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Abstract.
In 2011 the speaker began to work with finite Dirichlet series of length \(N\)
vanishing at \(N-1\) initial non-trivial
zeroes of Riemann's zeta function. Intensive multiprecision
calculations revealed several interesting phenomena. First,
such series approximate with great accuracy the values of the product
\((1-2\cdot2^{-s})\zeta(s)\) for a large range of \(s\)
lying inside the critical strip and
also to the left of it (even better approximations can be obtained
by dealing with ratios of
certain finite Dirichlet series). In particular the series vanish also very close to many other non-trivial zeroes
of the zeta function (informally,
one can say that ``initial non-trivial zeroes know about subsequent non-trivial zeroes''). Second, the coefficients of such series encode prime
numbers in several ways.
So far no theoretical explanation was given to the observed phenomena.
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