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
139
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
1
vote
0
answers
48
views
Asymptotic formula for sum of $\frac{1}{n^\lambda}$ over square free $n\leq x$
I'd like to get an asymptotic formula for $\sum_{\substack{n\leq x\\ n\text{ square free}}}\frac{1}{n^l}$ for $l>0$.
We know that $\sum_{\substack{n\leq x\\ n\text{ square free}}}1= cx+O(\sqrt{x})$....
- 521
2
votes
2
answers
111
views
Asymptotic Formula of Selberg
I'm new to asymptotic operation so I need help to understand it. As I know $\mathcal{O(x)}$ is a set of functions. In Selberg's paper about elementary proof of prime number theorem there is that ...
user1127460
-1
votes
0
answers
33
views
Problems in analytic number theory
There is one computation I am struggling with. I quote "Problems in analytic number theory" page 129:
"It is clear that as $\sigma \rightarrow 0^{+}$ log$\zeta(1+\sigma) = log(\frac {1}{...
- 99
3
votes
0
answers
45
views
Number of representations of an integer by a binary quadratic form
In a paper by Heath-Brown, upon having to estimate the number of solutions $(x,y)\in\mathbb Z^2\cap[-B,B]^2$ to the equation $Q(x,y)=k$ with $Q$ an integer-coefficient non-degenerate quadratic form ...
- 157
0
votes
0
answers
39
views
Normal Order of Distinct Prime Factor $\omega(n)$
Define $\omega(n)$ as number of distinct prime factors $n$ has, that is if $n=p_1^{a_1}... p_k^{a_k}$, then $\omega(n)=k$.
It is commonly understood that normal order of $\omega(n)$ is $\log\log(n)$, ...
- 43
0
votes
1
answer
35
views
lower bound of $\sum_{n=1}^x \frac{\mu(n)}{n}$
Denote by $\mu$ the Mobius function. Poussin showed that
$$
\sum_{n=1}^x \frac{\mu(n)}{n} = O(1/\log x),
$$
and there are further improvements since. I wonder what is known about lower bound of ...
- 175
1
vote
1
answer
95
views
Non negativity involving sequences
Define for $n\in\mathbb{N}$ $$a_n=\left[n\sum_{k=1}^{n}\frac{1}{k^5}\right]$$ where $[x]$ denotes the greatest integer $\leq x$.
Prove that $$(n+1)a_n-n a_{n+1}+1\geq 0 \ \ \forall n\geq 1$$
$$(n+1)...
- 928
6
votes
2
answers
279
views
$\lim_{n\to\infty}\left\{n\sum_{k=1}^n\frac{1}{k^5}\right\}$
I need to calculate the limit (if it exist) $$\lim_{n\to\infty}\left\{n\sum_{k=1}^n\frac{1}{k^5}\right\}$$ where $\{x\}$ denotes the fractional part of $x$ and $n\in\mathbb{N}$.
By definition of ...
- 928
1
vote
0
answers
32
views
How do you parameterize simultaneous solutions to equations with expressions like "$ x +2 \left\lfloor\frac{x}{3}\right\rfloor + 1 - [3 \mid x]$"?
Let all functions be integer functions herein. I.e. $\Bbb{Z}\to\Bbb{Z}$ or $\Bbb{N}\to\Bbb{Z}$ where appropriate.
I found this jewel of floor functions.
So that made me wonder whether, we can solve ...
0
votes
0
answers
91
views
Some questions about Fousseraus proof of $\pi(x)=o(x).$
Below is a well known Corollary from Analytic number theory and a proof (excerpt) by G. Fousserau (1892) which I have found here: Narkiewicz. (2000). The Development of Prime Number Theory on page 13.
...
- 2,308
1
vote
1
answer
63
views
Write the sum in terms of the Riemann zeta function
I believe it is a question from JHMT.
Write the sum $\sum_{a=1}^\infty\sum_{b=1}^\infty\frac{gcd(a,b)}{(a+b)^3}$ in terms of Riemann zeta function. The answer should be $-Z(2)+\frac{Z(2)^2}{Z(3)}$...
