# Tagged Questions

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

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### What are the fertile areas of research in Analytic Number Theory?

My professor once told me that Analytic Number Theory was "dead," which at the time was something of a disappointment, and which I struggled to agree with. Surely any subject may appear inferrtile in ...
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### $|A(n)|<B$, $\lim_{s \to 0^{+}}\sum_{n=1}^{\infty}a_{n}n^{-s}=a$ imply $\lim_{x \to \infty}\sum_{n \leq x} a_{n}(1-\log{n}/\log{x})=a$

Suppose that $|A(n)|<B$ and $\lim_{s \to 0^{+}}\sum_{n=1}^{\infty}a_{n}n^{-s}=a$ where $A(x)=\sum_{n \leq x}a_{n}$. Then $$\lim_{x \to \infty}\sum_{n \leq x} a_{n}(1-\log{n}/\log{x})=a$$ What I ...
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### What is the analytic continuation of the Riemann Zeta Function

I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation. What is this other ...
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### Tying some pieces regarding the Zeta Function and the Prime Number Theorem together

I came across two remarks that I would appreciate help in making the connections. I) In Riemann's Explicit Formula: for $x > 1$ $\Pi = Li(x) - \sum_{\rho:\zeta(\rho)=0}Li (x^{\rho})- \log(2) +$ ...
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### Growth rate of product of smallest prime factors

For $n\in \mathbb{N}$, let $p(n)$ denote the smallest prime dividing $n$. Then consider the function $f:\mathbb{N}\rightarrow \mathbb{N}$ defined by $f(n)= \prod_{k=1}^{n}p(k)$. Question: What is ...
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### Proving the Bernoulli number relation $(1+B)^n=B^n$

We know that we can generate the Bernoulli numbers using the relation $(1+B)^n=B^{[n]}$ where $B_n$ is $n$th Bernoulli number. But how we can prove this works? Thanks to all. Edit 2: is there a ...
I was playing around with prime numbers and a question came into my mind: Let $S(n)$ denote the sum of square roots of primes from $2$ to the $n$th prime number. Are there infinitely many numbers $n$ ...
In February 2013, Sadegh Nazardonyavi and Semyon Yakubovich posted on arxiv: Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$. I have a question about Theorem 2.27 on page 22. My ...