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On the Brun--Titchmarsh theorem. I

Ping Xi, Junren Zheng

TL;DR

This paper advances the Brun–Titchmarsh upper bound for the prime-counting function $\pi(x;q,a)$ in the range $q\sim x^{\varpi}$ by constructing explicit constants $C(\varpi)$ for general and smooth moduli, improving prior barriers especially around $\varpi\ge 9/20$ and near $2/3$ under factorization assumptions.The authors develop a unified framework combining linear sieve methods with bilinear and trilinear forms, and they leverage trace-function technology, arithmetic exponent pairs, and spectral bounds for Kloosterman sums to control remainder terms in the sieve, obtaining sharper constants and new ranges.Two main methodological strands are used: multiplicative-characters approaches yield tighter results in the small-$\varpi$ regime, while additive-characters and Poisson/Kloosterman-sum machinery drive improvements for larger $\varpi$, including eta-smooth moduli and special exponent-pair configurations.Beyond the Brun–Titchmarsh bounds themselves, the work connects to Landau–Siegel zeros, twisted moments of Dirichlet $L$-functions, and average-variant results, highlighting the theoretical and potential computational consequences for the distribution of primes in arithmetic progressions and related zero-density phenomena.

Abstract

The classical Brun--Titchmarsh theorem gives an upper bound, which is of correct order of magnitude in the full range, for the number of primes $p\leqslant x$ satisfying $p\equiv a\bmod q$. We strengthen this inequality for different ranges of $\log q/\log x$, improving upon previous works by Motohashi, Goldfeld, Iwaniec, Friedlander and Iwaniec, and Maynard for general or special moduli. In particular, we are able to beat Iwaniec's barrier $q<x^{9/20-}$, and improve all existing inequalities in the range $x^{9/20}\ll q<x^{1/2-}$ by utilizing bilinear or trilinear structures in the remainder terms of linear sieve. The proof is based on various estimates for character and exponential sums, which we derive by appealing to arithmetic exponent pairs and bilinear forms with algebraic trace functions from $\ell$-adic cohomology, trilinear forms with Kloosterman fractions, and sums of Kloosterman sums from spectral theory of automorphic forms, as well as large value theorem for Dirichlet polynomials.

On the Brun--Titchmarsh theorem. I

TL;DR

This paper advances the Brun–Titchmarsh upper bound for the prime-counting function $\pi(x;q,a)$ in the range $q\sim x^{\varpi}$ by constructing explicit constants $C(\varpi)$ for general and smooth moduli, improving prior barriers especially around $\varpi\ge 9/20$ and near $2/3$ under factorization assumptions.The authors develop a unified framework combining linear sieve methods with bilinear and trilinear forms, and they leverage trace-function technology, arithmetic exponent pairs, and spectral bounds for Kloosterman sums to control remainder terms in the sieve, obtaining sharper constants and new ranges.Two main methodological strands are used: multiplicative-characters approaches yield tighter results in the small-$\varpi$ regime, while additive-characters and Poisson/Kloosterman-sum machinery drive improvements for larger $\varpi$, including eta-smooth moduli and special exponent-pair configurations.Beyond the Brun–Titchmarsh bounds themselves, the work connects to Landau–Siegel zeros, twisted moments of Dirichlet $L$-functions, and average-variant results, highlighting the theoretical and potential computational consequences for the distribution of primes in arithmetic progressions and related zero-density phenomena.

Abstract

The classical Brun--Titchmarsh theorem gives an upper bound, which is of correct order of magnitude in the full range, for the number of primes satisfying . We strengthen this inequality for different ranges of , improving upon previous works by Motohashi, Goldfeld, Iwaniec, Friedlander and Iwaniec, and Maynard for general or special moduli. In particular, we are able to beat Iwaniec's barrier , and improve all existing inequalities in the range by utilizing bilinear or trilinear structures in the remainder terms of linear sieve. The proof is based on various estimates for character and exponential sums, which we derive by appealing to arithmetic exponent pairs and bilinear forms with algebraic trace functions from -adic cohomology, trilinear forms with Kloosterman fractions, and sums of Kloosterman sums from spectral theory of automorphic forms, as well as large value theorem for Dirichlet polynomials.
Paper Structure (31 sections, 32 theorems, 314 equations, 2 figures, 2 tables)

This paper contains 31 sections, 32 theorems, 314 equations, 2 figures, 2 tables.

Table of Contents

  1. Backgrounds and main results
  2. Main results
  3. Ingredients of proof and comparisons with previous works
  4. Connections with Landau--Siegel zeros
  5. Dirichlet polynomials and character sums
  6. General trace functions and arithmetic exponent pairs
  7. Trace functions over finite fields
  8. Composite trace functions and arithmetic exponent pairs
  9. Sieve methods
  10. Linear sieve
  11. Sifting primes in $\pi(x;q,a)$
  12. Proofs of Theorems \ref{['thm:BT-generalmoduli<1/2']} and \ref{['thm:BT-smoothmoduli<1/2']}
  13. Approaching by multiplicative characters
  14. Concluding Theorem \ref{['thm:BT-generalmoduli<1/2']}
  15. Concluding Theorem \ref{['thm:BT-smoothmoduli<1/2']}
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $\varpi\in[9/20,1/2[.$ Then we may take in eq:BT with some constant $C^*(\varpi)\geqslant0.$ In general, we may choose and for prime $q$ we may choose Here $\vartheta$ denotes the exponent towards the Ramanujan--Petersson conjecture for $\mathrm{GL}_2(\mathbf{Q}).$ In particular, one may take $\vartheta=7/64$ thanks to Kim and Sarnak KS03.

Figures (2)

  • Figure 1: $C(\varpi)$ for $\varpi<1/2$
  • Figure 2: $C(\varpi)$ for $\varpi>1/2$

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • Lemma 2.3
  • Theorem 2.2
  • ...and 24 more