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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Infinite product representation of a function in terms of its non-trivial zeroes?
From Wikipedia's Weierstrass Factorization Theorem, I learned that every entire function can be represented as a product involving its zeroes. Examples are the sine and cosine function. The Riemann ...
- 7,118
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votes
1
answer
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Is Riemann Zeta Function symmetrical about the real axis?
From wikipedia,
http://en.wikipedia.org/wiki/Riemann_zeta_function
"Furthermore, the fact that $\zeta(s) = \zeta(s^*)^*$ for all complex s ≠ 1 ($s^*$ indicating complex conjugation) implies that the ...
- 1,913
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3
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On Dirichlet series and critical strips
(I'll keep this one short)
Given a Dirichlet series
$$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$
where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero ...
7
votes
1
answer
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Nonnegativity of the quadratic Dirichlet L-function $L(\tfrac{1}{2},\chi)$ under GRH
I have been looking for a proof of the statement:
"Assume the Generalized Riemann Hypothesis. Let $d$ be a fundamental discriminant and $\chi_d$ the associated primitive quadratic character. Then, $$L(...
- 221
59
votes
6
answers
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How hard is the proof of $\pi$ or $e$ being transcendental?
I understand that $\pi$ and $e$ are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
- 4,496
54
votes
8
answers
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Riemann zeta function at odd positive integers
Starting with the famous Basel problem, Euler evaluated the Riemann zeta function for all even positive integers and the result is a compact expression involving Bernoulli numbers. However, the ...
- 3,049
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3
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Proving $\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$
How to prove this: $$\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} \geq \frac{1}{2}\log{x} -\log{\log{x}}$$
From Apostol's number theory text i know that $$\sum\limits_{p \leq x} \frac{1}{p} = \log{\log{...
anonymous
24
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3
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Motivation for Hecke characters
The context is the definition of Hecke Größencharakter:
http://en.wikipedia.org/wiki/Hecke_character
This is supposed to generalize the Dirichlet $L$-series for number fields. Dirichlet characters ...
angiquesophie
7
votes
1
answer
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Why is width of critical strip what it is?
For Riemann zeta function and $L$-functions of number fields, the width of critical strip is $1$. For $L$-functions of modular forms of weight $k$, the width of the critical strip is $k$.
Why is ...
angiquesophie
1
vote
1
answer
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Showing $e^{\psi(x)}= \text{lcm}[ 1,2,\cdots , \lfloor{x\rfloor}]$
Let $$ \theta(x) = \sum\limits_{p \leq x} \log{p} \quad \ ; \ \psi(x)=\sum\limits_{n=1}^{\infty} \theta(x^{1/n})$$ then how does one prove $$e^{\psi(x)}= \text{lcm}[ 1,2,\cdots , \lfloor{x\rfloor}]$$
anonymous
0
votes
2
answers
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Bounding the series $\sum_{m\leq x,m\neq n}\frac{1}{|\log(m/n)|}$
I am trying to reproduce the following bound:
$\sum_{1\leq m\leq x, m\neq n}\frac{1}{|\log(m/n)|}=O(x\log(x))$,
for $x\geq 2$ and some $n$, $1\leq n\leq x$ (the implicit constant shouldn't depend on ...
- 221
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1
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Efficiently calculating the logarithmic integral with complex argument
My number theory library of choice doesn't implement the logarithmic integral for complex values. I thought that I might take a crack at coding it, but I thought I'd ask here first for algorithmic ...
- 32.3k
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Why is the following evaluation of Apery's Constant wrong and do you have suggestions on how, if at all, this method could be improved?
Please let me summarize the method by which L. Euler solved the Basel Problem and how he found the exact value of $\zeta(2n)$ up to $n=13$. Euler used the infinite product
$$
\displaystyle f(x) = \...
- 7,118
11
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answer
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Continued Fraction expansion of $\tan(1)$
Prove that the continued fraction of $\tan(1)=[1;1,1,3,1,5,1,7,1,9,1,11,...]$. I tried using the same sort of trick used for finding continued fractions of quadratic irrationals and trying to find a ...
- 605
4
votes
3
answers
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Markov-Hurwitz equation
Prove that the Markov-Hurwitz equation $x^2+y^2+z^2=dxyz$ is solvable in positive integers iff d= 1 or 3. Of course the reverse direction is easy, just set x=y=z=1, d=3. But I really have no idea ...
- 605