|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'504'585
Articles rated: 2609
24 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Bounding $S(t)$ and $S_1(t)$ on the Riemann hypothesis | | Emanuel Carneiro
; Vorrapan Chandee
; Micah B. Milinovich
; | | Date: |
6 Sep 2013 | | Abstract: | Let $pi S(t)$ denote the argument of the Riemann zeta-function, $zeta(s)$,
at the point $s=frac{1}{2}+it$. Assuming the Riemann hypothesis, we present
two proofs of the bound $$ |S(t)| leq left( frac{1}{4} + o(1)
ight) frac{log t}{log log t} $$ for large $t$. This improves a result of
Goldston and Gonek by a factor of 2. The first method consists in bounding the
auxiliary function $S_1(t) = int_0^{t} S(u) du$ using extremal functions
constructed by Carneiro, Littmann and Vaaler. We then relate the size of $S(t)$
to the size of the functions $S_1(tpm h)-S_1(t)$ when $hasymp 1/loglog t$.
The alternative approach bounds $S(t)$ directly, relying on the solution of the
Beurling-Selberg extremal problem for the odd function $f(x) =
arctanleft( frac{1}{x}
ight) - frac{x}{1 + x^2}$. This draws upon recent
work by Carneiro and Littmann. | | Source: | arXiv, 1309.1526 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER