Questions tagged [analytic-number-theory]
Questions on the use of the methods of real/complex analysis in the study of number theory.
3,997
questions
3
votes
0
answers
42
views
How to prove a property of the product of Eisenstein Series
I want to prove that, if $$ E_i(\tau) = \frac{1}{2\zeta(2i)}\sum_{(m,n)\in\mathbb{Z}^2-\{(0,0)\}}(m\tau+n)^{-k}$$
is the normalized Eisenstein Series of weight $i$, then $E_i(\tau)E_j(\tau)\neq E_{i+j}...
- 39
-3
votes
0
answers
19
views
what is the value of d/ds (integral(0,infinity) sin(s arctan t)/((1-t^2)(s/2) (e^(2pi t) -1)) dt) [closed]
I was reading about the Riemann zeta function and found the integral form of it ζ(s)= 1/(s-1) + 1/2 + 2(integral(0, infinity) sin(s arctan t)/((1-t^2)(s/2) (e^(2pi t) -1)) dt). When I was trying to ...
5
votes
1
answer
82
views
Given nonzero $p(x)\in\mathbb Z[x]$. Are there infinitely many integers $n$ such that $p(n)\mid n!$ is satisfied?
This problem comes from a famous exercise in elementary number theory:
Prove that there are infinitely many $n\in\mathbb Z_+$ such that $n^2+1\mid n!$.
I know a lot of ways to do this. A fairly easy ...
- 786
1
vote
1
answer
140
views
Given the primes, how many numbers are there?
I have a concrete question and a more abstract version of the same. There are lots of theorems that take information on the number of elements and translate it to information on the primes; I’m ...
- 32.3k
-1
votes
1
answer
70
views
How to give this sum a bound?
Let $x,y\in\mathbb{Z},$ consider the sum below
$$\sum_{x,y\in\mathbb{Z}\\ x\neq y}\frac{1}{|x-y|^{4}}\Bigg|\frac{1}{|x|^{4}}-\frac{1}{|y|^{4}}\Bigg|,$$
is there anything I could do to give this sum a ...
- 329
1
vote
1
answer
66
views
Sums involving reciprocal of primes
I am interested in obtaining an upper bound for
$$
\sum_{j=1}^N 1/p_j
$$
and upper+ lower bound for
$$
\sum_{p|K} 1/p.
$$
Here $p_j$ is the j-th prime and $p$ is prime.
I was able to find an ...
- 2,913
2
votes
0
answers
38
views
Prime ideals dividing the Artin conductor
Let $L/K$ be a Galois extension of number fields, and let $(\phi,V)$ be a representation of $\operatorname{Gal}(L/K)$. Let $\mathfrak{f}_\phi$ be the Artin conductor of this representation, which is ...
- 787
2
votes
1
answer
55
views
General form of Jacobi Theta Transformation $\sum_{n \in \mathbb{Z}} e^{n^2 \pi / x} = \sqrt{x} \sum_{n \in \mathbb{Z}} e^{n^2 \pi x} $
I was looking into the functional equation of $\zeta(s)$ and at one point the proof uses the Jacobi Theta Transformation:
$$\sum_{n \in \mathbb{Z}} e^{n^2 \pi / x} = \sqrt{x} \sum_{n \in \mathbb{Z}} ...
- 730
7
votes
1
answer
711
views
What's the easiest way to prove the Riemann Zeta function has any zeros at all on the critical line?
I learned in my intro to complex analysis class that the Riemann Zeta has trivial zeros at the negative even integers; this follows from the Riemann Zeta Reflection identity and the behavior of the $\...
- 1,290
2
votes
1
answer
53
views
The "Euler Product formula" for general multiplicative functions
For the totient function $\phi$, we have the well known "Euler's product formula" (as named on Wikipedia) $$\phi(n) = n \prod_{p | n} \left( 1 - \frac{1}{p} \right)$$
This is easy to show ...
1
vote
0
answers
57
views
Periodic zeta function
Let $e(x)=e^{2\pi ix}$ and let $$F(x,s)=\sum _{n=1}^\infty \frac {e(nx)}{n^s}$$ be the periodic zeta function.
What is the functional equation for the periodic zeta function ?: I can find a statement ...
- 1,662
-2
votes
0
answers
52
views
Average of exponential-weighted Dirichlet function
I am looking for some references on the following type of averages
\[
\int_{0}^{1}\left\vert\sum
\frac{a\left(n\right){\rm e}^{2\pi{\rm i}nt}
}{n^{s}}\right\vert^{2}{\rm d}t,
\]
where $a(n)$ is any ...
- 62
1
vote
1
answer
55
views
$H(x)$ approximates $\pi(x)$ pretty well. But what are the drawbacks, when compared with Riemann's $R(x)$?
I'm aware of the Gram series which is equivalent to $R(x)$ (Riemann prime counting function):
$$ R(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}li(x^{1/n}). $$
Over the interval $x=2$ to $x=10^4$ the average ...
- 754
2
votes
2
answers
215
views
Is $\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$ not convergent?
I need to calculate the limit (if it exist) $$\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$$ where $\{x\}$ denotes the fractional part of $x$, $n!$ is the factorial of $n$...
- 928
1
vote
1
answer
30
views
Study of weighted average or sum of a multiplicative function
In the study of a multiplicative function $f$, sometimes the weighted average of the function $\sum_{1\leq n\leq x} (1-\frac{n}{x}) f(n)$ is studied instead of the sum $\sum_{1\leq n\leq x} f(n)$. Why ...
- 521