# Tagged Questions

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### About the implicit funtion in a holomorphic situation.

Let $f(x,y)$ be a polonomial with integral coefficients which has a zero $(a,b)\in \mathbb{R}^2$ such that the partial derivative respect to $y$ at this point is nonzero. Then by the implicit function ...
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### Integral = $\pi/2$ !!

I am trying to calculate the integral $$I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}dx.$$ (I have literature on this, if people want). Note, we can write the ...
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### Zero to power Zero (Zero ^ Zero) indeterminable or not? [duplicate]

I want to know Zero power to Zero equal to 1 or Indeterminable. I think it cannot be exist. Please explain with proper mathematical definitions.
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### Ratio of maximal to minimal jump in the set of angle multiples

Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times S^1\to\mathbb{R}$ be the distance function given by the arc length. Let $\theta\in S^1$ be an element of infinite order, that is ...
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### A problem with cosine function

I try to understand something from number theory and the author gave this as an excersise: Prove that $z\longmapsto 2\sqrt{p}\cos z$ is a bijection of a set ...
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### On finding the zeros of a polynomial

What is the zero (real) of the polynomial $$x^{k+1}-2x^{k}+1=0$$ If there is such, how can I find it or what method can I use?
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### What is the explicit formula for the nth prime? [duplicate]

The explicit formulas for the second chebyshev function or the prime counting function (in terms of Riemann zeta zero's) are well known. But what is the explicit formula for the nth prime ? For ...
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### Higher dimensional analogues of the argument principle?

I know there are higher dimensional analogues of the argument principle. (See http://en.wikipedia.org/wiki/Variation_of_argument) But I do not have books about it and I cannot find anything of value ...
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### Solving an integral coming from Perron's formula

In analytic number theory, Perron's formula says that $$\sum_{1 \leq k < n} a_k + \frac{1}{2}a_n = \int_{c - i\infty}^{c+i\infty} f(s)\frac{n^s}{s}ds,$$ where $f(s) = \sum_{k \geq 1} a_k/k^s$ ...
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### How does it follow $s\int_1^{\infty}\frac{\psi(x)}{x^{s+1}}dx$?

I have two relations: 1)$-\frac{\zeta'(s)}{\zeta(s)}=\sum_{1}^{\infty}\frac{\Lambda(n)}{n^s}$. 2)$\psi(x)=\sum_{n\leq x}\Lambda(n)$. From these two how does it follow that ...
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### A case where $z^z = 0$ where $z$ is complex number

Is there any case where $z^z = 0$ where $z$ is complex number? The case excludes the case where $z=0$.
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### Analytically continue a function with Euler product

I would like to estimate the main term of the integral $$\frac{1}{2\pi i} \int_{(c)} L(s) \frac{x^s}{s} ds$$ where $c > 0$, $\displaystyle L(s) = \prod_p \left(1 + \frac{2}{p(p^s-1)}\right)$. ...
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### complex analysis poles and residues

I am trying to understand a lemma on the (end of the first page - second page) on this link: http://www.math.uga.edu/~pollack/infprimes-final.pdf Basically, they end up with \sum_{d \geq ...
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### Dirichlet L-series and Gamma function question

Could someone help me, please, with this exercise? Consider a sequence of complex numbers $\{a_n\}$ such that $a_n=a_m$ iff $n\cong m$ mod $q$ for some positive integer $q$. Define the ...
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### When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
### How to show $e^{2 \pi i \theta}$ is not algebraic.
I was wondering if someone could possibly help me figure out how to show $e^{2 \pi i \theta}$ is not algebraic when $\theta$ is irrational. Thanks!
Is there a "naturally occurring" function $f$ which is meromorphic in the complex plane such that the poles of $f$ on the real axis are precisely at the primes? I say "naturally occurring" since we ...