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

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1
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1answer
490 views

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 ...
0
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1answer
76 views

$|A(n)|<B$, $\lim_{s \to 0^{+}}\sum_{n=1}^{\infty}a_{n}n^{-s}=a$ 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 ...
0
votes
1answer
441 views

The solutions of $x+2y+3z=n,(x,y,z\in \mathbb N)$

Define $f(n)=\dfrac{n^2}{48}+\dfrac{n}{8}(1+x_2(n))+X(n),$ where $$X(n)=-\dfrac{7}{48}+\dfrac{9}{16}x_2(n)+\dfrac{1}{4}x_2(\lfloor \dfrac{n+1}{2} \rfloor)+\dfrac{1}{3}x_3(n),$$ and $\begin{eqnarray} ...
2
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0answers
86 views

Equation involved in generating function of divisor function [duplicate]

There is an identity between the divisor function of the odd numbers and the "odd" divisor function of power $3$(I don't know if there is a name for function for this type, if there is , sorry for my ...
7
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1answer
253 views

Prove that $\sum_{\substack{0<k<3^n\\3\nmid k}}\sigma{(3^n-k)}\sigma{(k)}=6\cdot27^{n-1}$

A similar problem to this problem (ccorn has given a nice answer to it). Prove that $$\sum_{\substack{0<k<3^n\\3\nmid k}}\sigma{(3^n-k)}\sigma{(k)}=6\cdot27^{n-1},$$ where $\sigma(N)$ is ...
6
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2answers
261 views

Ratio of sum of Euler's totient to $n$: $\lim_{n \to \infty} {\log \left( \sum_{k=2}^n \varphi(k) \right) \over \log(n)}$

This is more a casual/recreational question... It seems to me, that the limit as given in the subject line $$\lim_{n \to \infty} {\log \left( \sum_{k=2}^n \varphi(k) \right) \over \log(n)} = \log_n ...
1
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1answer
54 views

Closed forms for $\lim_{x\rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$

Im looking for closed forms for $\lim_{x \rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$ where $x$ is a positive real, $a$ is a given real, $p$ is the set of primes such that the ...
0
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1answer
84 views

To which extent distribution of Riemann non-trivial zeros follow a gauss process?

I am trying to clearer and preciser understand to which extent the distribution of the non-trivial zeros of the Riemann $\zeta$-function follow a Gauss process? Yet, what I figured out from readnigs,...
1
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0answers
43 views

What is known about meromorphic functions agreeing with $\pi(n)$?

Let $f$ be a meromorphic function in some region containing the positive real axis such that $f(n) = \pi(n)$ for all but finitely many positive integers $n$, where $\pi(n)$ is the number of primes ...
2
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1answer
206 views

Exercise 1.1.10 from Murty's Problems in Analytic Number Theory

The question is Suppose that $$\sum_{k=1}^\infty d_3(k)|f(kx)|<\infty,$$ where $d_3(k)$ denotes the number of factorizations of $k$ as a product of three numbers. Show that if $$g(x)=\sum_{...
5
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0answers
275 views

Where's my mistake applying Perron's Formula?

I applied Perron's Formula to Riemann Zeta Function and got a weird result. First, I started with a simple definition of Riemann Zeta Function, $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$$ where $\...
9
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4answers
9k views

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 ...
8
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2answers
452 views

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 $x > 1$ $\Pi = Li(x) - \sum_{\rho:\zeta(\rho)=0}Li (x^{\rho})- \log(2) +$ ...
2
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2answers
107 views

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 ...
2
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0answers
191 views

Entropy Rate of a sequence of Random Variables with Distributions related to Primes

Let us consider a stochastic process $\mathcal{X}=\{X_i\}_{i \in \mathbb{N} }$ where $X_i$'s are independent and $X_i$ is distributed as $$X_i=p_k \ \mbox{w. p.}\frac{p_k}{\sum_{l=1}^{i}p_l},\ 1\leq k\...
1
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1answer
35 views

Proving that $\sum_{i=2}^{M}\frac{\pi(x^{1/i})}{i}=O(x^{1/2})+O(Mx^{1/3})$

How do I prove that $$\sum_{i=2}^{M}\frac{\pi(x^{1/i})}{i}=O(x^{1/2})+O(Mx^{1/3}).$$ I tried to use Prime Number theorem for $\pi(x)$ and then approximating the summation by integral, but when I used ...
4
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1answer
93 views

Primes in binary

Let $$S_n(k)=\{1\leq m\leq n: m\ \mbox{has $k$ ones in its binary representation and $m$ is prime}\}\ \\ \forall \ n\geq 2^k-1,\ k\geq1.$$ Let $\pi(x)$ be the prime number function. Then what can be ...
21
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1answer
465 views

What is a zeta function?

