Burgess-quality subconvexity exponent for Dirichlet $L$-functions

Description of constant

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

Known upper bounds

Known lower bounds

Additional comments and links

• 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

• [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.”

    Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.