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 Context

This paper addresses the asymptotic and structural behavior of exponential sums

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

where $f : \mathbb{N} \to \mathbb{U}$ is a $1$-bounded multiplicative function, and $e(n\alpha) := \exp(2\pi i n\alpha)$. Understanding the maximum size and typical structure of these sums in various ranges for $\alpha$ and for classes of $f$ remains central in analytic number theory, with deep connections to character sums, the distribution of smooth numbers, the pretentious approach to multiplicative functions, and the study of prime patterns. The work extends the reach of classical results such as Montgomery-Vaughan (1977), La Bretèche-Granville, and Tenenbaum, providing new optimal bounds and precise structure theorems across both minor and major arcs.

Main Results

The principal contributions of the paper are threefold:

1. Structure theorems for large exponential sums in the minor and major arcs: The paper classifies precisely which $f$ and which values of $\alpha$ lead to maximal size for $S_f(x, \alpha)$, proving that unless $f$ correlates strongly with a character twist or phase modulation (in an explicit pretentious sense), the sum must be significantly smaller than the maximal possible order.
2. Sharp upper bounds for exponential sums over smooth numbers and logarithmically weighted sums: The authors generalize classical bounds to the friable setting and for sums weighted by $f : \mathbb{N} \to \mathbb{U}$0, showing that maximal phenomena occur only in constrained algebraic settings, and proving tight quantitative bounds up to explicit factors in terms of prime divisors of the modulus.
3. New technical results and conjectures for mean values and distribution in arithmetic progressions: The paper develops new Halász-type inequalities, pretentious large sieve inequalities, and explores conjectures for the distribution of large exponential sums over smooth numbers in arithmetic progressions, demonstrating optimality up to polylog factors in wide parameter regimes.

Detailed Discussion

Unweighted Exponential Sums: Classification and Bounds

The classical Montgomery-Vaughan bound states that for $f : \mathbb{N} \to \mathbb{U}$1 $f : \mathbb{N} \to \mathbb{U}$2-bounded multiplicative, and $f : \mathbb{N} \to \mathbb{U}$3 near $f : \mathbb{N} \to \mathbb{U}$4 with $f : \mathbb{N} \to \mathbb{U}$5,

$f : \mathbb{N} \to \mathbb{U}$6

when $f : \mathbb{N} \to \mathbb{U}$7. The authors rigorously show that both terms are independently sharp: the first is attained by special $f : \mathbb{N} \to \mathbb{U}$8 with local bias at large primes; the second is only possible for $f : \mathbb{N} \to \mathbb{U}$9 with primitive Dirichlet character $1$0 modulo $1$1, and $1$2 extremely well-approximated by $1$3.

On minor arcs—defined via $1$4 and moderately large $1$5—they demonstrate that unless the values $1$6 for large $1$7 align (or "pretend") as $1$8 for some small $1$9, the sum is reduced to $e(n\alpha) := \exp(2\pi i n\alpha)$0 (Corollary 1). Conversely, all large examples arise from this bias. These precise cutoffs are demonstrated using structured random model constructions and couplings of $e(n\alpha) := \exp(2\pi i n\alpha)$1.

For completely multiplicative $e(n\alpha) := \exp(2\pi i n\alpha)$2, the authors extend a delicate decomposition argument to prove

$e(n\alpha) := \exp(2\pi i n\alpha)$3

for $e(n\alpha) := \exp(2\pi i n\alpha)$4, confirming the optimality of the bound outside a logarithmic window.

Sums over Smooth (Friable) Numbers

Let $e(n\alpha) := \exp(2\pi i n\alpha)$5 denote the count of $e(n\alpha) := \exp(2\pi i n\alpha)$6-friable numbers up to $e(n\alpha) := \exp(2\pi i n\alpha)$7. The exponential sum restricted to $e(n\alpha) := \exp(2\pi i n\alpha)$8-smooth numbers,

$e(n\alpha) := \exp(2\pi i n\alpha)$9

is of central interest in the analytic study of friable numbers. The authors generalize the best possible bounds to

$\alpha$0

(up to $\alpha$1 if $\alpha$2 is highly composite), showing that each term is separately best possible in their parameter regimes.

They answer an open question of La Bretèche and Granville by removing extraneous $\alpha$3 factors in the main estimates (Theorem 2), and show that when $\alpha$4 lies on a major arc, the main term arises precisely as in the classical regime: from twists by characters or prime phase biases. The error estimates exploit intricate distribution properties of smooth numbers in arithmetic progressions, Bombieri-Vinogradov-type theorems for friable numbers, and mean-value reductions.

Logarithmically Weighted Sums and Pretentious Large Sieve

For logarithmically weighted sums,

$\alpha$5

the structure mirrors the unweighted case but presents new technical challenges due to the slow variation and sensitivity to lower order terms. The authors formulate a precise conjecture (matching Montgomery-Vaughan expectations): $\alpha$6 which they verify in all but a short logarithmic window, and classify all $\alpha$7 for which the bound is nearly attained: again, only those $\alpha$8 pretentious with respect to character twists achieve the upper order.

A significant technical development is the establishment of a "pretentious large sieve" (Theorem 3), valid for sums of the form

$\alpha$9

with a bound that depends sharply on the pretentious distance between $f$0 and all non-exceptional characters. The optimality and necessity of the various truncations and ranges are demonstrated by explicit constructions.

Examples and Optimality

Throughout, a series of carefully constructed model examples elucidates the necessity and sharpness of hypotheses. For instance, hybrid constructions illustrate the separation between minor arc phenomena (local phase pretentiousness at large primes) and major arc structure (character twist), and the existence of hybrid functions at intermediary cases.

Implications and Further Directions

Practically, the results unify and extend the understanding of the size and distribution of exponential sums with "generic" multiplicative coefficients, putting precise constraints on when bias or "pretentiousness" is necessary for large sums. This work has immediate applications in the analysis of character sums, the spectral theory of multiplicative functions, and additive-combinatorial aspects of prime patterns.

Theoretically, this paper both solidifies and extends the "pretentious" philosophy in multiplicative number theory, sharply quantifying the transition from randomness to structure. The optimal exponential sum bounds over friable numbers have ramifications for the circle method in thin sets, additive problems with smooth numbers, and mean-value theory.

Future work suggested by this research includes:

• Resolving the tight bounds for the remaining exceptional ranges in the logarithmic case.
• Extending the pretentious large sieve framework further, possibly with additional arithmetic weights or analytic twists.
• Application to problems such as three-prime sums over friable numbers, or the distribution of multiplicative functions in short intervals and arithmetic progressions.

Conclusion

This paper achieves a comprehensive optimal classification for large exponential sums with multiplicative coefficients, explicating the necessary structure for extremal behavior and providing sharp, general quantitative bounds. The technical tools developed, including the pretentious large sieve and mean-value reductions, are likely to be influential across analytic number theory, especially in contexts involving friable numbers, multiplicative function theory, and distribution in minor and major arcs. The results close several longstanding gaps and signal further advances in the analysis of arithmetic exponential sums.

Reference:

Granville, A. and Lamzouri, Y. "Large values of exponential sums with multiplicative coefficients" (2604.02306).