Timeline for Tools to prove lower bounds in analytic number theory
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 10, 2023 at 16:58 | comment | added | EGME | @Peter Humphries. Yes, there are such expression, although I had not thought of using them thus far. What you just sent might be what I was hoping to find out about. I need to study it. Thank you. | |
| Apr 10, 2023 at 16:45 | comment | added | Peter Humphries | I'm not sure I quite understand what you mean. Is there an explicit expression in terms of sums of zeroes? Then sometimes you can get quite good bounds conditional on standard conjectures. See for example my answer here: mathoverflow.net/a/260540/3803 | |
| Apr 10, 2023 at 16:35 | comment | added | EGME | @Peter Humphries It is this function I need to bound away from the $x$-axis. An $\Omega$ type of bound won't do, however, since you want both the lim sup and the lim inf to be on the same side of the $x$-axis. You want the whole function to be bounded away from the axis. | |
| Apr 10, 2023 at 16:34 | comment | added | EGME | @Peter Humphries Thank you. What you are commenting in those posts is quite relevant to what I am trying to do. I have a function for which I can prove, using Landau's lemma, that if it is of constant sign, RH follows and not(LI) follows (a similar case to L). However, the function which I have in mind hardly goes up and down, let alone oscillate. You can almost prove that it is of constant sign by induction, but this doesn't quite work; and there are other some such approaches, each with its own problem. I conjecture that it's bounded below the $x$-axis by $-\sqrt{x}.$ (contd.) | |
| Apr 10, 2023 at 16:04 | comment | added | Peter Humphries | Not Perron's formula, but rather Landau's lemma; see Chapter 15 of Montgomery-Vaughan. I answered some related questions using these techniques: mathoverflow.net/a/266415/3803, mathoverflow.net/a/368511/3803 | |
| Apr 10, 2023 at 15:40 | history | asked | EGME | CC BY-SA 4.0 |