A modified CFKRS recipe correctly predicts the secondary main term in the second moment of Dirichlet L-functions along cosets by incorporating the non-independence of root numbers and coefficients.
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For h from a Hardy field with polynomial growth, h(Ω(n)), h(ω(n)), and h(Ω(q_n)) are uniformly distributed mod 1 precisely when h deviates from rational polynomials according to one of two explicit growth conditions.
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Remarks on the distribution of Dirichlet $L$-functions along cosets
A modified CFKRS recipe correctly predicts the secondary main term in the second moment of Dirichlet L-functions along cosets by incorporating the non-independence of root numbers and coefficients.
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Weighted averages of arithmetic functions and applications to equidistribution
For h from a Hardy field with polynomial growth, h(Ω(n)), h(ω(n)), and h(Ω(q_n)) are uniformly distributed mod 1 precisely when h deviates from rational polynomials according to one of two explicit growth conditions.