i would like to know what is the exact relation between prime numbers and the zeta function. Thank you in advance !
56198
In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?
92419
Someone claimed that a number, multiplied by the number after it plus 17 is always prime, and showed several cases. I'm not a complete amateur in Number Theory, and I know that $17*18+17=17*19$, so it does not work for $n\equiv0(\mod17)$ but does it always work for other $n$ values? If not, can someone give me a counter example that I can show that person so they can correct their statement?
9136
I am a Biology major (please don't shame me!) but I really enjoy mathematics. Recently I have been reading about this conjecture and its importance in understanding the distribution of primes. After being studied for so long, how come all attempts have failed? Why is this such a difficult thing to study and work on? Would finding a closed form of $$\sum_1^\infty \frac {1}{n^s}$$ help the issue? Have any attempts been made at this at all? Thank you very much for your time educating me and your help!!
111145
Related to the question :  It is well known that   $$\zeta(2n) = f(2n) \pi^{2n}$$  where $f(n)$ is an function in rationals: (the denominator = OEIS : 6, 90, 945, 9450).  Apery proved that $\zeta(3)$ is irrational, and presumably $f(2n+1)$ is so for all (but the latter is open).  But what else is known about $f(2n+1)$? What sort of pattern does it share (if any) with $f(2n)$? Numerically does it fit well? Is there any clue/progress since Apery as to its form? Of course, $f(2n+1)$ may in effect divide out the $\pi$, meaning that somehow $\pi$ is not involved in $\zeta(2n+1)$ in any essential way.
84288
I have recently read an article about the prime number theorem, in which Mathematicians Erdos and Selberg had claimed that proving $\lim \frac{p_n}{p_{n+1}}=1$, where $p_k$ is the $k$th prime, is a very helpful step towards proving the prime number theorem, although I don't know how, primarily because I have not gone through the proof of the theorem even once.  Anyway, I was trying to prove the result $\lim \frac{p_n}{p_{n+1}}=1$ by really elementary methods, as is always my habit, and quite recently I learned of a theorem that the sum of the reciprocals of the primes diverges (I am only a beginner). The idea that suddenly struck me is that this theorem implies $\limsup\frac{p_n}{p_{n+1}}=1$, by a simple application of the ratio test and the fact that $\frac{p_n}{p_{n+1}} &lt; 1$ for all $n$. So, is there any simple way to show that $\liminf\frac{p_n}{p_{n+1}}=1$ or at least $\ge1$, so that the result is proved?  Another beautiful result that came to me, follows from the theorem on divergence of $\sum\frac{1}{p}$, that given any $h &gt; 1$ there are infinitely many $n$ such that $p_n &lt; h^n$. Are there any other results with really simple proofs (easy to understand even for beginners like me) having deep consequences in the theory of the distribution of primes? I'd really like to hear them!
99869
I am a graduate student and have just finished Burton's book on number theory. Now I want to read further on prime numbers. Does anyone have any suggestion?
80634
I'm asked to find all integers $n$ such that $\phi(n)+\sigma(n)=2n.$ I know that when $n$ is a prime, $\phi(n)+\sigma(n)=(n-1)+(n+1)=2n.$ My guess is that $n$ can be primes only, and I want to derive a contradiction when $n$ is a composite number. But I can only show that when $n=pq$ ($p$, $q$ distinct primes), $\phi(n)+\sigma(n)=2pq+2=2n+2.$ How do I generalize from here?
104287
Show there are infinitely many primes that are equivalent to $1 \pmod{8}$.  Hello there! I have been trying to do this problem for a pretty long time with no avail.    I noticed that this is really similar to Euclid's proof that there are infinitely many primes.  However, I can't find a way to use that here.  Can anyone help me?
113587
Hello everyone this is my first question here so if there are any suggested edits  feel free to participate! Is it true that for every $n$ there two prime numbers $p$ and $q$ with $n\mid p-q$? I have verified this empirically but  am not sure if there is a proof for this
13021
How to prove that there exists a prime number between $n$ and $ n!$, for all $ n&gt; 2$?    ( gives a much better bound, but this question is about obtaining a self-contained proof.)
2372
Is this series known to converge, and if so, what does it converge to (if known)?  Where $p_n$ is prime number $n$, and $p_1 = 2$,  $$\sum\limits_{n=1}^\infty \frac{1}{p_n}$$
50103
I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation. What is this other function that they use explicitly given?  Also if I were to take one of these non trivial zeroes and plug it into the original definition would my answer tend towards zero as I evaluate the series?
69457
So a couple of days ago the $17$ million digit number $2^{57885161}-1$ was beaten by the $22$ million digit number $2^{74207281}-1$ at being the largest known prime number.  Are there any specific (purely mathematical) implications of the fact that this particular number is prime? In particular, does this resolve something other than the question if $2^{74207281}-1$ is prime, in the style of some newly discovered theorem resolving previous conjectures? Is it perhaps a counterexample to something so far believed to be true?  Edit: I'm aware of the many non-(strictly mathematical) practical implications and reasons to continue the search for large primes (cryptography etc), but I'm not aware of any strictly mathematical uses of particular numbers being prime, which is why (and what) I'm asking.
81964
Why is $1$ not considered a prime number?  Or, why is the definition of prime numbers given for integers greater than $1$?
72506
I had an idea and was wondering if it works. I seem to have gotten away quite cheaply. The many multiplications on the right side made me consider the prime divisors:   $p\mid (ah,bk)\Rightarrow p\mid ah \land p\mid bk \Rightarrow \left(p\mid a\lor p\mid h\right)\land\left(p\mid b\lor p\mid k\right)\Rightarrow \left(p\mid a\land p\mid b\right)\lor\left(p\mid a\land p\mid k\right)\lor\left(p\mid h\land p\mid b\right)\lor\left(p\mid h\land p\mid k\right)$.   Which matches exactly the right side! I am almost tempted to replace the implications with equivalences, use the definition of the gcd and call it a day! What do you think?  Reference  Apostol, Introduction to Analytic Number Theory, exercise 1.24.
100457
The number $30$ has a curious property:  All numbers co-prime to it, which are between $1$ and $30$ (non-inclusive) are all prime numbers!  I tried searching(limited search, of course) for numbers $\gt 30$ that have this property, but could not find any.  Are there any such numbers $\gt 30$?
2587
A question that has been puzzling me for quite some time now:  Why is the value of the Riemann Zeta function equal to $0$ for every even negative number?  I assume that even negative refers to the real part of the number, while its imaginary part is $0$.  So consider $-2$ for example:  $f(-2) =  \sum_{n=1}^{\infty}\frac{1}{n^{-2}} =  \frac{1}{1^{-2}}+\frac{1}{2^{-2}}+\frac{1}{3^{-2}}+\dots =  1^2+2^2+3^2+\dots =  \infty$  What am I missing here?