Timeline for Riemann–Von Mangoldt formula
Current License: CC BY-SA 4.0
18 events
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| Nov 17, 2021 at 22:03 | comment | added | Jesse Elliott | @2734364041. If you replace the first non-integral terms with $1+\frac{1}{\pi}\theta(T)$, where $\theta$ is an equality, then you can get an exact formula for $N(T)$ for $T$ not an imaginary part of a zero of $\zeta(s)$. Why not do that? I'm not sure why may answer got downvotes. The formula I gave is correct. | |
| Nov 6, 2021 at 19:25 | history | edited | 2734364041 | CC BY-SA 4.0 |
Explicated the "arg" term using an integral. This is for additional clarity.
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| Nov 2, 2021 at 2:24 | comment | added | Jesse Elliott | Never mind, I found a reference that writes out the full expansion and doesn't use undefined notation. | |
| Nov 1, 2021 at 13:16 | comment | added | Jesse Elliott | I'm just asking about notation. I know the argument principle, I was just wondering what is the precise definition of $\Delta$. CHANGE IN ARGUMENT is not a definition, and nor does MathWorld define $\Delta$. I guess you want me to reverse engineer the definition. | |
| Nov 1, 2021 at 3:44 | comment | added | 2734364041 | @JesseElliott mathworld.wolfram.com/VariationofArgument.html or youtube.com/watch?v=79-ESkh5_f0 | |
| Nov 1, 2021 at 3:23 | comment | added | Jesse Elliott | So you are integrating the logarithmic derivative of the principal value of the argument of $\xi(s)$, but not dividing by $2 \pi$? "Change in argument" has no precise meaning for me. You mean something like in the total change theorem of calculus? Is the $1/(2\pi i)$ in the operator expression for $\Delta$? | |
| Oct 31, 2021 at 22:54 | comment | added | 2734364041 | (Sorry for the caps, I don't know how to italicize or embolden in the comments.) | |
| Oct 31, 2021 at 22:48 | comment | added | 2734364041 | @JesseElliott Regarding $\pi$ vs. $2\pi$: You're looking in a different part of the proof. Look at the bottom of p. 97. Regarding the notation $\Delta$: Like I said, $\Delta$ denotes CHANGE IN ARGUMENT, which can be expressed as an integral (this is the argument principle). | |
| Oct 31, 2021 at 20:50 | comment | added | Jesse Elliott | @2734364041. So $\Delta$ means integral? If so, why don't they just write integrals? Also, can you write out the series for me? Because I can't figure out what it is from the Davenport book. If you do I will accept your answer. Also, Davenport writes $\pi N(T)$, not $2\pi N(T)$. | |
| Oct 31, 2021 at 0:40 | comment | added | KConrad | That use of $\Delta$ is basically referring to the argument principle: if you want to count the number of zeros of an analytic function $f$ around a rectangle $R$, you integrate $f'/f$ once around the boundary counterclockwise (assuming $f$ itself is nonvanishing on $R$, which is a tricky issue when $f(s) = \zeta(s)$ and $R$ passes through the critical strip). In particular, $N(T)$ is counting zeros with multiplicity (since that is how the argument principle works). We expect all zeros of $\zeta(s)$ to be simple, but that's not proved or even known to follow from the Riemann hypothesis. | |
| Oct 30, 2021 at 18:01 | comment | added | 2734364041 | @JesseElliott People often write $\Delta$ to denote "change". So when Davenport writes $2\pi N(T) = \Delta_R \arg\xi(s)$, the RHS means the change in argument of $\xi(s)$ as one traverses the rectangle $R$ (in the positive direction) with vertices $2$, $2+iT$, $-1+iT$, and $-1$. | |
| Oct 30, 2021 at 17:57 | comment | added | 2734364041 | @JesseElliott As I said in my answer, if you look at the proof, the $O(T^{-1})$ can be expanded further using the expansions for arctan (Taylor series) and gamma (lower order terms in Stirling's formula). | |
| Oct 30, 2021 at 8:52 | comment | added | Jesse Elliott | OK I found the book. What does his $\Delta$ mean in the explicit formula $\pi N(T) = \Delta_L \operatorname{Arg} \xi(s)$? I've searched and searched but can't find the definition of $\Delta$ anywhere. | |
| Oct 30, 2021 at 8:35 | comment | added | Jesse Elliott | I mean, is it possible to develop the $O(T^{-1})$ term further? | |
| Oct 30, 2021 at 8:27 | comment | added | Jesse Elliott | Many thanks! I don't have that book. Can you give the whole expansion? And is that the principal branch of arg? | |
| Oct 30, 2021 at 6:24 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing (bracket scaling)
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| Oct 30, 2021 at 4:35 | history | edited | 2734364041 | CC BY-SA 4.0 |
deleted 10 characters in body
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| Oct 30, 2021 at 4:29 | history | answered | 2734364041 | CC BY-SA 4.0 |