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Dr Sacha Mangerel's Outputs (18)

On Equal Consecutive Values of Multiplicative Functions (2024)
Journal Article
Mangerel, A. P. (2024). On Equal Consecutive Values of Multiplicative Functions. Discrete Analysis, 12, https://doi.org/10.19086/da.125450

Let f : N → C be a multiplicative function for which ∑ p: | f (p)|̸ =1 1 p = ∞. We show under this condition alone that for any integer h ̸ = 0 the set {n ∈ N : f (n) = f (n + h) ̸ = 0} has logarithmic density 0. We also prove a converse result, alon... Read More about On Equal Consecutive Values of Multiplicative Functions.

Large sums of high‐order characters (2023)
Journal Article
Mangerel, A. P. (2024). Large sums of high‐order characters. Journal of the London Mathematical Society, 109(1), Article e12841. https://doi.org/10.1112/jlms.12841

Let χ $\chi$ be a primitive character modulo a prime q $q$ , and let δ > 0 $\delta > 0$ . It has previously been observed that if χ $\chi$ has large order d ⩾ d 0 ( δ ) $d \geqslant d_0(\delta)$ then χ ( n ) ≠ 1 $\chi (n) \ne 1$ for some n ⩽ q δ $... Read More about Large sums of high‐order characters.

Three conjectures about character sums (2023)
Journal Article
Granville, A., & Mangerel, A. P. (2023). Three conjectures about character sums. Mathematische Zeitschrift, 305(3), Article 49. https://doi.org/10.1007/s00209-023-03374-8

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya–Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess’ estimate f... Read More about Three conjectures about character sums.

Beyond the Erdős discrepancy problem in function fields (2023)
Journal Article
Klurman, O., Mangerel, A. P., & Teräväinen, J. (2024). Beyond the Erdős discrepancy problem in function fields. Mathematische Annalen, 389(3), 2959-3008. https://doi.org/10.1007/s00208-023-02700-z

We characterize the limiting behavior of partial sums of multiplicative functions f:Fq[t]→S1. In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined using long intervals, short... Read More about Beyond the Erdős discrepancy problem in function fields.

Divisor-bounded multiplicative functions in short intervals (2023)
Journal Article
Mangerel, A. P. (2023). Divisor-bounded multiplicative functions in short intervals. Research in the Mathematical Sciences, 10(12), https://doi.org/10.1007/s40687-023-00376-0

We extend the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function f in typica... Read More about Divisor-bounded multiplicative functions in short intervals.

Short Character Sums and the Pólya–Vinogradov Inequality (2022)
Journal Article
Mangerel, A. P. (2022). Short Character Sums and the Pólya–Vinogradov Inequality. The Quarterly Journal of Mathematics, 71(4), 1281–1308. https://doi.org/10.1093/qmath/haaa031

We show in a quantitative way that any odd primitive character χ modulo q of fixed order g ≥ 2 satisfies the property that if the Pólya–Vinogradov inequality for χ can be improved to max1≤t≤q∣∣∣∣∑n≤tχ(n)∣∣∣∣=oq→∞(q√logq) then for any ɛ > 0 one may ex... Read More about Short Character Sums and the Pólya–Vinogradov Inequality.

Squarefrees are Gaussian in short intervals (2022)
Journal Article
Gorodetsky, O., Mangerel, A., & Rodgers, B. (2023). Squarefrees are Gaussian in short intervals. Journal für die reine und angewandte Mathematik, 2023(795), 1-44. https://doi.org/10.1515/crelle-2022-0066

We show that counts of squarefree integers up to X in short intervals of size H tend to a Gaussian distribution as long as H ! 1 and H D Xo.1/. This answers a question posed by R. R. Hall in 1989. More generally, we prove a variant of Donsker’s theor... Read More about Squarefrees are Gaussian in short intervals.

Additive functions in short intervals, gaps and a conjecture of Erdős (2022)
Journal Article
Mangerel, A. P. (2022). Additive functions in short intervals, gaps and a conjecture of Erdős. Ramanujan Journal, 59(4), 1023-1090. https://doi.org/10.1007/s11139-022-00623-y

With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomäki–Radziwiłł theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of a correspondi... Read More about Additive functions in short intervals, gaps and a conjecture of Erdős.

Large odd order character sums and improvements of the P\'olya-Vinogradov inequality (2022)
Journal Article
Lamzouri, Y., & Mangerel, A. P. (2022). Large odd order character sums and improvements of the P\'olya-Vinogradov inequality. Transactions of the American Mathematical Society, 375, 3759-3793. https://doi.org/10.1090/tran/8607

For a primitive Dirichlet character modulo q, we dene M() = maxt j P nt (n)j. In this paper, we study this quantity for characters of a xed odd order g 3. Our main result provides a further improvement of the classical Polya-Vinogradov inequality in... Read More about Large odd order character sums and improvements of the P\'olya-Vinogradov inequality.

Squarefree Integers in Arithmetic Progressions to Smooth Moduli (2021)
Journal Article
Mangerel, A. P. (2021). Squarefree Integers in Arithmetic Progressions to Smooth Moduli. Forum of Mathematics, Sigma, 9, Article e72. https://doi.org/10.1017/fms.2021.67

Let ε>0 be sufficiently small and let 0<η<1/522 . We show that if X is large enough in terms of ε , then for any squarefree integer q≤X196/261−ε that is Xη -smooth one can obtain an asymptotic formula with power-saving error term for the number of sq...

On the orbits of multiplicative pairs (2020)
Journal Article
Klurman, O., & Mangerel, A. P. (2020). On the orbits of multiplicative pairs. Algebra & Number Theory, 14(1), 155–189. https://doi.org/10.2140/ant.2020.14.155

We characterize all pairs of completely multiplicative functions
fg:N→T, where T denotes the unit circle, such that

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
{(f(n),g(n+1))}n≥1 ≠T×T.

In so doing, we settle an old conjecture of Zoltán Daróczy and Imre Kátai. Read More about On the orbits of multiplicative pairs.

On the bivariate Erdős–Kac theorem and correlations of the Möbius function (2019)
Journal Article
Mangerel, A. P. (2019). On the bivariate Erdős–Kac theorem and correlations of the Möbius function. Mathematical Proceedings of the Cambridge Philosophical Society, 169(3), 547-605. https://doi.org/10.1017/s0305004119000288

Given a positive integer n let ω (n) denote the number of distinct prime factors of n, and let a be fixed positive integer. Extending work of Kubilius, we develop a bivariate probabilistic model to study the joint distribution of the deterministic ve... Read More about On the bivariate Erdős–Kac theorem and correlations of the Möbius function.

Rigidity theorems for multiplicative functions (2018)
Journal Article
Klurman, O., & Mangerel, A. P. (2018). Rigidity theorems for multiplicative functions. Mathematische Annalen, 372(1-2), 651–697. https://doi.org/10.1007/s00208-018-1724-6

We establish several results concerning the expected general phenomenon that, given a multiplicative function f:N→C
, the values of f(n) and f(n+a) are “generally” independent unless f is of a “special” form. First, we classify all bounded completel... Read More about Rigidity theorems for multiplicative functions.