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Register a new userExceptional characters and prime numbers in sparse sets
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Authors
Jori Merikoski
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We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less arithmetic information is required to prove non-trivial lower bounds. As an application of our method, assuming the existence of infinitely many exceptional characters we show that there are infinitely many primes of the form $a^2+b^8$.
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For a nonincreasing function $psi$, let $textrm{Exact}(psi)$ be the set of complex numbers that are approximable by complex rational numbers to order $psi$ but to no better order. In this paper, we obtain the Hausdorff dimension and packing dimension of $textrm{Exact}(psi)$ when $psi(x)=o(x^{-2})$. We also prove that the lower bound of the Hausdorff dimension is greater than $2-tau/(1-2tau)$ when $tau=limsup_{xtoinfty}psi(x)x^2$ small enough.
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