Questions tagged [analytic-number-theory]
Questions on the use of the methods of real/complex analysis in the study of number theory.
3,997
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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
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Inequalities for the Dirichlet eta function at non-trivial Riemann zeta zeros.
I am interested in these inequalities for sufficiently large $n$:
$$\Large \left(\Re\left(\sum _{k=1}^n (-1)^{k+1} x^{\log (k) c}\right)\right)^2 \leq \left( \Re\left(x^{\log \left(n+\frac{1}{2}\right)...
- 7,438
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What does this author mean by a simple compactness argument?
In the book "Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis", in the first chapter they say that a sequence of points {$u_n$} is uniformly distributed if:
...
- 81
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Degree of extension of the field of coefficients of modular forms
I am beginning to study modular forms and I came across an inequality defining the bound for the degree of extension of the field of coefficients of a modular form $f\in S_k(\Gamma_1(N))$. This goes ...
- 720
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Clarification about argument why $\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}$ diverge
I would like to clarify some aspects in this answer by Noam D. Elkies proving divergence of
$$\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}.$$
Firstly, do I understand the general strategy ...
- 7,499
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Differently defined Cesàro summability implies Abel summability
I am trying to solve the Exercise 10 of Section 5.2 of the book `Multiplicative Number Theory I. Classical Theory' by Montgomery & Vaughan. In the exercise, they define the Cesàro summability of ...
- 1,145
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Closed form of $\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))}\, dx\,dy\,du\,dv\,dw$
I need closed form for the integral $$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
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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
$$
\sum \limits _{B<b\leq 2B} \tau(b)b^{...
- 21
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Are Hecke L-functions associated to Artin L-functions primitive?
I'm reading a famous paper by Lagarias & Odlyzko on effective versions of the Chebotarev density theorem. There is one thing about Artin L-functions that is a little perplexing to me.
Let $L/K$ be ...
- 787
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Does this function in $3$b$1$b has a name?
I was watching this video from $3$Blue$1$Brown channel, at minute 21:00 he introduced the following function:
$$
\chi(n)=
\begin{cases}
0 & \text{if } n=2k \\
1 & \text{if } n=4k+1\\
-1 & \...
- 6,100
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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. ...
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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
$$\sum_{a <n <b} n^{-\frac{1}{2}-it} \ll (a/t)^{...
- 31
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Partition of a number as the sum of k integers, with repetitions but without counting permutations.
The Hardy-Littlewood circle method (with Vinogradov's improvement) states that given a set $A \subset \mathbb{N}\cup \left \{ 0 \right \} $ and given a natural number $n$, if we consider the sum:
$$f(...
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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
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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