In my readings, I've come across a wide variety of objects called zeta functions. For example, the Ihara zeta function, Igusa local zeta function, Hasse-Weil zeta function, etc. My question is simple: ...
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2answers
787 views

A problem about the largest prime factor of $n^2+1$

Let $f(n)$ be the largest prime factor of $n$. The image of function $g(n)=\sqrt{f(n^2+1)}$ is like this: Question: If we want to draw a horizontal line which bisects the points from $n=1$ to $...
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0answers
98 views

estimate $\sum_{x<p\le x+y} \log{p}/p$

In his paper the prime number theorem via the large sieve, A. Hildebrand made use of the following inequality $$\sum_{x<p\le x+y} \frac{\log{p}}{p} \le (2+o(1))\log{\frac{x+y}{x}}$$ where $x\ge y$ ...
2
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1answer
76 views

Question on Wolstenholme's theorem

In one of T. Apostol's student textbooks on analytic number theory (i.e., Introduction to Analytic Number Theory, T. Apostol, Springer, 1976) Wolstenholme's theorem is stated (Apostol, Chapt. 5, page ...
2
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0answers
106 views

Duffin-Schaeffer theorem/conjecture (counter)example

By the "easy" direction of Duffin-Schaeffer conjecture, it is known that if (*)$\sum_{q=1}^{\infty}\phi(q)f(q) < \infty $ (when $\phi(q)$ is euler totient function) then almost all numbers are not ...
3
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0answers
157 views

question about riemann zeta function

How can one prove that $$\zeta (2n)=\frac{(-1)^{n-1}2^{2n-1}\pi ^{2n}B_{2n}}{(2n)!}$$ where $n\in N$ and how can one prove that $$\zeta (2n)=\frac{(-1)^{n}2^{2n-2}\pi ^{2n}E_{2n-1}}{(2n-1)!(2^{2n}-...
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0answers
68 views

Order of summation of Moebius function with summations of fractional parts as coefficients

I want to show that $\displaystyle\sum_{i=0}^n\left(\mu(i)\sum_{j=1}^{\lfloor\frac{n}{i}\rfloor}\{jx\}\right)=O(n)$ for $x\in (0,1)$. I have tried to use the result that $\displaystyle\sum_{i=0}^n\...
3
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2answers
135 views

Order of a function related to divisors

Let $f(n)=\max(\{d(ab):\ a,b\le n\})$ where $d(m)$ is the number of divisors of $m.$ What is the order of $f$? In particular I'm looking for an asymptotic upper bound.
5
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0answers
389 views

Maximum length of sequence of non-coprimes of $N$ - least upper bound for Jacobsthal's function

I am looking at the length of the longest sequences of adjacent integers that are not coprime to $N$ for very large $N$. Let $F_N$ be the set of integers less than $N$ which are not coprime with $N$: ...
2
votes
1answer
219 views

Challenging the Chebychev function / prime number theorem?

The prime number theorem accords with the following equation for the first Chebychev function that: $$\lim_{x\rightarrow\infty}\frac{\vartheta(x)}{x}=1 \qquad (1)$$ According to Muñoz García, E. and ...
5
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1answer
127 views

Number of prime factors of difference of two numbers

As is the custom, define $\omega(m)$ to be number of distinct primes dividing $m$. Also, let $P(m)$ represent set of primes divisors of $m$. Let $S=\{p_1,p_2,\ldots,p_n\}$ be a set of $n$ distinct ...
6
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0answers
133 views

Inverting the Riemann zeta function in $s>1$

Let $s>1$ be a positive real and the Riemann zeta fucntion be defined for $s>1$ as $$ \zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}. $$ I am looking for an inversion formula for the zeta ...
0
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1answer
66 views

How to introduce an integer function into $\zeta$ function instead of $n$

I have a problem that I am struggling with since long and probably it is simple but I can not get through. So your help would be very welcome. Known that Riemann $\zeta$ function is defined as sum ...
2
votes
1answer
142 views

The relation of $\zeta$-function and $p^k$ for $Re(s) \le 1$?

