Burgess Estimates in Number Theory

• Burgess estimates are bounds for short non-principal Dirichlet character sums that become nontrivial when the sum length exceeds q^(1/4 + ε).
• They employ multiplicative shifts, Hölder’s inequality, and high-moment analysis to transform short-sum problems into higher-moment estimates with concrete error terms.
• Explicit constant versions of these bounds extend applications to least quadratic non-residues, least primitive roots, and subconvexity for Dirichlet L-functions.

Burgess estimates are short character-sum bounds for non-principal Dirichlet characters, together with a broader family of Burgess-like inequalities for mixed character sums and $L$-functions. In the prime-modulus setting, the characteristic form is

$$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$

with $r>1$, and the estimate is nontrivial as soon as $N>q^{1/4+\varepsilon}$. This $q^{1/4}$-threshold is the defining feature of the Burgess method, and it underlies classical results on least quadratic non-residues, least primitive roots, and $q$-aspect subconvexity for Dirichlet $L$-functions (Kerr et al., 2017, Jain-Sharma et al., 2020, Francis, 2022).

1. Classical statement and Burgess range

For a prime modulus $q$, a nontrivial multiplicative character $\chi \bmod q$, and an integer $r>1$, the Burgess bound takes the form

$$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$0

Its characteristic exponent structure is $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$1: increasing $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$2 improves the exponent of $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$3 toward $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$4, while worsening the exponent of $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$5. In the standard conductor-aspect formulation, for a non-principal character $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$6 with conductor $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$7, one has

$$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$8

provided either the conductor is cube-free or $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$9 (Kerr et al., 2017, Jain-Sharma et al., 2020).

The significance of the estimate is its short-sum range. Pólya–Vinogradov yields $r>1$0, which is nontrivial only once $r>1$1 reaches the square-root scale. Burgess estimates remain nontrivial as soon as $r>1$2, which is the regime traditionally called the Burgess range. In the quadratic case, this translates into $r>1$3 for $r>1$4, and in analytic applications it gives the classical Burgess subconvexity exponent $r>1$5 for $r>1$6 (Fromm et al., 2017, Francis, 2022).

2. Core mechanism of the Burgess method

The standard Burgess architecture combines multiplicative shifts, Hölder’s inequality, and high moments of short character sums. A representative Burgess-style moment is

$r>1$7

and an explicit upper bound of the form

$r>1$8

is obtained by expanding the $r>1$9-th power and applying a Weil-type estimate to the resulting character sums in polynomial arguments (McGown, 2010).

The analytic heart is the separation between generic and exceptional polynomial configurations. Generic configurations are controlled by Weil-type bounds of size $N>q^{1/4+\varepsilon}$0; exceptional configurations, for which the relevant polynomial becomes an $N>q^{1/4+\varepsilon}$1-th power modulo $N>q^{1/4+\varepsilon}$2, are handled trivially but counted combinatorially. The payoff appears after Hölder’s inequality converts the original short sum into a higher-moment problem plus a counting problem for multiplicative congruences (McGown, 2010, Kerr et al., 2017).

Modern refinements sharpen the same template rather than replacing it. One such refinement averages over

$N>q^{1/4+\varepsilon}$3

that is, over integers with no small prime factors, rather than over a full interval. This improves the counting of solutions to congruences of the form

$N>q^{1/4+\varepsilon}$4

and thereby reduces the logarithmic loss in the final Burgess inequality (Kerr et al., 2017).

3. Explicit constants and concrete arithmetic consequences

A major later development is the replacement of asymptotic $N>q^{1/4+\varepsilon}$5-notation by explicit constants. For prime moduli $N>q^{1/4+\varepsilon}$6, Treviño proved that for any positive integer $N>q^{1/4+\varepsilon}$7,

$N>q^{1/4+\varepsilon}$8

for every non-principal character $N>q^{1/4+\varepsilon}$9. For $q^{1/4}$0 and $q^{1/4}$1, an explicit version with $q^{1/4}$2 is also available, with constants $q^{1/4}$3 tabulated for $q^{1/4}$4 (Treviño, 2014).

These explicit forms support explicit least non-residue problems. In the same work, if $q^{1/4}$5 is prime and $q^{1/4}$6 divides $q^{1/4}$7, then for

$q^{1/4}$8

the least $q^{1/4}$9-th power non-residue modulo $q$0 is at most $q$1. The argument combines an explicit Burgess inequality with a lower-bound device of Vinogradov type for sums $q$2 under the hypothesis that all integers $q$3 are $q$4-th power residues (Treviño, 2014).

A different explicit consequence concerns the maximum length $q$5 of an interval on which a non-principal Dirichlet character $q$6 is constant. McGown proved

$q$7

with explicit numerical forms

$q$8

This is a Burgess-type estimate for consecutive residues or non-residues, obtained by comparing upper and lower bounds for $q$9 and making the constants completely explicit (McGown, 2010).

4. Refinements and structural extensions

The Burgess framework has been pushed in several orthogonal directions: sharper logarithmic factors, composite moduli, polynomial additive phases, multidimensional boxes, and additive-combinatorial sets such as generalized arithmetic progressions.

