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We study the number of non-trivial simple zeros of the Dedekind zeta-function of a quadratic number field in the rectangle
. We prove that such a number exceeds
is sufficiently large. This improves upon the classical lower bound
established by Conrey et al [Simple zeros of the zeta function of a quadratic number field. I. Invent. Math.86 (1986), 563–576].
be a regular indefinite integral quadratic form with
, and let
be an integer. Denote by
the set of
-adic units in
. It is established that
has solutions in primes if (i) there are positive real solutions, and (ii) there are local solutions in
for all prime
denote the number of positive integers
, which cannot be represented as the sum of four squares of primes. We establish that
, thus improving on an earlier result of Harman and the first author, where the exponent
appears in place of
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