The constant-2 Brun–Titchmarsh theorem below $x \ge q^8$: widening the admissible range for general moduli

Maynard (arXiv:1201.1777; Acta Arithmetica 157 (2013)) proved that the count $\pi(x;q,a)$ of primes at most $x$ in a residue class $a \pmod{q}$ satisfies $\pi(x;q,a) < 2\,\mathrm{Li}(x)/\phi(q)$ whenever $x \ge q^8$ and $q$ is large, with an effectively computable threshold; the constant 2 is best possible without excluding real zeros ('Siegel zeros') of Dirichlet $L$-functions near 1, since such a zero roughly doubles the count in suitable residue classes. He also proved companion lower bounds at $x \ge q^8$: unconditionally and effectively up to a factor about $q^{1/2}/\log q$, and of the expected order $x/(\phi(q)\log x)$ either ineffectively (via Siegel's theorem) or effectively under the exclusion of $\eta$-Siegel zeros. The open problem is to widen this upper bound without any Siegel-zero input and for general moduli: prove $\pi(x;q,a) < 2\,\mathrm{Li}(x)/\phi(q)$ for $x \ge q^u$ with some explicit $u < 8$ (the folklore expectation, endorsed in Xi–Zheng's 2024 account, is that it holds whenever $q$ is at most a small power of $x$). The only post-2012 extension of the constant-2 range is for special moduli: for squarefree $q$ with all prime factors at most $q^\eta$, Xi–Zheng (2024) lowered the threshold to $x > q^{12/5}$. For general moduli, $x \ge q^8$ remains the best known.

Background

Let $\pi(x;q,a)$ denote the number of primes $p \le x$ with $p \equiv a \pmod{q}$, where $(a,q)=1$; let $\phi$ be Euler's totient function and $\mathrm{Li}(x) = \int_2^x dt/\log t$. The prime number theorem for arithmetic progressions predicts $\pi(x;q,a) \sim \mathrm{Li}(x)/\phi(q)$: primes should equidistribute over the $\phi(q)$ invertible residue classes modulo $q$. This asymptotic is proved uniformly only for very small moduli (the Siegel–Walfisz theorem, valid for $q \le (\log x)^N$ for any fixed $N$). The generalized Riemann hypothesis (GRH) — the conjecture that all nontrivial zeros of the relevant $L$-functions have real part $1/2$ — would give the asymptotic uniformly for $q \le x^{1/2-\delta}$, and it is widely conjectured to hold for $q \le x^{1-\delta}$.

For all $q < x$, unconditional upper bounds come from the Brun–Titchmarsh theorem, whose standard form (due to Montgomery and Vaughan via the large sieve — an inequality that controls sums over all invertible residue classes modulo $q$ simultaneously) is $\pi(x;q,a) \le \frac{2}{1-\log q/\log x}\cdot\frac{x}{\phi(q)\log x}$. Writing $\varpi = \log q/\log x \in (0,1)$, so that $x = q^{1/\varpi}$, the literature surveyed here studies the admissible constant $C(\varpi)$ in the inequality $\pi(x;q,a) \le (C+o(1))\,x/(\phi(q)\log x)$. A sequence of works from the 1970s to the 1990s (tabulated in Xi–Zheng's 2024 paper) reduced $C(\varpi)$ below the Montgomery–Vaughan value $2/(1-\varpi)$ in several ranges of $\varpi$, using linear-sieve and bilinear-form techniques, but every unconditional result still yields $C > 2$ whenever $\varpi$ is bounded away from 0.

The reason 2 is a natural floor is the possible existence of Siegel zeros. A Dirichlet character $\chi$ mod $q$ is a multiplicative function on the residue classes mod $q$, and its Dirichlet $L$-function $L(s,\chi)$ encodes the distribution of primes among residue classes. A 'Siegel zero' is a real zero of $L(s,\chi)$, for a real character $\chi$, lying exceptionally close to the point $s=1$, say at $1-\lambda_1/\log q$ with $\lambda_1$ small. By the explicit formula, such a zero biases the primes toward the classes with $\chi(a)=-1$, making $\pi(x;q,a)$ about $(1+e^{-\lambda_1 M})$ times its expected size when $x = q^M$ — nearly twice as large when $\lambda_1$ is small. It was already observed in the 1970s (recounted by Xi–Zheng, Section 1.3) that any uniform bound with constant $2-\delta$ in the bounded-$M$ regime would imply zero-free regions excluding Siegel zeros; hence, without progress on the Siegel zero problem, the best possible constant is exactly 2, and it has been folklore that $C = 2$ should be admissible for $\varpi$ below some absolute constant.

