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pro vyhledávání: '"Mangerel, Alexander P."'
Autor:
Mangerel, Alexander P.
Let $\lambda$ be the Liouville function. Assuming the Generalised Riemann Hypothesis for Dirichlet $L$-functions (GRH), we show that for every sufficiently large even integer $N$ there are $a,b \geq 1$ such that $$ a+b = N \text{ and } \lambda(a) = \
Externí odkaz:
http://arxiv.org/abs/2412.17199
Autor:
Mangerel, Alexander P., You, Yichen
Let $\chi$ be a primitive character modulo $q$, and let $\delta > 0$. Assuming that $\chi$ has large order $d$, for any $d$th root of unity $\alpha$ we obtain non-trivial upper bounds for the number of $n \leq x$ such that $\chi(n) = \alpha$, provide
Externí odkaz:
http://arxiv.org/abs/2405.00544
Autor:
Mangerel, Alexander P.
Let $\lambda$ denote the Liouville function. We show that for all sufficiently large integers $N$, the (non-trivial) convolution sum bound $$ \left|\sum_{1 \leq n < N} \lambda(n) \lambda(N-n)\right| < N-1 $$ holds. This (essentially) answers a questi
Externí odkaz:
http://arxiv.org/abs/2404.12117
Autor:
Mangerel, Alexander P.
Motivated by questions about the typical sizes of gaps $|f(n+1)-f(n)|$ in the sequence $(f(n))_n$, where $f$ is an integer-valued multiplicative function, we investigate the set of solutions $$ \{n \in \mathbb{N} : f(n+a) = f(n) + b\}, \quad ab \neq
Externí odkaz:
http://arxiv.org/abs/2311.11636
Autor:
Mangerel, Alexander P.
Publikováno v:
Discrete Analysis, 2024:12
Let $f: \mathbb{N} \to \mathbb{C}$ be a multiplicative function for which $$ \sum_{p : \, |f(p)| \neq 1} \frac{1}{p} = \infty. $$ We show under this condition alone that for any integer $h \neq 0$ the set $$ \{n \in \mathbb{N} : f(n) = f(n+h) \neq 0\
Externí odkaz:
http://arxiv.org/abs/2306.09929
Let $f:\mathbb{N}\to \mathbb{D}$ be a multiplicative function. Under the merely necessary assumption that $f$ is non-pretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts $h_1,h_2$ the two-poi
Externí odkaz:
http://arxiv.org/abs/2304.05344
Autor:
Mangerel, Alexander P.
Let $\chi$ be a primitive character modulo a prime $q$, and let $\delta > 0$. It has previously been observed that if $\chi$ has large order $d \geq d_0(\delta)$ then $\chi(n) \neq 1$ for some $n \leq q^{\delta}$, in analogy with Vinogradov's conject
Externí odkaz:
http://arxiv.org/abs/2207.14377
Autor:
Mangerel, Alexander P.
Let $f$ be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let $\{\lambda_f(n)\}_n$ be its sequence of normalized Fourier coefficients. We show that if $K/ \mathbb{Q}$ is any number field, and $\mathcal{N}_K$ denotes the collection of
Externí odkaz:
http://arxiv.org/abs/2206.01947
Publikováno v:
Math. Ann. 389(3), 2959-3008, 2024
We characterize the limiting behavior of partial sums of multiplicative functions $f:\mathbb{F}_q[t]\to S^1$. In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined using long i
Externí odkaz:
http://arxiv.org/abs/2202.10370
We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the P\'{o}lya-Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess' estima
Externí odkaz:
http://arxiv.org/abs/2112.12339