Fractional hypergeometric functions for function fields and weak transcendence in positive characteristic

Saturday, Sep 23 On some results and problems involving Hardy’s function Z(t) Aleksandar Ivić Serbian Academy of Arts and Sciences, Belgrade Hardy’s classical function Z(t) is defined, for real t, as Z(t) := ζ( 1 2 + it)χ( 1 2 + it)−1/2, ζ(s) = χ(s)ζ(1− s), This talk will cover some problems involving Hardy’s function, including moments at points where |Z(t)| is maximal, large values of Hardy’s function, and distribution of positive and negative values of Z(t). Congruent numbers, quadratic forms and K2 Hourong Qin Department of Mathematics, Nanjing University, Nanjing A celebrated theorem due to Tunnell gives a criterion for a positive integer to be congruent (under the BSD). We show that if a square-free and odd (respectively, even) positive integer n is a congruent number, then #{(x, y, z) ∈ Z|n = x + 2y + 32z} = #{(x, y, z) ∈ Z|n = 2x + 4y + 9z − 4yz}, respectively, #{(x, y, z) ∈ Z3| 2 = x + 4y + 32z} = #{(x, y, z) ∈ Z3| 2 = 4x + 4y + 9z − 4yz}.