to the Siegel modular group Sp_n(Z) and let f(N) be its Fourier coefficient with index N. Making use of Rankin's convolution, one proves the estimate (1) $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2} - \tfrac{1}{2}\delta (n)} ),$$ where $$\delta (n) = \frac{{n + 1}}{{\left( {n + 1} \right)\left( {2n + \tfrac{{1 + ( - 1)^n }}{2}} \right) + 1}}.$$ Previously, for n ≥ 2 one has known Raghavan's estimate $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2}} )$$ In the case n=2, Kitaoka has obtained a result, sharper than (1), namely: (2) $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2} - \tfrac{1}{4} + \varepsilon } ).$$ At the end of the paper one investigates specially the case n=2. It is shown that in some cases the result (2) can be improved to, apparently, unimprovable estimates if one assumes some analogues of the Petersson conjecture. These results lead to a conjecture regarding the optimal estimates of f(N), n=2.     

Technology and tools for the rapid quality diagnostic and identification of liquids based on the electrophysical method of states

A new electrophysical method is proposed for liquid products rapid diagnosis of quality and identification. The method is widely prospective for practical application.     

Behavior of Automorphic <Emphasis Type="Italic">L</Emphasis>-Functions at the Points s=1 and s=1/2

Let S_k(N)⁺ be the set of primitive cusp forms of even weight k for Γ₀(N) and let L(s, sym ²f) be the symmetric square L-function L(s, f) of a form f ∈ S_k(N)⁺. The moments of the variable L(1, sym ²f), f ∈ S₂(N)⁺, are computed for N = p, and the corresponding limiting distribution is determined in N-aspect. Let f ∈ S_k(1)⁺, g ∈ S_l(1)⁺, and ω_f = Γ(k - 1)/(4π)^k-1 〈f, f〉. Asymptotic formulas for and as k → ∞ are obtained. Bibliography: 17 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 149–167.     

Distribution of numbers with a prescribed number of distinct prime divisors

One gives a new quantitative result regarding the uniform distribution of numbers with a prescribed number of distinct prime divisors.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, pp. 158–164, 1979.     

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.     

Some diophantine systems

Using known results concerning the representation of binary forms as sums of three or four squares of linear forms we derive an asymptotic formula connected with the representation of integers in an imaginary quadratic number field R as sums of three or four squares of integers from R.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 155–160, 1982.     

Effects of beam loading on the parameters of the accelerating systems of cyclic electron accelerators in the steady state and during injection

Conditions are derived for matching in the accelerating system, and also an expression for the input resistance of the cavity with allowance for beam loading in the steady state. The loading effect alters the field frequency during injection, and so there may be a substantial fall in the injection efficiency.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, Vol. 12, No. 1, pp. 74–80, January, 1969.