Keijo O. Väänänen

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This paper considers rational integer values of Kloosterman sums over finite fields of characteristic <i>p</i> &gt; 3. We shall prove two main results. The first one is a congruence relation satisfied by possible integer values. One consequence is that there are no Kloosterman zeroes in the case of characteristic <i>p</i> &gt; 3, which generalizes recent(More)
We investigate arithmetic properties of values of the entire function F (z) = Fq(z; λ) = ∞ ∑ n=0 z ∏n j=1(q j − λ) , |q| > 1, λ / ∈ q Z>0 , that includes as special cases the Tschakaloff function (λ = 0) and the q-exponential function (λ = 1). In particular, we prove the non-quadraticity of the numbers Fq(α; λ) for integral q, rational λ and α / ∈ −λqZ>0 ,(More)
For fixed complex q with |q| > 1, the q-logarithm Lq is the meromorphic continuation of the series ∑ n>0 z / q −1 , |z| < |q|, into the whole complex plane. IfK is an algebraic number field, one may ask if 1, Lq 1 , Lq c are linearly independent over K for q, c ∈ K× satisfying |q| > 1, c / q, q2, q3, . . .. In 2004, Tachiya showed that this is true in the(More)
In this paper, the algebraic independence of values of the functionGd(z) := ∑ h≥0 z dh/(1− zdh ), d>1 a fixed integer, at non-zero algebraic points in the unit disk is studied. Whereas the case of multiplicatively independent points has been resolved some time ago, a particularly interesting case of multiplicatively dependent points is considered here, and(More)
We give explicitly the number of rational places of certain function fields in terms of the reciprocals of the zeros of the function fields in question. The results are then compared with the Hasse-Weil bounds by using the approximation theorems of Dirichlet and Kronecker and it turns out that in many of these function fields the number of rational places(More)
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