Shiu's Brun–Titchmarsh Theorem

• Shiu’s Brun–Titchmarsh theorem is a short-interval estimate for sums of nonnegative multiplicative functions in arithmetic progressions, generalizing the prime-counting Brun–Titchmarsh inequality.
• It imposes bounded prime-power conditions and mild global growth to control the behavior of multiplicative functions, with an Euler-product correction term that reflects their average size.
• The theorem underpins applications in divisor function estimates and smooth number counts, and it has been extended to accommodate larger multiplicative classes.

Shiu’s Brun–Titchmarsh theorem is a short-interval upper bound for nonnegative multiplicative functions in arithmetic progressions. In the formulation recalled by Wright, if $f$ belongs to the class $M$ of nonnegative multiplicative functions with bounded prime-power values and mild global growth, if $0<\alpha<1$, $0<\kappa<\tfrac12$, $x^\kappa\le y\le x$, $k<y^{1-\alpha}$, and $(a,k)=1$, then for all sufficiently large $x$,

$$\sum_{\substack{x\le n<x+y\ n\equiv a\!\!\!\pmod k}} f(n) \ll \frac{y}{\phi(k)\log x} \exp\!\Bigg(\sum_{\substack{p<x\ p\nmid k}}\frac{f(p)}{p}\Bigg).$$

This is “Brun–Titchmarsh type” because it preserves the short-interval scale $y$, the progression density factor $M$0, and a logarithmic denominator, while replacing the prime indicator by a broad multiplicative class (Wright, 24 Aug 2025).

1. Statement and admissible function class

Shiu’s original setup, as restated in modern work, introduces the class $M$1 of nonnegative multiplicative functions $M$2 satisfying two conditions. First, there exists a constant $M$3 such that for every prime $M$4 and integer $M$5,

$M$6

Second, for every $M$7 there exists $M$8 such that for every $M$9,

$0<\alpha<1$0

Under these hypotheses, and in the range

$0<\alpha<1$1

Shiu’s theorem gives the upper bound displayed above. The dependence of “sufficiently large” is only on $0<\alpha<1$2 and the constants $0<\alpha<1$3 (Wright, 24 Aug 2025).

The theorem is local in the variable $0<\alpha<1$4: it controls a sum over the shifted interval $0<\alpha<1$5, not merely an initial segment $0<\alpha<1$6. It is also uniform in the reduced residue class $0<\alpha<1$7 throughout the short-interval regime $0<\alpha<1$8 with $0<\alpha<1$9. In this sense it is stronger than global progression bounds of the form $0<\kappa<\tfrac12$0, because it resolves distribution inside moving intervals.

The multiplicative-function factor

$0<\kappa<\tfrac12$1

is the theorem’s characteristic correction term. It records the average size forced by the prime values of $0<\kappa<\tfrac12$2, and it is this factor that allows the prime-indicator case to be replaced by a general nonnegative multiplicative weight.

2. Position within Brun–Titchmarsh theory

The classical prime-counting Brun–Titchmarsh problem asks for upper bounds on

$0<\kappa<\tfrac12$3

or on $0<\kappa<\tfrac12$4. In an explicit arbitrary-interval form, Yamada proves that for all integers $0<\kappa<\tfrac12$5, all integers $0<\kappa<\tfrac12$6 with $0<\kappa<\tfrac12$7, and all real $0<\kappa<\tfrac12$8 with $0<\kappa<\tfrac12$9,

$x^\kappa\le y\le x$0

This is a prime-counting theorem uniform in the starting point $x^\kappa\le y\le x$1 and interval length $x^\kappa\le y\le x$2, but it concerns primes only (Yamada, 2023).

Shiu’s theorem occupies a different place. It is not merely an arbitrary-interval prime bound, and it is not a smoothing theorem for real-variable weights. Rather, it transfers the Brun–Titchmarsh paradigm from primes to multiplicative functions. The central object is

$x^\kappa\le y\le x$3

with $x^\kappa\le y\le x$4 multiplicative and nonnegative, and the output retains the expected short-interval scale $x^\kappa\le y\le x$5 up to the Euler-product-like correction above (Wright, 24 Aug 2025).

A related but different direction is the weighted Brun–Titchmarsh inequality for primes. For an interval $x^\kappa\le y\le x$6, coprime integers $x^\kappa\le y\le x$7, and a nonnegative weight $x^\kappa\le y\le x$8, one has

$x^\kappa\le y\le x$9

where

$k<y^{1-\alpha}$0

This is a direct weighted analogue of the prime Brun–Titchmarsh inequality, but its generalization is in the direction of smooth real-variable weights rather than multiplicative functions (Büthe, 2014).

