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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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AoPS Volume 1+2
A7456321   2
N 5 hours ago by OGMATH
I've recently been reading the AoPS Volume 1 and 2, and I can't seem to find the answers for any of the exercises. Is there a separate solutions manual I have to buy, or am I just blind and the answers are in the book? Thanks in advance!

EDIT: I couldn't find a forum to post this in so I just posted it in site support. Hope that's ok!
2 replies
A7456321
5 hours ago
OGMATH
5 hours ago
J
H
Introducing myself at AoPS, and what's your magic wand?
asuth_asuth   1176
N Today at 4:18 PM by MathNinja9
Hi!

I'm Andrew Sutherland. I'm the new Chief Product Officer at AoPS. As you may have read, Richard is retiring and Ben Kornell and I are working together to lead the company now. I'm leading all the software and digital stuff at AoPS. I just wanted to say hello and introduce myself! I'm really excited to be part of the special community that is AoPS.

Previously, I founded Quizlet as a 15-year-old high school student. I did Course 6 at MIT and then left to lead Quizlet full-time for a total of 14 years. I took a few years off and now I'm doing AoPS! I wrote more about all that on my blog: https://asuth.com/im-joining-aops

I have a question for all of you. If you could wave a magic wand, and change anything about AoPS, what would it be? All suggestions welcome! Thank you.
1176 replies
asuth_asuth
Mar 30, 2025
MathNinja9
Today at 4:18 PM
J
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All Topics Marked
alpha31415   1
N Yesterday at 6:10 PM by jmr2010
Excuse me!
for I am a frashman in aops.I wonder how I could get back to mark all my read.All topics in the forum became as if they had been marked.Would someone please help me deal with it???
1 reply
alpha31415
Yesterday at 12:39 PM
jmr2010
Yesterday at 6:10 PM
J
H
k profile pictures
happymoose666   1
N Thursday at 11:24 PM by RollingPanda4616
I'm pretty sure this question was raised before but why are their two different avatars?
1 reply
happymoose666
Thursday at 11:10 PM
RollingPanda4616
Thursday at 11:24 PM
J
H
The Riemann Zeta Function
aoum   2
N May 2, 2025 by aoum
The Riemann Zeta Function: A Central Object in Mathematics

The Riemann Zeta Function $\zeta(s)$ is one of the most important functions in mathematics, deeply connected to number theory, complex analysis, and mathematical physics. Its study has led to profound insights into the distribution of prime numbers and the structure of the complex plane.

1. Definition

For complex numbers $s$ with real part greater than $1$, the Riemann zeta function is defined by the absolutely convergent series:

$$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}. $$
That is,

$$ \zeta(s) = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \cdots $$
This definition converges when $\Re(s) > 1$.

2. Analytic Continuation

The function $\zeta(s)$ can be extended to a meromorphic function on the entire complex plane, except for a simple pole at $s=1$. The extension is achieved using techniques like:

[list]
[*] The functional equation,
[*] Mellin transforms,
[*] Dirichlet series manipulations.
[/list]

3. Functional Equation

The Riemann zeta function satisfies a remarkable symmetry, given by the functional equation:

$$ \zeta(s) = 2^s \pi^{s-1} \sin\left( \frac{\pi s}{2} \right) \Gamma(1-s) \zeta(1-s), $$
where $\Gamma(s)$ is the Gamma function.

This equation connects the values of $\zeta(s)$ at $s$ and $1-s$, and it is crucial for the study of its zeros.

4. Euler Product Formula

One of Euler's most important discoveries was that $\zeta(s)$ has an infinite product representation over prime numbers when $\Re(s) > 1$:

$$ \zeta(s) = \prod_{p \, \text{prime}} \frac{1}{1 - p^{-s}}. $$
This shows the deep connection between $\zeta(s)$ and the distribution of prime numbers. It expresses the fundamental theorem of arithmetic (unique prime factorization) analytically.

5. Special Values

At positive even integers:

$$ \zeta(2) = \frac{\pi^2}{6}, \quad \zeta(4) = \frac{\pi^4}{90}, \quad \zeta(6) = \frac{\pi^6}{945}, \quad \text{etc.} $$
At negative integers:

$$ \zeta(-n) = -\frac{B_{n+1}}{n+1}, $$
where $B_n$ are the Bernoulli numbers.

For example:

$$ \zeta(-1) = -\frac{1}{12}, \quad \zeta(-3) = \frac{1}{120}. $$
Note that $\zeta(0) = -\frac{1}{2}$.

