An open question in analytic number theory is: How do the non-trivial zeros of a Dirichlet L-function behave in the critical strip? One might wonder whether the zeros bunch up or if they are well-spaced. In particular, is it possible for the zeros to lie in a vertical arithmetic progression? Let a and b be positive real constants and consider the arithmetic progression {0.5+i(a+kb)} where k ranges between the integers 0 to N. In joint work with Greg Martin, we show that many terms of this arithmetic progression are not zeros of a fixed Dirichlet L-function.
Suppose that X is an affine embedded variety over Z; in other words, we
have some polynomials in n variables with integer coefficients and we
are interested in the set of zeros. For simplicity, let us assume that
there is only a single polynomial f. Fix a prime p and a natural number
k, and consider the solutions of f=0 in the ring Z/p^{k}Z:
X( Z/p^{k}Z) =
{(a_{1},...,a_{n})ε Z/p^{k}Z : f(a_{1},...,a_{n}) = 0}
If we write integers in base p, then each element of X( Z/p^{k}Z)
corresponds to an n-tuple of the last k digits of a_{1},...,a_{n})ε Z such
that the last k digits of f(a_{1},...,a_{n}) are equal to 0.
Now, for any tuple (a_{i})in X( Z/p^{k}Z) we can ask how each a_{i} can be
extended by an additional digit to give an element of X( Z/p^{k+1}Z).
Starting with the "tuple of empty numbers" in X( Z/p^{0}Z) and adding
digits one by one, we get a tree T(X) of possibilities: the nodes of
T(X) at distance k from the root (the element of X( Z/p^{0}Z) are exactly
the elements of X( Z/p^{k}Z), and each element of X( Z/p^{k}Z) is connected to
the element of X( Z/p^{k-1}Z) which one obtains by forgetting one digit
of every coordinate.
The question I want to consider in this talk is: What can one say about
the structure of the tree T(X) (for arbitrary |
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J-C. Schlage-Puchta
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