The standard misconception is therefore twofold. First, Shiu’s theorem is not just a restatement of the prime-counting arbitrary-interval bound. Second, it is not the same as a weighted prime theorem with a Sobolev weight. Its defining feature is the multiplicative structure of $k<y^{1-\alpha}$1, not merely the geometry of the interval or the smoothness of an external weight.

3. Proof architecture

In Wright’s account of Shiu’s original method, the argument begins from an Euler-product-type estimate. For $k<y^{1-\alpha}$2,

$k<y^{1-\alpha}$3

Using the bounded prime-power hypothesis $k<y^{1-\alpha}$4, Shiu’s framework shows that

$k<y^{1-\alpha}$5

and hence

$k<y^{1-\alpha}$6

This is the basic multiplicative majorant that feeds the later decomposition (Wright, 24 Aug 2025).

The proof then partitions the interval sum into four classes $k<y^{1-\alpha}$7 according to the factorization structure of $k<y^{1-\alpha}$8. Three classes are handled by sieve and smooth-number arguments. The delicate class is the one with many small prime factors, where a modified Rankin-type estimate introduces a decisive negative term: $k<y^{1-\alpha}$9 The appearance of $(a,k)=1$0 is the mechanism that compensates for configurations with too many prime factors (Wright, 24 Aug 2025).

The sieve input is encoded through the counting function

$(a,k)=1$1

for which Shiu’s lemma gives

$(a,k)=1$2

This is the short-interval progression sieve estimate that interacts with the multiplicative decomposition. The overall architecture is therefore neither a short-interval prime number theorem nor a Kloosterman-sum argument; it is a sieve-and-mean-value framework adapted to multiplicative weights (Wright, 24 Aug 2025).

4. Later extensions: larger functions and smooth support

A direct modern extension is Wright’s theorem for larger multiplicative functions. He replaces Shiu’s bounded prime-power condition by

$(a,k)=1$3

for all primes $(a,k)=1$4 and all $(a,k)=1$5, together with the global growth condition

$(a,k)=1$6

for every $(a,k)=1$7. If $(a,k)=1$8 with

$(a,k)=1$9

then for any $x$0,

$x$1

The interval and progression range is unchanged, but the price of allowing $x$2 to grow is the loss of the full $x$3 denominator (Wright, 24 Aug 2025).

Wright also develops a smooth-supported version. If $x$4 is $x$5-smooth-supported and

$x$6

then there exists a constant $x$7 such that

$x$8

If $x$9, there exists $$\sum_{\substack{x\le n<x+y\ n\equiv a\!\!\!\pmod k}} f(n) \ll \frac{y}{\phi(k)\log x} \exp\!\Bigg(\sum_{\substack{p<x\ p\nmid k}}\frac{f(p)}{p}\Bigg).$$0 and a constant $$\sum_{\substack{x\le n<x+y\ n\equiv a\!\!\!\pmod k}} f(n) \ll \frac{y}{\phi(k)\log x} \exp\!\Bigg(\sum_{\substack{p<x\ p\nmid k}}\frac{f(p)}{p}\Bigg).$$1 such that for $$\sum_{\substack{x\le n<x+y\ n\equiv a\!\!\!\pmod k}} f(n) \ll \frac{y}{\phi(k)\log x} \exp\!\Bigg(\sum_{\substack{p<x\ p\nmid k}}\frac{f(p)}{p}\Bigg).$$2,

$$\sum_{\substack{x\le n<x+y\ n\equiv a\!\!\!\pmod k}} f(n) \ll \frac{y}{\phi(k)\log x} \exp\!\Bigg(\sum_{\substack{p<x\ p\nmid k}}\frac{f(p)}{p}\Bigg).$$3

The paper states that one may take

$$\sum_{\substack{x\le n<x+y\ n\equiv a\!\!\!\pmod k}} f(n) \ll \frac{y}{\phi(k)\log x} \exp\!\Bigg(\sum_{\substack{p<x\ p\nmid k}}\frac{f(p)}{p}\Bigg).$$4

These results show that Shiu’s theorem remains structurally stable even when the multiplicative function is larger or concentrated on smooth numbers (Wright, 24 Aug 2025).

