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Petersburg Department of Steklov Institute of Mathematics

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PREPRINT 15(I)/1999

A. I. Vinogradov
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THE RANKIN CONVOLUTIONS AND
DIRICHLET SERIES. I

This preprint was accepted September , 1999

Contact:
` A. I. Vinogradov `

ABSTRACT:
In the present paper, the Riemann hypothesis is proved for
classical Dirichlet series with quadratic characters, including
the Riemann zeta function, which occurs in this set as a
Dirichlet series with trivial quadratic character modulo 1.
The paper consists of two parts. In part I, a spectral-number
technique is elaborated and it is shown how this technique works
with an example of effectivization of the Siegel theorem, i.e.,
the Riemann hypothesis is proved for the Siegel zero. In part
II, this proof is carried over to all of the remaining
nontrivial zeros.
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