Over F_2(t), the number of parameters y of height 2^M for which the conic x0^2+x0x1+yx1^2=t x2^2 has a rational point is asymptotically c 2^{2M}/M^{1/2}, with an explicit Euler product constant c.
Character sums over products of prime polynomials
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We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit new phenomena concerning Chebyshev-type biases of such sums when the number of irreducible factors is very large.
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Rational points in a family of conics over $\mathbb{F}_2(t)$
Over F_2(t), the number of parameters y of height 2^M for which the conic x0^2+x0x1+yx1^2=t x2^2 has a rational point is asymptotically c 2^{2M}/M^{1/2}, with an explicit Euler product constant c.