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 $q$. <a href="#Ked2007-def-character">[Ked2007-def-character]</a>

Such a character is called primitive if it is not induced from a smaller level. <a href="#Ked2007-def-primitive">[Ked2007-def-primitive]</a>

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 $\Re (s)>1$. <a href="#Ked2007-def-L">[Ked2007-def-L]</a>

Define the pointwise subconvexity exponent $${\mu }_{\mathrm{Dir}}:=\inf \{\theta \ge 0:L(1/2,\chi ){\ll }_{\epsilon }{q}^{\theta +\epsilon }\text{for all primitive }\chi (\mathrm{mod}q)\}.$$ This formulation is in the same conductor aspect used in modern Dirichlet subconvexity work. <a href="#PY2020-primitive-conductor">[PY2020-primitive-conductor]</a> <a href="#PY2020-central-values-q-aspect">[PY2020-central-values-q-aspect]</a>

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 $q$, $$L(1/2,\chi ){\ll }_{\epsilon }{q}^{3/16+\epsilon }.$$ <a href="#PY2020-burgess-3-16">[PY2020-burgess-3-16]</a> 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 $1/6$. <a href="#PY2020-weyl-cubefree">[PY2020-weyl-cubefree]</a> <a href="#PY2020-weyl-def">[PY2020-weyl-def]</a>

• Wikipedia page on Burgess inequality

References

• <a id="Bur1963"></a>[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

• <a id="Ked2007"></a>[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
• <a id="Ked2007-def-character"></a>[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."
• <a id="Ked2007-def-primitive"></a>[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."
• <a id="Ked2007-def-L"></a>[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 ${\prod }_p(1-\chi (p)p^{-s}{)}^{-1}$."

• <a id="PY2020"></a>[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
• <a id="PY2020-primitive-conductor"></a>[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 $q$."
• <a id="PY2020-central-values-q-aspect"></a>[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 $q\to \infty$..."
• <a id="PY2020-burgess-3-16"></a>[PY2020-burgess-3-16] loc: arXiv v1 PDF p.1, Introduction, equation (1.2) sentence quote: "In 1963, Burgess [B] showed ... $L(1/2,\chi ){\ll }_{\epsilon }{q}^{3/16+\epsilon }$."
• <a id="PY2020-weyl-def"></a>[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 $L(1/2,\pi )\ll Q(\pi {)}^{1/6+\epsilon }$ the Weyl bound."
• <a id="PY2020-weyl-cubefree"></a>[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."

What counts as progress

• A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
• A formal proof (Lean) of a known bound, or a precise error in a claimed one.
• New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.