The most important conceptual point in these extensions is that the parameter range for $$\sum_{\substack{x\le n<x+y\ n\equiv a\!\!\!\pmod k}} f(n) \ll \frac{y}{\phi(k)\log x} \exp\!\Bigg(\sum_{\substack{p<x\ p\nmid k}}\frac{f(p)}{p}\Bigg).$$5 is preserved. The theorem is strengthened by enlarging the admissible multiplicative class, not by enlarging the short-interval regime.

5. Nearby theorems that are not Shiu’s theorem

Several results in the Brun–Titchmarsh literature are closely related but non-equivalent. Motohashi’s large-sieve theorem gives a progression-counting bound with denominator $$\sum_{\substack{x\le n<x+y\ n\equiv a\!\!\!\pmod k}} f(n) \ll \frac{y}{\phi(k)\log x} \exp\!\Bigg(\sum_{\substack{p<x\ p\nmid k}}\frac{f(p)}{p}\Bigg).$$6, namely a restricted-range refinement of the classical global problem, but it contains no short-interval formula of the form

$$\sum_{\substack{x\le n<x+y\ n\equiv a\!\!\!\pmod k}} f(n) \ll \frac{y}{\phi(k)\log x} \exp\!\Bigg(\sum_{\substack{p<x\ p\nmid k}}\frac{f(p)}{p}\Bigg).$$7

and does not mention Shiu (Motohashi, 2012). Maynard’s theorem proves that one may take $$\sum_{\substack{x\le n<x+y\ n\equiv a\!\!\!\pmod k}} f(n) \ll \frac{y}{\phi(k)\log x} \exp\!\Bigg(\sum_{\substack{p<x\ p\nmid k}}\frac{f(p)}{p}\Bigg).$$8 in the classical bound for $$\sum_{\substack{x\le n<x+y\ n\equiv a\!\!\!\pmod k}} f(n) \ll \frac{y}{\phi(k)\log x} \exp\!\Bigg(\sum_{\substack{p<x\ p\nmid k}}\frac{f(p)}{p}\Bigg).$$9 once $y$0, again addressing the up-to-$y$1 problem rather than short intervals (Maynard, 2012).

Yamada’s explicit arbitrary-interval theorem is closer in shape to the prime-specialized side of Shiu’s setting, since it treats $y$2 uniformly for every $y$3, but it is still a prime-counting statement: $y$4 It supplies explicit constants for the prime case, but it does not handle general multiplicative functions (Yamada, 2023).

A different non-equivalence appears in prime number theorem refinements for arithmetic progressions. The short-interval progression asymptotic

$y$5

in the admissible regime of the main term implies only a weak Brun–Titchmarsh-type upper bound for weighted prime counts, and it requires a lower bound such as

$y$6

in the displayed application. That is asymptotic-derived and range-restricted, not a uniform Shiu theorem (Thorner et al., 2021).

These distinctions matter because “Brun–Titchmarsh theorem” is now used for several adjacent statements. Shiu’s theorem is the multiplicative-function short-interval theorem. Prime-counting theorems, weighted-prime analogues, and global progression estimates may be methodologically close, but they are not interchangeable with it.

6. Applications and current significance

The contemporary importance of Shiu’s theorem is visible in the applications of its modern extensions. Wright applies the enlarged theorem to high powers of divisor functions and to smooth numbers in short intervals. In the divisor-function application, the key observation is that for fixed $y$7 and

$y$8

with $y$9,

$M$00

so these weights fall inside the enlarged prime-power hypotheses (Wright, 24 Aug 2025).

For smooth numbers, the theorem feeds into lower bounds for

$M$01

Under the additional assumption

$M$02

Wright proves that there exists a constant $M$03 such that

$M$04

and restates the conclusion in the form

$M$05

This shows that the Shiu framework is not only a uniform upper-bound device; it is also a structural input for sharp local density statements concerning arithmetic sets much sparser than the primes (Wright, 24 Aug 2025).

Taken together, these developments position Shiu’s Brun–Titchmarsh theorem as a foundational result in the local theory of multiplicative functions. Its defining contribution is the transfer of Brun–Titchmarsh control from primes to nonnegative multiplicative weights in short intervals and arithmetic progressions. Later work has broadened the admissible classes, added smooth-support corrections through the Dickman–de Bruijn function, and clarified the boundary between multiplicative, weighted, and prime-only versions of the theorem, but the core structure remains the original short-interval multiplicative estimate.