Cochrane 2026 mixed incomplete character sums rational functions

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Todd Cochrane, Andrew Granville, Junren Zheng, Mixed incomplete character sums of rational functions with smooth moduli. arXiv preprint (2026). arXiv:2601.10927.

Graham and Ringrose proved power savings for incomplete character sums over intervals of length N = q^Delta when q is squarefree and q^xi-smooth, but their admissible smoothness xi was only quadratically small in Delta. Theorem 1 shows the smoothness parameter can be taken essentially as large as the interval length, namely q may be N^(1-epsilon)-smooth in the relaxed class N(y) of moduli with at most one prime factor in (y, y^2] and all other prime power divisors at most y, and simultaneously generalizes the estimate to mixed sums of the form sum over n in I of chi(f(n)) e(g(n)/q) for fixed rational functions f and g, giving a bound of order N/q^eta except in the degenerate case where g is a polynomial of degree below 1/delta, when the saving is in terms of the conductor q' of the primitive character inducing chi^(r_f); the paper argues this exceptional form is best possible. Corollaries record consequences: Corollary 2 gives sum over n in I of chi(n) << N^(1-eta) for smooth moduli, Corollary 3 bounds |L(1+it, chi)| in terms of the largest prime power divisors of q, and Corollary 4 gives a strong Brun-Titchmarsh inequality. The method adapts Heath-Brown's q-analog of van der Corput to iterated smooth factorizations of the modulus.

Source: https://arxiv.org/abs/2601.10927. The arXiv record (https://arxiv.org/abs/2601.10927, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

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Results to transcribe.

• Theorem 1: For fixed f, g in Q(x), not both constant, any character chi mod q with q in N(y), y = q^delta, coprime to an explicit modulus, and any interval I of length N with q >= N >= y^(1+epsilon), the mixed sum of chi(f(n)) e(g(n)/q) over I is << N/q^eta, except when g is a polynomial of degree < 1/delta, where the bound is N/(q')^eta with q' the conductor of the primitive character inducing chi^(r_f).
• Corollary 1: For non-constant f in Q(x), any character chi mod q and any integer b, under the hypotheses on q and I of Theorem 1, the sum of chi(f(n)) e(bn/q) over I is << N/Q^eta, where Q is the conductor of the primitive character inducing chi^(r_f).
• Corollary 2: For q in N(y) with y = q^delta, coprime to an explicit modulus, any non-principal chi mod q and any interval I of length N, sum over n in I of chi(n) << N^(1-eta) whenever q >= N >= y^(1+epsilon).
• Corollary 3: For q = P_1 P_2 ... with P_i the prime power divisors in decreasing order and chi a non-principal character mod q, |L(1+it, chi)| <= max{(1/2) log P_1(q), log P_2(q)} + o(log q) when P_1 is prime, and <= log P_1(q) + o(log q) otherwise, for t = q^o(1).
• Corollary 4: A strong Brun-Titchmarsh inequality for smooth moduli q in N(q^delta), deduced from Corollary 2.