Abstract. It is proved that Epstein's zeta-function ζ Q (s), related to a positive definite integral binary quadratic form, has a zero 1/2+iγ with T ≤ γ ≤ T +T 3/7+ε for sufficiently large positive numbers T . This is an improvement of the result by M. Jutila and K. Srinivas (Bull. London Math. Soc. 37 (2005) 45-53).To Professor Matti Jutila with deep regards

International audience Hardy's theorem for the Riemann zeta-function ζ(s) says that it admits infinitely many complex zeros on the line (s) = 1 2. In this note, we give a simple proof of this statement which, to the best of our knowledge, is new.

We prove that all imaginary biquadratic fields and cyclic quartic fields of class number 1 are Euclidean.

No abstract