We investigate the intersections of the curve R ∋ t → ζ( 1 2 + it) with the real axis. We show unconditionally that the zeta-function takes arbitrarily large positive and negative values on the critical line.

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We consider the sum of squared absolute values of the Dirichlet L-function taken at the nontrivial zeros of another Dirichlet L-function.From Theorem 1.1 we immediately deduce the following corollary. Corollary 1.2. Let ψ mod Q and χ mod q be primitive Dirichlet characters and χ = ψ. Assume that all nontrivial zeros of L(s, χ) lie on the critical line. Then the sequence (L(ρ χ , ψ)), where ρ χ runs over all nontrivial zeros of L(s, χ), is unbounded. Theorem 1.1 is related to the following theorem proved in [11]. Int. J. Number Theory 2013.09:945-963. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SAN DIEGO on 02/03/15. For personal use only.

A. Fujii, and later J. Steuding, considered an asymptotic formula for the sum of values of the Dirichlet L-function taken at the nontrivial zeros of another Dirichlet L-function. Here we improve the error term of this asymptotic formula.