- 17
1
vote
1
answer
87
views
The sum $f(x)= \sum_{p \le x} e^{\frac{1}{p\log p}}$ is very close to $\pi(x)$. Why is that?
The prime counting function $\pi(x)$ counts the number of primes less than a given $x$. There are other counting functions like Chebyshev's functions which count sums of logarithms of primes up to a ...
- 754
2
votes
0
answers
65
views
Dedekind zeta function of abelian number fields is a product of Dirichlet L-functions
Can somebody give an explicit (and self-contained) proof of the fact that the Dedekind zeta function of an abelian number field $K$ is a product of Dirichlet L-functions? I.e.,
$$
\zeta_K(s)=\prod_{\...
- 787
1
vote
1
answer
72
views
A special kind of Poisson summation formula
There is a Poisson summation formula as follows:
Let $V$ be a smooth function with compact support on $\mathbb{R}$. For $X > 1$ and $q > 1$, we have
$$\sum_{n\equiv a~\mathrm{ mod }~q}{V}\left( ...
- 39
4
votes
0
answers
127
views
If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then $b\neq p^k$
If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then prove that $b\neq p^k$ where $p$ is any prime and $k\in\mathbb{N}$
Take $a,b\in\mathbb{N}$ such that $(a,b)=1$. Now if $b=p^k$ then $$p^k\zeta(5)=a$$ So, by ...
- 928
1
vote
1
answer
51
views
Proposition 16.5.4 in Ireland-Rosen
We aim to show that if $\chi$ is a complex Dirichlet character mod $m$, then $L(1, \chi) \neq 0$. Assuming otherwise, we easily prove that if $F(s) = \prod_{\chi}L(s, \chi)$, where the product is over ...
- 4,429
1
vote
0
answers
91
views
Inequalities for the Dirichlet eta function at non-trivial Riemann zeta zeros.
I am interested in these inequalities for sufficiently large $n$:
$$\Large \left(\Re\left(\sum _{k=1}^n (-1)^{k+1} x^{\log (k) c}\right)\right)^2 \leq \left( \Re\left(x^{\log \left(n+\frac{1}{2}\right)...
- 7,438
5
votes
1
answer
133
views
What does this author mean by a simple compactness argument?
In the book "Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis", in the first chapter they say that a sequence of points {$u_n$} is uniformly distributed if:
...
- 81
2
votes
1
answer
108
views
Degree of extension of the field of coefficients of modular forms
I am beginning to study modular forms and I came across an inequality defining the bound for the degree of extension of the field of coefficients of a modular form $f\in S_k(\Gamma_1(N))$. This goes ...
- 720
0
votes
0
answers
82
views
Clarification about argument why $\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}$ diverge
I would like to clarify some aspects in this answer by Noam D. Elkies proving divergence of
$$\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}.$$
Firstly, do I understand the general strategy ...
- 7,499
2
votes
1
answer
70
views
Differently defined Cesàro summability implies Abel summability
I am trying to solve the Exercise 10 of Section 5.2 of the book `Multiplicative Number Theory I. Classical Theory' by Montgomery & Vaughan. In the exercise, they define the Cesàro summability of ...
- 1,145
4
votes
1
answer
140
views
Closed form of $\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))}\, dx\,dy\,du\,dv\,dw$
I need closed form for the integral $$I:=\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))} \, dx\,dy\,du\,dv\,dw$$
Then $$0<I<\int_{(0,1)^5} \frac{1}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)...
- 928
1
vote
1
answer
41
views
Upper Bound on Average Number of Divisiors
I am working with Iwaniec's "Fourier coefficients of modular forms of half-integral weight". For the first estimate on p.397 he seems to have used
$$
\sum \limits _{B<b\leq 2B} \tau(b)b^{...
- 21
3
votes
0
answers
49
views
Are Hecke L-functions associated to Artin L-functions primitive?