Direction	Representative estimate	Source
Prime-modulus logarithmic refinement	$L$0 for $L$1 and $L$2	(Kerr et al., 2017)
Explicit composite modulus, $L$3	$L$4 for primitive $L$5, $L$6	(Jain-Sharma et al., 2020)
Explicit cubefree moduli, $L$7	Explicit bounds with constants $L$8 and arithmetic factors built from $L$9, $q$0, $q$1	(Hasanalizade et al., 21 Nov 2025)
Mixed sums with polynomial phase	Bounds for $q$2 via Vinogradov mean value estimates	(Heath-Brown et al., 2014)
Multi-dimensional mixed sums	Bounds for $q$3, with saving $q$4 when $q$5	(Pierce, 2014)
Rank-2 generalized arithmetic progressions	$q$6 for proper rank-2 GAPs with $q$7	(Alsetri et al., 9 Sep 2025)

The logarithmic refinement to $q$8 is obtained by averaging over integers with no small prime factors rather than over an interval. This improves the estimate for the multiplicative congruence count that sits inside the Burgess method, while preserving the characteristic Burgess exponents in $q$9 and $\chi \bmod q$0 (Kerr et al., 2017).

For composite moduli, explicit treatments become more arithmetic. The explicit bound for primitive $\chi \bmod q$1 with

$\chi \bmod q$2

shows how the dependence on the factorization of $\chi \bmod q$3 enters through $\chi \bmod q$4 and $\chi \bmod q$5. The later cubefree-modulus work makes the full Burgess parameter $\chi \bmod q$6 explicit by introducing

$\chi \bmod q$7

together with auxiliary factors $\chi \bmod q$8 and $\chi \bmod q$9, and proves explicit Burgess inequalities for all $r>1$0 when $r>1$1 is cubefree (Jain-Sharma et al., 2020, Hasanalizade et al., 21 Nov 2025).

The mixed-sum generalizations show that Burgess estimates are not confined to pure multiplicative sums. For

$r>1$2

the additive phase is controlled by Vinogradov mean value theory, and under optimal VMVT one recovers Burgess-quality savings $r>1$3 at lengths $r>1$4. In dimensions $r>1$5, embedding a given polynomial into a smaller translation-dilation invariant system can improve the resulting exponent, a phenomenon the paper describes as novel to dimensions two and higher (Heath-Brown et al., 2014, Pierce, 2014).

The recent rank-2 GAP result extends the Burgess threshold $r>1$6 from intervals to proper generalized arithmetic progressions in $r>1$7. Its core input is the multiplicative energy bound

$r>1$8

for rank-2 GAPs, together with new geometry-of-numbers bounds for Bohr sets (Alsetri et al., 9 Sep 2025).

5. Subconvexity and automorphic Burgess-like bounds

Burgess estimates control central values of Dirichlet $r>1$9-functions through partial summation and Phragmén–Lindelöf interpolation. An explicit version for prime moduli $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$00, primitive non-principal $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$01, and $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$02 is

$$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$03

which realizes the classical Burgess exponent $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$04 with explicit constants. For $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$05, one also has

$$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$06

These bounds are obtained by inserting explicit Burgess character-sum estimates on the lines $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$07 and interpolating across the critical strip (Francis, 2022).

Over general number fields, the same exponent pattern appears in conductor aspect. For Hecke characters $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$08 over a number field $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$09,

$$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$10

where $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$11 is the analytic conductor and $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$12 is a Ramanujan parameter for $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$13. This is explicitly described as a Burgess-like subconvex bound for $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$14 (Wu, 2016).

For twists of $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$15 cusp forms by Hecke characters, the analogous statement is

$$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$16

with $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$17. Under Ramanujan, $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$18, this becomes

$$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$19

which the paper explicitly calls a Burgess bound in the $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$20 setting (Wu, 2012).

A plausible implication is that “Burgess estimate” has become both a specific theorem about short Dirichlet character sums and a template for conductor-aspect subconvexity exponents derived from the same structural input: amplification, higher moments, and square-root bounds for complete character sums.

6. Interconnections, conditional improvements, and related usages

One striking line of work shows that Burgess and Pólya–Vinogradov are not independent phenomena. If the Pólya–Vinogradov inequality were improved to

$$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$21

for all even primitive quadratic characters, then for odd primitive quadratic characters $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$22 one would obtain

$$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$23

which is essentially the conjectural Burgess range $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$24 rather than $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$25. This implication is qualitative in the original argument and is made explicit by the later formula

$$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$26

where $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$27 is the improved Pólya–Vinogradov constant and $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$28 holds for all $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$29 (Fromm et al., 2017, Bordignon et al., 2020).

The phrase also has a separate combinatorial use in the theory of the Erdős–Burgess constant. For the multiplicative semigroup $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$30 of $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$31, one obtains the lower bound

$$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$32

with equality when $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$33 is a prime power or a product of pairwise distinct primes. In that literature, this lower bound is explicitly described as a Burgess-type estimate, but it belongs to semigroup combinatorics rather than to analytic character-sum theory (Hao et al., 2018).

This terminological split suggests two established meanings. In analytic number theory, Burgess estimates are short character-sum and subconvexity bounds centered on the $$\sum_{M<n\le M+N}\chi(n)=O\!\left(N^{1-1/r}q^{(r+1)/4r^2}(\log q)\right),$$34-threshold. In combinatorial semigroup theory, Burgess-type estimates concern idempotent products and the Erdős–Burgess constant. The shared name reflects D. A. Burgess’s influence in two distinct but mathematically structured settings (Hao et al., 2018).