OP-12077OpenNumber theory

Burgess-quality subconvexity exponent for Dirichlet $L$ -functions ( $C_{62b}$ )

Burgess-quality subconvexity exponent for Dirichlet $L$ -functions: determine $C_{62b}$ or improve its bounds; the repository records 0 (lower) and $\frac{3}{16}$ (upper).

MSC 11M06 ·

\zeta (s)

and

L(s, \chi)

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓

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Problem statement

A Dirichlet character of level $q$ is an arithmetic function $\chi$ that is multiplicative, is defined by a character on $(\mathbb{Z}/q\mathbb{Z})^\ast$ on integers coprime to $q$ , and is $0$ on integers not coprime to $q$ . [Ked2007-def-character]

Such a character is called primitive if it is not induced from a smaller level. [Ked2007-def-primitive]

For a Dirichlet character $\chi$ , the associated Dirichlet $L$ -function is

L(s,\chi)\ :=\ \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s},

with Euler product for $\Re(s)>1$ . [Ked2007-def-L]

Define the pointwise subconvexity exponent

\mu_{\mathrm{Dir}}\ :=\ \inf\Bigl\{\theta\ge 0:\ L(1/2,\chi)\ll_{\varepsilon} q^{\theta+\varepsilon}\ \text{for all primitive }\chi\ (\bmod\ q)\Bigr\}.

This formulation is in the same conductor aspect used in modern Dirichlet subconvexity work. [PY2020-primitive-conductor] [PY2020-central-values-q-aspect]

We define

C_{62b}\ :=\ \mu_{\mathrm{Dir}},

the Burgess-quality subconvexity exponent for Dirichlet $L$ -values at the central point.

Burgess proved that, for all primitive characters $\chi$ modulo $q$ ,

L(1/2,\chi)\ \ll_{\varepsilon}\ q^{3/16+\varepsilon}.

[PY2020-burgess-3-16] In particular, $C_{62b}\le 3/16$ .

The best established range currently is

0\ \le\ C_{62b}\ \le\ \frac{3}{16}.

Lean 4

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Background

Cube-free conductors. Petrow-Young prove Weyl-exponent subconvexity for Dirichlet $L$ -functions of cube-free conductor. In this context, the Weyl exponent is $1/6$ . [PY2020-weyl-cubefree] [PY2020-weyl-def]
Wikipedia page on Burgess inequality

References (from the repository page)

[Bur1963] Burgess, D. A. On character sums and $L$ -series. II. Proceedings of the London Mathematical Society (3) 13 (1963), no. 1, 524-536. DOI: https://doi.org/10.1112/plms/s3-13.1.524. Google Scholar
[Ked2007] Kedlaya, Kiran S. Dirichlet characters and Dirichlet L-series (MIT 18.785 course notes, 2007). PDF: https://kskedlaya.org/18.785/lfunc.pdf. Google Scholar
[Ked2007-def-character] loc: PDF p.1, Section 1 ("Dirichlet characters"), opening paragraph quote: "For $N$ a positive integer, a Dirichlet character of level $N$ is an arithmetic function $\chi$ ... and is zero on integers not coprime to $N$ ; such a function is completely multiplicative."
[Ked2007-def-primitive] loc: PDF p.1, Section 1, paragraph beginning "Sometimes a Dirichlet character..." quote: "We say the character is imprimitive in this case and primitive otherwise."
[Ked2007-def-L] loc: PDF p.1, Section 2 ("L-series"), first paragraph and displayed equation (1) quote: "The Dirichlet series associated to a Dirichlet character $\chi$ ... denoted $L(s,\chi)$ . Since $\chi$ is completely multiplicative, $L(s,\chi)$ formally factors as $\prod_p (1-\chi(p)p^{-s})^{-1}$ ."
[PY2020] Petrow, Ian; Young, Matthew P. The Weyl bound for Dirichlet $L$ -functions of cube-free conductor. Annals of Mathematics 192 (2020), no. 2, 437-486. DOI: https://doi.org/10.4007/annals.2020.192.2.3. arXiv PDF: https://arxiv.org/pdf/1811.02452v1.pdf. Google Scholar
[PY2020-primitive-conductor] loc: arXiv v1 PDF p.2, Section 1.1 ("Statement of results"), first sentence quote: "Let $q$ be a positive integer, and $\chi$ be a primitive Dirichlet character of conductor $q$ ."
[PY2020-central-values-q-aspect] loc: arXiv v1 PDF p.1, Introduction paragraph beginning "Estimating the Dirichlet..." quote: "Estimating the Dirichlet $L$ -functions $L(1/2,\chi)$ of conductor $q$ as $q\to\infty$ ..."
[PY2020-burgess-3-16] loc: arXiv v1 PDF p.1, Introduction, equation (1.2) sentence quote: "In 1963, Burgess [B] showed ... $L(1/2,\chi)\ll_{\varepsilon} q^{3/16+\varepsilon}$ ."
[PY2020-weyl-def] loc: arXiv v1 PDF p.1, Introduction paragraph beginning "Today we call..." quote: "Today we call a subconvex bound of the form $L(1/2,\pi)\ll Q(\pi)^{1/6+\varepsilon}$ the Weyl bound."
[PY2020-weyl-cubefree] loc: arXiv v1 PDF p.1, Abstract quote: "We prove a Weyl-exponent subconvex bound for any Dirichlet $L$ -function of cube-free conductor."

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0; best known upper bound $\frac{3}{16}$ . Values marked with an asterisk have minimal verification.

Known upper bounds

Bound	Reference	Comments
$3/16$	[Bur1963]	Burgess bound (as quoted in modern sources). [PY2020-burgess-3-16]

Known lower bounds

Bound	Reference	Comments
$0$		Trivial from the definition $C_{62b}\ge 0$ .

References

[1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Burgess-quality subconvexity exponent for Dirichlet $L$-functions ($C_{62b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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Source & evidence

https://teorth.github.io/optimizationproblems/constants/62b.html

Constant page from github.com/teorth/optimizationproblems at commit 2c1968c, Apache-2.0. The maintainers review submissions for plausibility and replicability but do not certify correctness; asterisked bounds have minimal verification.

Source date: Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differ
Imported: 2026-10-10

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Record

Difficulty: HardResisted sustained effort by specialists.
Posed by: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers) and contributors
Posed: 2026
Source: Curated problem listOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOriginal statement ↗
Citation: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Burgess-quality subconvexity exponent for Dirichlet $L$-functions ($C_{62b}$)
Added: 5 Nov 2026
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Fields

Number theory