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pro vyhledávání: '"Harper, Adam J."'
Autor:
Harper, Adam J.
We prove conjecturally sharp upper bounds for the Dirichlet character moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r$, and $0 \leq q \leq 1$ is real. In particular,
Externí odkaz:
http://arxiv.org/abs/2301.04390
Autor:
Harper, Adam J.
We investigate the sums $(1/\sqrt{H}) \sum_{X < n \leq X+H} \chi(n)$, where $\chi$ is a fixed non-principal Dirichlet character modulo a prime $q$, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erd\H{o}s, and more recently Lamzouri, prov
Externí odkaz:
http://arxiv.org/abs/2203.09448
Autor:
Harper, Adam J.
We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum_{n \leq x} f(n)| \geq \sqrt{x} (\log\log x)^{1/4+o(1)}$. This is the first such bound that
Externí odkaz:
http://arxiv.org/abs/2012.15809
Autor:
Harper, Adam J.
We investigate the ``partition function'' integrals $\int_{-1/2}^{1/2} |\zeta(1/2 + it + ih)|^2 dh$ for the critical exponent 2, and the local maxima $\max_{|h| \leq 1/2} |\zeta(1/2 + it + ih)|$, as $T \leq t \leq 2T$ varies. In particular, we prove
Externí odkaz:
http://arxiv.org/abs/1906.05783
Autor:
Harper, Adam J.
This is the text to accompany my Bourbaki seminar from 30th March 2019, on the maximum size of the Riemann zeta function in "almost all" intervals of length 1 on the critical line. It surveys the conjecture of Fyodorov--Hiary--Keating on the behaviou
Externí odkaz:
http://arxiv.org/abs/1904.08204
Autor:
Harper, Adam J.
Publikováno v:
Alg. Number Th. 13 (2019) 2277-2321
We determine the order of magnitude of $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ up to factors of size $e^{O(q^2)}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, for all real $1 \leq q \leq \frac{c\log x}{\log\log x}$. In t
Externí odkaz:
http://arxiv.org/abs/1804.04114
Publikováno v:
Math. Annalen 374 (1) (2019), 517-551
We continue to investigate the race between prime numbers in many residue classes modulo $q$, assuming the standard conjectures GRH and LI. We show that provided $n/\log q \rightarrow \infty$ as $q \rightarrow \infty$, we can find $n$ competitor clas
Externí odkaz:
http://arxiv.org/abs/1711.08539
We prove a sharp version of Hal\'asz's theorem on sums $\sum_{n \leq x} f(n)$ of multiplicative functions $f$ with $|f(n)|\le 1$. Our proof avoids the "average of averages" and "integration over $\alpha$" manoeuvres that are present in many of the ex
Externí odkaz:
http://arxiv.org/abs/1706.03755
Publikováno v:
Compositio Math. 155 (2019) 126-163
Hal\'asz's Theorem gives an upper bound for the mean value of a multiplicative function $f$. The bound is sharp for general such $f$, and, in particular, it implies that a multiplicative function with $|f(n)|\le 1$ has either mean value $0$, or is "c
Externí odkaz:
http://arxiv.org/abs/1706.03749
Autor:
Harper, Adam J.
We determine the order of magnitude of $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0 \leq q \leq 1$. In the Steinhaus case, this is equivalent to determining the order of $\l
Externí odkaz:
http://arxiv.org/abs/1703.06654