Large values of exponential sums with multiplicative coefficients

Abstract: In 1977 Montgomery and Vaughan gave tight bounds for exponential sums of the form $\sum_{n\leq x}f(n)e(nα)$ where $f$ is a $1$-bounded multiplicative function and $α\in\mathbb R$, close to the conjectured $\ll \frac{x}{\sqrt{q}}+ \frac{x}{\log x}$ where $α$ is best approximated by $|α-a/q|\leq 1/(qx)$, showing their results to be best-possible'' by observing that the first part of their bound is more-or-less attained when $f(n)=χ(n), α=\frac aq$ where $χ$ is a primitive character mod $q$, and the second part when $f(p)=e(-αp)$ for all large primes $p$. La Bretèche and Granville proved that when $α$ lies on a major arc the exponential sum is significantly smaller unless $f$pretends to be'' $χ(n)n^{it}$ for some character $χ$ and real number $|t|<\log x$; and herein we prove that when $α$ lies on a minor arc, the exponential sum is significantly smaller unless $f(p)$ pretends to be $e(-hpα)$ for primes $p\leq x$ for some bounded integer $h$. We also study exponential sums $\sum_{n\leq x, P^+(n)\leq y} f(n) e(nα)$ restricted to $y$-smooth (or $y$-friable) integers $n$. We conjecture that this sum is $\ll \frac{Ψ(x, y)}{\sqrt{q}}+ \frac{\sqrt{xy}}{\log x} $ in a wide range of parameters, show that if true this is best possible, and prove an upper bound in a wide range that is only slightly weaker than the conjecture. Finally we study the logarithmically weighted exponential sums $\sum_{n\leq x} \frac{f(n)}{n} e(nα)$. We conjecture that this sum is $\ll \frac{\log x}{\sqrt{q}}+\log q$ in a wide range of parameters, show that if true this is best possible, and prove an upper bound in a wide range that is only slightly weaker than the conjecture. Along the way, we will prove various technical results about multiplicative functions which may be of use elsewhere.

• The paper establishes precise structure theorems for exponential sums, showing that maximal growth occurs only when multiplicative functions exhibit significant pretentious bias.
• Sharp upper bounds are derived for sums over smooth numbers and logarithmically weighted sums, clarifying optimal behavior over both minor and major arcs.
• Novel technical tools, including a pretentious large sieve and Halász-type inequalities, are introduced, paving the way for further advances in analyzing multiplicative functions.

Large Values of Exponential Sums with Multiplicative Coefficients

Introduction and Motivation

This work delivers a comprehensive analysis of the large values attained by exponential sums of the form

$\sum_{n \leq x} f(n) e(n\alpha),$

where $f : \mathbb{N} \to \mathbb{U}$ is a multiplicative function bounded in modulus by 1, and $\alpha \in [0,1)$. Such exponential sums are foundational objects in analytic number theory, with deep connections to the distribution of arithmetic sequences, the study of character sums, pseudorandomness, and pretentiousness in multiplicative functions.

The classical bounds of Montgomery and Vaughan prescribe that for $f$ multiplicative and $f(n)$ uniformly bounded, one has

$$\sum_{n \leq x} f(n) e(n \alpha) \ll \frac{x}{\sqrt{q}} + \frac{x}{\log x},$$

when $\alpha$ is well approximated by $a/q$ with $(a,q)=1$, $|\alpha - a/q| \leq 1/(qx)$, and $f : \mathbb{N} \to \mathbb{U}$0 not excessively large. These bounds are known to be sharp in specific regimes and are intimately related to Pólya–Vinogradov-type inequalities and to the behavior of character sums. However, Montgomery and Vaughan's analysis does not fully capture the landscape for general multiplicative coefficients, nor for sums restricted to smooth numbers, nor for logarithmic weights.

This work addresses these gaps: extending the understanding of the structure of possible extremal examples, determining precisely when the main terms can be achieved, refining upper bounds in a variety of regimes, and providing optimal lower bounds in terms of precise structural pretentious constraints on the coefficients $f : \mathbb{N} \to \mathbb{U}$1, both for all natural numbers and for those supported on friable (smooth) integers.

Main Results: Unweighted Sums

Structure of Exceptional $f : \mathbb{N} \to \mathbb{U}$2

One of the central contributions is a structural classification of the scenarios under which

$f : \mathbb{N} \to \mathbb{U}$3

can be large. The results sharpen Montgomery–Vaughan by showing that, away from "major arcs" (i.e., when $f : \mathbb{N} \to \mathbb{U}$4 is large), such exponential sums are small unless $f : \mathbb{N} \to \mathbb{U}$5 aligns with a precise imitation of additive twists on the primes:

• If the sum is large, then for many large primes, $f : \mathbb{N} \to \mathbb{U}$6 for some small $f : \mathbb{N} \to \mathbb{U}$7.
• For major arcs, large sums arise only when $f : \mathbb{N} \to \mathbb{U}$8 pretends to be $f : \mathbb{N} \to \mathbb{U}$9 for some primitive Dirichlet character $\alpha \in [0,1)$0 and small real $\alpha \in [0,1)$1.

The analysis leverages the pretentious distance introduced by Granville and Soundararajan, with quantitative estimates specifying how close $\alpha \in [0,1)$2 must be to a "structured" function to support large exponential sums. Furthermore, in all minor arc situations, the only significant contributions arise from terms where the integer argument has a large prime factor, and the large sum comes from a structural bias on $\alpha \in [0,1)$3 for these primes.