Maynard proved the first unconditional result of this strength: $\pi(x;q,a) < 2\,\mathrm{Li}(x)/\phi(q)$ for all $x \ge q^8$ with $q$ at least an effectively computable constant (Theorem 1 of Maynard's paper). His method is analytic rather than sieve-theoretic, adapting the technology of Linnik's theorem on the least prime in an arithmetic progression (whose best known constant, $L = 5.2$, is due to Xylouris, building on Heath-Brown). Via the explicit formula — the identity expressing $\pi(x;q,a)$ as a main term plus a weighted sum over the zeros of the Dirichlet $L$-functions modulo $q$ — the problem reduces to bounding weighted sums over zeros of Dirichlet $L$-functions with imaginary part of order $O(1)$, controlled by three classical inputs: a zero-free region, the Deuring–Heilbronn zero-repulsion phenomenon (an exceptional Siegel zero repels all other zeros of the $L$-functions mod $q$ away from $s=1$), and Heath-Brown's log-free zero-density estimate (zeros near 1 of many different $L(s,\chi)$ cannot cluster). The threshold 8 reflects the current numerically explicit strength of these inputs: the decisive step is to show that a certain weighted sum $\sum_{\chi,\rho} e^{-M\lambda_\rho}$ over zeros, with $\lambda_\rho = (1-\Re\rho)\log q$, is less than 1 at $M = 7.999$. The same paper proves companion lower bounds at $x \ge q^8$: unconditionally and effectively up to a factor about $q^{1/2}/\log q$ (Theorem 2); of the expected order $x/(\phi(q)\log x)$ either ineffectively via Siegel's theorem (Theorem 3) or effectively under the hypothesis that no $\eta$-Siegel zero exists, i.e. no real zero in the interval $[1-\eta/\log q,\,1]$ (Theorem 4, which then gives a two-sided bound with the constant-2 upper bound).

The 2024–2026 state of the art (Xi–Zheng) improves $C(\varpi)$ for general $q$ in the range $\varpi \in [9/20,\,1/2[$ — the regime where $C > 2$ must hold anyway — using deep estimates from the spectral theory of automorphic forms and $\ell$-adic cohomology; and for squarefree moduli all of whose prime factors are at most $q^\eta$ ($\eta$-smooth moduli), it extends the constant-2 range to $\varpi \in [1/8,\,5/12[$, i.e. $x > q^{12/5}$. For general moduli, however, $x \ge q^8$ remains the widest known range for the constant-2 bound, and no unconditional effective lower bound of the expected order is known in any $x \ge q^u$ regime without Siegel-zero input or ineffectivity.

Problem Statement

The problem is to widen the unconditional constant-2 Brun–Titchmarsh theorem of the source paper, without any input on Siegel zeros, for general moduli. With the notation above:

Primary target (the constant-2 threshold). Prove that for some explicit $u$ with $1 < u < 8$ — and, at the strongest, for every fixed $u > 1$, as predicted by the folklore expectation that $\pi(x;q,a) \le 2\,\mathrm{Li}(x)/\phi(q)$ whenever $q$ is at most a small power of $x$ — there exists an effectively computable $q_0(u)$ such that $\pi(x;q,a) < 2\,\mathrm{Li}(x)/\phi(q)$ (equivalently, $\pi(x;q,a) \le (2+o(1))\,x/(\phi(q)\log x)$) uniformly for all $q \ge q_0(u)$, all $x \ge q^u$, and all $(a,q)=1$. The statement must concern general moduli $q$: no smoothness, factorization, or primeness restriction, and no exclusion of Siegel zeros. The known reference points are $u = 8$ for general $q$ (Maynard, Theorem 1) and $u = 12/5$ for squarefree $q$ with all prime factors at most $q^\eta$ (Xi–Zheng, Theorem 1.5).

