# Tagged Questions

Questions on the use of the methods of real/complex analysis in the study of number theory.

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### Intuition for Class Numbers

So I've been thinking about the analytic class number formula lately, and class numbers in general and I'm trying to develop a good intuition for them. My basic question, which may be too general/...
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### Is there only one analytic continuation of the Riemann zeta function?

If I were to manipulate the zeta function in a 'new way' would I end up with an analytic continuation that is equal to the one know or something completely new for values less than 1 and complex ...
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### Are there asymptotically more nonabelian groups of order $p^k$ than there are abelian groups of order $\leq p^k$?

Let $\alpha(n)$ denote the number of isomorphism classes of abelian groups of order $n$ and $\alpha^\prime(n)=\sum_{m=1}^n\alpha(m)$. Similarly, define $f(p^k)$ to be the number of isomorphism ...
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### Questions regarding the Riemann-Siegel $\theta$ Function

My questions are a request, please, for help in understanding some comments in the wikipedia article discussing the Riemann-Siegel $\theta$ function http://en.wikipedia.org/wiki/Riemann%E2%80%...
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### Where are the resources on the prime number theorem?

I am looking for resources which explain the prime number theorem to 18 year old students. I am not seeking a proof of the result but something which will have an impact and motivate a student to ...
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### Arithmetical Functions Sum, $\sum_{d|n}\sigma(d)\phi(\frac{n}{d})$ and $\sum_{d|n}\tau(d)\phi(\frac{n}{d})$

$$\sum_{d|n}\sigma(d)\phi\left(\frac{n}{d}\right)=n\tau(n) ,\\ \sum_{d|n}\tau(d)\phi\left(\frac{n}{d}\right)=\sigma(n)$$ The problem (7.4.15) of Burton's Elementary Number Theory has been request to ...
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### Partial summation: integral version

In a book about analytic number theory, I found two lemmas about partial summation. The first one is the discrete version of partial summation (See http://en.wikipedia.org/wiki/Summation_by_parts) ...
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### Prime harmonic series

We have following identity: ($p$ is a prime number) $$\left(1+\frac{1}{p}\right)\sum_{k=0}^n\frac{1}{p^{2k}}=\sum_{k=0}^{2n+1}\frac{1}{p^k}$$ Now, How to derive the following inequality from the above ...
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### Some identities with the Riemann zeta function

Can someone either help derive or give a reference to the identities in Appendix B, page 27 of this, http://arxiv.org/pdf/1111.6290v2.pdf Here is a reproduction of Appendix B from Klebanov, Pufu, ...
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### Between Mertens' theorems

It is well-known that $$\sum_{p\le x}\frac{\log p}{p}=\log x+O(1)$$ and $$\sum_{p\le x}\frac1p=\log\log x+M+o(1).$$ What is the order of $$\sum_{p\le x}\frac{\sqrt{\log p}}{p}$$ ?
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### Proofs of trivial zeros of zeta function?

I know that the trivial zeros of zeta function are negative even integers . I have seen the wiki-proof using the functional equation of zeta function, I might have seen a proof using Bernoulli ...
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### Newman's “Natural proof”(Analytic) of Prime Number Theorem (1980)

I am trying to understand this short proof by newmann. I faced some problems while grasping this very proof. Please help me out. 1 . I am not clear, why in step (1)'s proof he says that from unique ...
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### Prove that $\sum_{n=1}^\infty \frac{\sigma_a(n)}{n^s}=\zeta(s)\zeta(s-a)$

I would appreciate a hint concerning how to surpass the roadblock I've encountered in my attempt at a proof below. A nicer proof than mine would also help (Edit: The latter part is now done by Gerry ...
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### Elementary Proof of Landau's count on number representable as sum of two squares

In Analytic Number Theory by Iwaniec and Kowalski, there is an elementary proof of Landau's result of $\#\{n \le x: \exists a, b,\ s.t.\ n = a^2 + b^2\} \sim Cx/\sqrt{\log x}$ with an explicit ...
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### Goldbach weak conjecture verification

I found from http://en.wikipedia.org/wiki/Talk:Goldbach%27s_weak_conjecture that Goldbach's weak conjecture might have been proven but the proof has not been peer reviewed yet. What results of the ...
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### Prove that the number of primes satisfying this is $\log(n)$

Let $\textrm{ord}(n)$ denote the number of primes $p$ such that order of $10$ modulo $p$ is $n$. Prove that $\textrm{ord}(n) \sim \log(n)$.
Given $L(s,\chi)$ has a zero of order $m\ge1$ at $s=1+it$, prove that for this t we have: (a) $\frac{L'}{L}(\sigma+it,\chi)=\frac{m}{\sigma-1}+O(1)$ as $\sigma \to 1^{+}$ and (b) there exists an ...
With $p_n = n^{th}$ prime and $f(s):=\sum_{n=1}^\infty 1/p_n^s$ when the series converges. What is the status of the following questions: What is the abscissa of convergence of $f$? What are the ...