Optimal Bounds

The classical Montgomery–Vaughan bound,

$\alpha \in [0,1)$4

is shown to be best possible, up to constants, in all relevant ranges except possibly when $\alpha \in [0,1)$5. This is realized by constructing explicit examples of $\alpha \in [0,1)$6 matching the extremal behavior on each term. In the regime $\alpha \in [0,1)$7 the bounds can be extended to all multiplicative $\alpha \in [0,1)$8, not just completely multiplicative ones, via a reduction to the squarefree case and detailed analysis of convolutions involving powerful numbers.

Sums over Smooth Numbers

The paper extends the consideration of exponential sums to $\alpha \in [0,1)$9-smooth numbers, i.e., integers $f$0 with $f$1. Here, new bounds are proven: $f$2 where $f$3 denotes the count of $f$4-smooth numbers and $f$5 is as above.

For $f$6, the first term dominates, while for larger $f$7 and moderate $f$8, the bound by $f$9 is optimal (again up to small factors and with precise range conditions). The additional $f(n)$0 is necessary in the sparse regime. The analysis is delicate, requiring the interplay of the distribution of friable numbers, the behavior of Ramanujan sums, short interval sieving, and estimates from the distribution in arithmetic progressions.

Strong lower bounds show that these estimates are sharp across the full range of the regimes considered.

Logarithmically Weighted Sums

The paper investigates logarithmic weights: $f(n)$1 and develops a corresponding theory. The main upper bound is

$f(n)$2

for all $f(n)$3. Here, as in the unweighted setting, the claim of optimality is justified: for $f(n)$4 a primitive character modulo $f(n)$5 and $f(n)$6, the main term does achieve $f(n)$7. On minor arcs, the sum is much smaller unless $f(n)$8 is structured appropriately.

Again, the structure theorem specifies that large sums can only arise if $f(n)$9 is close in pretentious distance to $$\sum_{n \leq x} f(n) e(n \alpha) \ll \frac{x}{\sqrt{q}} + \frac{x}{\log x},$$0, with $$\sum_{n \leq x} f(n) e(n \alpha) \ll \frac{x}{\sqrt{q}} + \frac{x}{\log x},$$1 small, or if $$\sum_{n \leq x} f(n) e(n \alpha) \ll \frac{x}{\sqrt{q}} + \frac{x}{\log x},$$2 pretends to be an additive twist on primes for some small modulus, extending the granularity and optimality familiar from the unweighted case to the logarithmically weighted context.

This framework is underpinned by a pretentious large sieve for logarithmically weighted multiplicative functions, as well as an analytic Halász-type theorem tailored to sums of this structure.

Key Techniques and Methods

• Pretentious Large Sieve: The development of a logarithmically weighted variant of the pretentious large sieve enables the authors to control mean values of multiplicative functions twisted by characters and is crucial for upper bounds on sums.
• Structural Decomposition: The use of additive and multiplicative convolutions, together with combinatorial arguments on supports consisting of powerful numbers and friable numbers, allows the decomposition of sums into tractable contributions.
• Short Interval Sieving and Smooth Number Arithmetic: The bounds rely on a mix of classical and recent results on the distribution of friable numbers, as well as effective Brun–Titchmarsh estimates and sieve bounds.
• Optimal Example Construction: Through probabilistic and explicit construction, the work demonstrates that the upper bounds are sharp (subject to small correction factors in the most delicate ranges).
• Analysis on Major and Minor Arcs: A dichotomy analysis based on the Diophantine approximation quality of $$\sum_{n \leq x} f(n) e(n \alpha) \ll \frac{x}{\sqrt{q}} + \frac{x}{\log x},$$3 identifies the regimes where large sums are possible and classifies the structure of $$\sum_{n \leq x} f(n) e(n \alpha) \ll \frac{x}{\sqrt{q}} + \frac{x}{\log x},$$4 in these regimes.

Implications and Further Directions

This manuscript substantially clarifies the landscape of exponential sums over multiplicative coefficients, especially regarding the structure necessary for extremal behavior. The results inform the "pretentious" approach in multiplicative number theory—quantifying how additive and multiplicative structure interact to yield large sums—and provide template methods for further investigation of L-functions, zeros of Dirichlet series, and equidistribution phenomena in number theory.

The developed framework, especially the technical innovations in large sieve inequalities for log-weighted sums and the classification of extremal scenarios for logarithmically weighted sums, opens paths for further research on:

• Distribution of multiplicative functions in short intervals/arithmetic progressions,
• Generalizations to sums over more restricted sets (e.g., primes, friable integers),
• Applications to bounded gaps between zeros of Dirichlet series,
• Sharper bounds in the regime $$\sum_{n \leq x} f(n) e(n \alpha) \ll \frac{x}{\sqrt{q}} + \frac{x}{\log x},$$5 or for more general sets of coefficients tied to automorphic forms.

Conclusion

This paper establishes both qualitative structural theorems and quantitatively sharp bounds for exponential sums with multiplicative coefficients, in unweighted, smooth-restricted, and logarithmically weighted settings. The precise classification of when large sums are possible, together with strong upper and lower bounds and the refinement of analytic tools, make this a comprehensive and influential contribution to the study of analytic and probabilistic behavior of multiplicative functions in exponential sums (2604.02306).