Maynard's companion lower bounds (Theorems 2–4) and his effective threshold constants $q_1,\dots,q_6$ are described in the Background as context, not as targets of this problem. The source states that its lower bounds are 'essentially the strongest possible for $\log x/\log q$ bounded without excluding the existence of an $\eta$-Siegel zero' (Maynard, Section 3), and the Background explains why: if an $\eta$-Siegel zero existed, an unconditional expected-order lower bound for every $(a,q)=1$ could not hold in the classes disfavored by the exceptional character, so such a bound is not a companion widening posed by the source. Likewise, the source's Section 5 remark that an explicit bound for the admissible range 'would require us to estimate the constant $C_3$' of the log-free zero-density estimate and 'would likely produce a very large bound' concerns the explicitness of the range exponent in the no-Siegel-zero case of its own proof — a difficulty the paper then circumvents — and the source does not pose computing explicit numerical thresholds for the modulus as an open problem.

The constant is known to be best possible: without excluding Siegel zeros, no uniform bound with constant below 2 can hold, so an answer must keep the constant exactly 2 and widen the range. Results that only improve constants $C > 2$ in other ranges of $\varpi$, that restrict to smooth or prime moduli, that average over moduli or residue classes, or that assume GRH or the absence of Siegel zeros, do not answer this problem.

References

1. J. Maynard, On the Brun–Titchmarsh theorem, arXiv:1201.1777 (2012), https://arxiv.org/abs/1201.1777; published in Acta Arithmetica 157 (2013), no. 3, 249–296, DOI 10.4064/aa157-3-3, https://doi.org/10.4064/aa157-3-3
2. H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), no. 2, 119–134, DOI 10.1112/s0025579300004708, https://doi.org/10.1112/s0025579300004708
3. D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proceedings of the London Mathematical Society (3) 64 (1992), 265–338, DOI 10.1112/plms/s3-64.2.265, https://doi.org/10.1112/plms/s3-64.2.265
4. T. Xylouris, On the least prime in an arithmetic progression and estimates for the zeros of Dirichlet L-functions, Acta Arithmetica 150 (2011), no. 1, 65–91, DOI 10.4064/aa150-1-4, https://doi.org/10.4064/aa150-1-4
5. Y. Motohashi, On some improvements of the Brun–Titchmarsh theorem. IV, arXiv:1201.3134 (2012), https://arxiv.org/abs/1201.3134
6. Y. Motohashi, An extension of the Linnik phenomenon, Proceedings of the Steklov Institute of Mathematics 280 (suppl. 2) (2013), 56–64, DOI 10.1134/S0081543813030061, https://doi.org/10.1134/S0081543813030061; also arXiv:1204.0149, https://arxiv.org/abs/1204.0149
7. P. Xi and J. Zheng, On the Brun–Titchmarsh theorem. I, arXiv:2404.01003 (2024), https://arxiv.org/abs/2404.01003; published in Transactions of the American Mathematical Society (2026), DOI 10.1090/tran/9727, https://doi.org/10.1090/tran/9727
8. P. Xi and J. Zheng, On the Brun–Titchmarsh theorem. II, International Mathematics Research Notices 2025 (2025), no. 21, Article rnaf337, DOI 10.1093/imrn/rnaf337, https://doi.org/10.1093/imrn/rnaf337; also arXiv:2504.12692, https://arxiv.org/abs/2504.12692
9. J. Thorner and A. Zaman, Refinements to the prime number theorem for arithmetic progressions, arXiv:2108.10878 (2021), https://arxiv.org/abs/2108.10878; published in Mathematische Zeitschrift 306 (2024), no. 3, Article 54, DOI 10.1007/s00209-023-03414-3, https://doi.org/10.1007/s00209-023-03414-3
10. L. Guth and J. Maynard, New large value estimates for Dirichlet polynomials, arXiv:2405.20552 (2024), https://arxiv.org/abs/2405.20552; published in Annals of Mathematics (2) 203 (2026), no. 2, DOI 10.4007/annals.2026.203.2.6, https://doi.org/10.4007/annals.2026.203.2.6
11. R. Li, An average Brun–Titchmarsh theorem and shifted primes with a large prime factor, arXiv:2508.18285 (2025), https://arxiv.org/abs/2508.18285
12. T. Yamada, Explicit improvements of the Brun-Titchmarsh theorem for arbitrary intervals, arXiv:2312.16090 (2023), https://arxiv.org/abs/2312.16090