Reasoning as in [i, pp. 97-100], making use of the obtained estimate and taking into account (otherwise, in the asymptotic formulas (i) and (2) the principal term will be smaller in ordes than the residual one), we arrive at formula (!).Theorem 2 is a straightforward consequence of Theorem 1 (see [i, pp. 100-106]).
LITERATURE CITED i.A.V. Malyshev, "Representations of integers by positive quadratic forms," Tr. Mat. Inst. Akad. Nauk SSSR, 65, Moscow--Leningrad (1962). SUMMATION FORMULAS FOR GENERAL KLOOSTERMAN SUMS N. V. Proskurin UDC 511.334 N. V. Kuznetsov's summation formula is generalized to the case of a discrete subgroup ~C$L2(~ ~ and a system of multiplicators ~ , satisfying certain not too restrictive conditions. In the arithmetic cases, when ~ is a congruence-subgroup in SLy(Z) , these conditions are satisfied. N. V. Kuznetsov's formula has been proved for the case ~=~L~(Z).~i. The results of the present paper constitute a generalization of N. V. Kuznetsov's summation formula for the Kloosterman sums [3]. Independent of Kuznetsov, but somewhat later, Bruggeman [8] has proved summation formulas. Both authors consider the Hilbert space of functions on the upper half plane, automorphic with respect to the full modular group S~zC~) , but their methods are different. Kuznetsov's idea consists in the computation by two different methods of the inner product of two real-analytic Poincar4 series. At first this inner product is computed in the same way as it is done in the theory of regular automorphic functions when one computes the inner product of an arbitrary cusp form with a Poincar~ series. It is here where the sum of the Kloosterman sums occur. The second method of the computation of the inner product is based on the expansion of automorphic functions in eigenfunctions of the Laplace--Beltrami operator. Here the Fourier coefficients of cusp forms of weight 0 and the Fourier coefficients of the Eisenstein series emerge. Even a simple comparison of the two obtained expression leads to some summation formula (see (4.13)[3] or (30), (31) of our paper). But for the proof of a summation formula with, possibly, a general weight function (see Theorems 2 and 7 in [3] or our Theorems 1-4) we need one more consideration, which will be elucidated in Sec. 6.
The cubic L-function is related to the cubic Kubota-Patterson theta function via the Mellin transformation. The cubic L-function obeys a functional equation of the Riemann type (with two gamma factors), but admits no expansion in an Euler product. In the paper, the cubic L-function is studied, and the distribution problem for the real parts of its zeros is considered. Some conjectures based on calculations are stated.
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