I'm reading a famous paper by Lagarias & Odlyzko on effective versions of the Chebotarev density theorem. There is one thing about Artin L-functions that is a little perplexing to me.
Let $L/K$ be ...
- 787
4
votes
1
answer
192
views
Does this function in $3$b$1$b has a name?
I was watching this video from $3$Blue$1$Brown channel, at minute 21:00 he introduced the following function:
$$
\chi(n)=
\begin{cases}
0 & \text{if } n=2k \\
1 & \text{if } n=4k+1\\
-1 & \...
- 6,100
0
votes
0
answers
94
views
Expressing a function in terms of the nontrivial zeros of the Riemann zeta function
Consider the function $\phi(x)$:
$$ \phi(x)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{2}}\left(\left\{ \frac{x}{n}\right\}-\frac{1}{2}\right)$$
$\left\{\cdot\right\}$ being the fractional part function. ...
2
votes
1
answer
104
views
Enquiry on a claim in Titchmarsh. [closed]
There is a claim on p.107 of Titchmarsh's ''The theory of the Riemann zeta function'', that if $0<a<b \leq 2a$ and $t>0$, then
the bound
$$\sum_{a <n <b} n^{-\frac{1}{2}-it} \ll (a/t)^{...
- 31
0
votes
0
answers
45
views
Partition of a number as the sum of k integers, with repetitions but without counting permutations.
The Hardy-Littlewood circle method (with Vinogradov's improvement) states that given a set $A \subset \mathbb{N}\cup \left \{ 0 \right \} $ and given a natural number $n$, if we consider the sum:
$$f(...
1
vote
1
answer
30
views
An upper bound of $S_f(N)$ using Dirichlet's approximation in Analytic Number Theory by Iwaniec and Kowalski page 199
On page 199 of 'Analytic Number Theory' by Iwaniec and Kowalski, it says that by Dirichlet's approximation theorem, there exists a rational approximation to $2\alpha$ of type $$\Bigl|2\alpha -\frac{a}{...
- 521
0
votes
1
answer
90
views
Understanding a key definition in Lagarias Odlyzko's paper on Chebotarev density theorem
Lagarias & Odlyzko has a 1977 paper where they prove effective versions of the Chebotarev density theorem. I am having trouble understanding equation (3.1).
Here, $L/K$ is a Galois extension of ...
- 787
3
votes
1
answer
143
views
$1^\alpha+2^\alpha+3^\alpha+\cdots+n^\alpha$
Let $\alpha>0$ and $m$ be a positive integers, use Euler's summation formula we can prove that there exists a constant $C$ such that
$$ \sum_{k=1}^nn^\alpha=\frac{n^{\alpha+1}}{\alpha+1}+\frac{n^...
- 915
1
vote
0
answers
46
views
Exercise 2 in Chapter 1 of Apostol's "Modular Functions and Dirichlet series in Number Theory"
I am reading the book in the title and I don't know how to prove the following (that is Exercise 2 in Chapter 1):
Suppose $f$ is an elliptic function (meaning that $f$ is meromorphic and there are two ...
- 1,136
2
votes
0
answers
53
views
Reconciling different ideal-theoretic definitions of Hecke Characters
I'm reading Chapter 7 of Neukirch's book on algebraic number theory, where the author defines a Größencharakter (6.1) as:
Let $\mathfrak{m}$ be an integral ideal of the number field $K$, and let $J^{\...
- 787
0
votes
1
answer
34
views
Question about the proof of distribution of $\Omega(n)-\omega(n)$
On page 68 of the book Multiplicative Number Theory by Montgomery and Vaughan, in the last step of the proof of Theorem 1.16, we have the following:
Here we want to show that $d_k$ above is the $k$-...
- 521
1
vote
0
answers
53
views
M/V Multiplicative NT : Theorem 11.3 and the Siegel zero
Two questions regarding Theorem 11.3 in the book of Montgomery & Vaughan Multiplicative Number Theory on the section "Case 4. Quadratic $χ$, real zeros.":
First the book supposes there ...
- 281