6. Zeros of the Zeta Function

The zeros of $\zeta(s)$ are of two types:

[list]
[*] Trivial zeros: Located at negative even integers $s = -2, -4, -6, \dots$.
[*] Non-trivial zeros: Located in the "critical strip" where $0 < \Re(s) < 1$.
[/list]

The famous Riemann Hypothesis conjectures that all non-trivial zeros lie on the "critical line" $\Re(s) = \frac{1}{2}$.

7. Applications of $\zeta(s)$

The Riemann zeta function appears in:

[list]
[*] Prime number theory: The distribution of primes.
[*] Random matrix theory: Models of quantum chaos.
[*] Physics: Statistical mechanics and quantum field theory.
[*] Probability: Connections to branching processes and the zeta distribution.
[*] Fractal geometry: Dimension computations involve zeta-like functions.
[/list]

8. Proof Sketch: $\zeta(2) = \frac{\pi^2}{6}$

One classic proof involves expanding $\sin(\pi x)$ as an infinite product:

$$ \sin(\pi x) = \pi x \prod_{n=1}^\infty \left( 1 - \frac{x^2}{n^2} \right), $$
Taking the logarithm and differentiating, and then comparing coefficients, yields:

$$ \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}. $$
Thus:

$$ \zeta(2) = \frac{\pi^2}{6}. $$
9. References

[list]
[*] Wikipedia: Riemann Zeta Function
[*] H. M. Edwards, Riemann's Zeta Function
[*] Titchmarsh, The Theory of the Riemann Zeta-Function
[*] AoPS Wiki: Riemann Zeta Function
[/list]
2 replies
aoum
Apr 26, 2025
aoum
May 2, 2025
J
No more topics!
H
J
H
changed URLs - is there a way to find very old topics?
spanferkel   2
N Mar 30, 2024 by Demetri
Hi,
I haven't visited this site for many years and see that the URLs have completely changed. If it was before
https://artofproblemsolving.com/Forum/viewtopic.php?p=2471680#p2471680
now it looks like
https://artofproblemsolving.com/community/c6h105686.
Sadly, the links in old posts haven't been updated. How to recover them?
E.g. I'd like to find the two threads quoted in #4 here:

https://artofproblemsolving.com/community/c6h46202p493480
The search function doesn't help, or can it?
Thanks in advance!
2 replies
spanferkel
Mar 30, 2024
Demetri
Mar 30, 2024
J
changed URLs - is there a way to find very old topics?
G H J
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spanferkel
1585 posts
spanferkel
#1 h  Mar 30, 2024, 9:23 PM • 1 Y
Y by AlienGirl05
Hi,
I haven't visited this site for many years and see that the URLs have completely changed. If it was before
https://artofproblemsolving.com/Forum/viewtopic.php?p=2471680#p2471680
now it looks like
https://artofproblemsolving.com/community/c6h105686.
Sadly, the links in old posts haven't been updated. How to recover them?
E.g. I'd like to find the two threads quoted in #4 here:

https://artofproblemsolving.com/community/c6h46202p493480
The search function doesn't help, or can it?
Thanks in advance!
This post has been edited 1 time. Last edited by spanferkel, Mar 30, 2024, 9:24 PM
Reason: typo
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The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
aidan0626
1924 posts
aidan0626
#2 h  Mar 30, 2024, 9:38 PM
Y by
spanferkel wrote:
Hi,
I haven't visited this site for many years and see that the URLs have completely changed. If it was before
https://artofproblemsolving.com/Forum/viewtopic.php?p=2471680#p2471680
now it looks like
https://artofproblemsolving.com/community/c6h105686.
Sadly, the links in old posts haven't been updated. How to recover them?
E.g. I'd like to find the two threads quoted in #4 here:

https://artofproblemsolving.com/community/c6h46202p493480
The search function doesn't help, or can it?
Thanks in advance!
the first link is this
i don't know about the second link tho
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Demetri
1376 posts
Demetri
#3 h  Mar 30, 2024, 9:38 PM
Y by
This has been reported before.
For more information visit this link:
https://artofproblemsolving.com/community/q1h2954840p26462050
Use search function to look for errors before reporting. :)
EDIT: After looking more closely the I have no idea how to get the second link though.
This post has been edited 1 time. Last edited by Demetri, Mar 30, 2024, 9:39 PM
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