# Tagged Questions

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### How many zero's does a general real entire function $f(z)$ have?

Let $f(z)$ be a real entire function. How do we find the number of solutions for $f(w)=0$ ? Can we express the number of zero's of $f$ in terms of its Taylor coëfficiënts ? Im not looking for the ...
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### Why are people more interested in the Riemann hypothesis than Goldbach's conjecture? [closed]

One of my friends, a math professor, told me almost every one of his colleagues (in the math department) had attempted to prove the Riemann hypothesis at some point in their life (maybe secretly). ...
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### Are there any linkage or transformation between some period transcendental numbers and algebraic irrational numbers?

Are there any linkage or transformation by combination of integral and algebraic function like in the definition of period number between some period transcendental numbers and algebraic irrational ...
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### What is the definition of period number and any relation between Abelian integral and such a kind of period number

I recall there is a kind of real or complex number called period number which is defined by integral and algebraic function.But now,I search it again and again having gotten no result.Now any one can ...
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### When is the fourier transform of a quasi-character $\hat c(\alpha)=|\alpha|c^{-1}(\alpha)$?

This is from lemma $2.4.2$ of Tate's thesis. Let $c$ be a quasi-character on $k^{*}$, the multiplicative group of a number field completed at a non-archimedian place. Lemma 2.4.2 For $c$ in the ...
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### If $\zeta$ is a function of characters what does it mean for it to be regular?

This is from lemma 2.4.1 of Tate's thesis. Lemma 2.4.1: A $\zeta$-function is regular in the "domain" of all quasi-characters of exponent greater than $0$. proof: We must show that for each ...
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### Conditionally convergent products

Can someone explain why this occurs. I came across this in a book by Titchmarsh. $$\prod_{n=2}^{\infty}\left(1-\frac{e^{in\theta}}{\log(n)}\right)$$ this sum does not converge for any rational ...
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### Number of lattice points in an annulus

Consider the lattice spanned by two nonzero complex numbers $\xi_{1}$ and $\xi_{2}$ such that their ratio is not real. Let $w = m\xi_{1} + n\xi_{2}$. Let $A(n)$ be the number of lattice points such ...
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### The series $2+3x+5x^2+7x^3+11x^4+…$

It occurred to me to ask whether the power series whose coefficients are the primes has non-zero radius of convergence, and if so, what kind of function it produces. Wikipedia has some bounds on the ...
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### path of the integral in the initial definition of gamma function

Can the path of the integral in the initial definition of gamma function be altered to a straight line starting from $0$ to $\infty;e^{ia},a<\pi/2$)?
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### special values of zeta function and L-functions

I was reading in some lectures notes about the Riemann zeta-function which takes on special values: $$\zeta(2) = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$ In fact, we can compute even values of the ...
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### Limit approaching a pole of $\phi(s)=-\frac{\zeta'(s)}{\zeta(s)} - \sum_p \frac{\log p}{p^s(p^s -1)}$

If: $$\phi(s) = -\frac{\zeta'(s)}{\zeta(s)} - \sum_p \frac{\log p}{p^s(p^s -1)},$$ where $\zeta(s)$ is the riemann zeta function, why is: $$\lim_{\epsilon \to 0} \epsilon\phi(1+\epsilon) = 1\quad ?$$ ...
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### $\theta(x) = O(x)$ in the prime number theorem

In the Newman short proof of the prime number theorem (http://www.maths.dur.ac.uk/~dma0hg/prime_number_theorem_zagier.pdf) Zagier states that the fact that $2^{2n} >= e^{\theta(2n) - \theta(n)}$ ...
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### Difficulty with a meromorphic extension.

I'm trying to understand the prime number theorem, but never having followed a course in complex analysis, I have some difficulties. (the article is this: ...
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### Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
I am reading about Prime Number Theorem from book by Ingham. As as application of PNT I found the following theorem: Now my doubt is at the step $\frac{\log(y)}{\log(x)}\rightarrow 1$, we can say ...