The von Mangoldt function: $$\Lambda(n) = \begin{cases} \log p &; \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1, \\ 0 &; \mbox{otherwise.} \end{cases}$$ establishes a ...
3
votes
1answer
185 views

Analytic continuation of Riemann Zeta funtion

I am reading about zeta function from book by Ingham. In that book the following theorem is given. I am unable to understand what does he mean by finite part of plane in the statement.
5
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0answers
149 views

Best upper bound on the number of divisors of $n$ that are larger than $N$.

I am looking for the best upper bound on $$\sum_{\substack{d | n\\ d \geq N}} 1.$$ I know that $$ d(n) = \sum_{\substack{d | n}} 1 \leq e^{O(\frac{\log n}{\log \log n})}. $$ For my application, I ...
2
votes
1answer
69 views

Lemma from arithmetic functions

Let $f$ arithmetic and $$H(f)=\lim_{x\rightarrow \infty}\frac{1}{x\log x}\sum_{n\leq x}f(n)\log n,$$ Then $H(f)$ exists if and only if $M(f)$ exists, and $M(f)=H(f)$ Where $$M(f)=\lim_{x\rightarrow \...
10
votes
3answers
309 views

Where is the fallacy in the argument using Prime Number Theorem

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 $\...
7
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1answer
82 views

Prove that the non-trivial root of $\sum_{k=1}^{2n} p_kx^k=0$ tends to $-1$

I looked at $$ \sum_{k=1}^{2n} p_kx^k=0, $$ where $p_k$ is the $k$th prime. I found that, next to the trivial root $x_0=0$, there is only one more root $x_n$ that tends towards $-1$, when $n$ ...
0
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2answers
75 views

Why is $\frac{1}{2\pi i} \int_C \left( \frac{x}{n} \right)^s \frac{ds}{s} = \theta(x-n) $?

I'm trying to understand the equation: $$\frac{1}{2\pi i} \int_C \left( \frac{x}{n} \right)^s \frac{ds}{s} = \theta(x-n).$$ Here $x\in \mathbb{R}, x\geq 0$, and $C = \{s:\operatorname{Re}(s) = \...
3
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1answer
221 views

Proving equivalences between prime counting functions.

If we have that: $$\theta(x)=\sum_{p\leq x}\log p,$$ and $$\psi(x)=\sum_{n\leq x}\Lambda(n)$$ Where $\Lambda(n)=\log p $ if $n=p^m$ and $\Lambda(n)=0$ in another case. How can I prove that : 1) $\...
10
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1answer
320 views

some standard estimates in Yitang Zhang's paper

I'm trying to understand Zhang's paper on prime gaps, but I can't figure out some "standard" estimates for which Zhang omitted details. As a layman in analytic number theory, I really need some hints (...
2
votes
1answer
272 views

Functional equation for Hecke $L$-series

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, Theorem II.10.3, we have Let $L(s,\psi)$ be the Hecke $L$-series attached to the Größencharakter $\psi$. Then $L(s,\psi)$ has ...
1
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1answer
166 views

definition of divisor functions

I have a question about the definition of divisor functions when I was reading primes in tuples by Goldston, Pintz, and Yıldırım: Let $\omega(q)$ denote the number of prime factors of a squarefree ...
8
votes
2answers
581 views

Generating functions and the Riemann Zeta Function

The generating function for the terms of the harmonic series: $\frac{1}{n}$ is $-\ln(1 - x)$. Does an ordinary generating function exist for the terms of the zeta function $\zeta(s) = \sum_{n=1}^\...
1
vote
2answers
90 views

Routine question about derivatives of automorphic forms being L^2

I consider Automorphic forms on $G = SL_2 (\mathbb{R})$, which are $\Gamma$-invariant, $K$-finite, $Z(g)$ finite, and of moderate growth. If I have such an automorphic form, which happens to be in $L^...
8
votes
1answer
858 views

Understanding Zhang's result of bounded prime gaps

Here is a link on the internet: https://www.dropbox.com/s/su3uak2a057yrqv/YitangZhang.pdf Can someone teach me how to use trivial estimation to reach (6.1) on page 24? Namely, how to impose $(d,P_0)&...
3
votes
1answer
141 views

Proving that Bombieri's Theorem implies Linnik's theorem

I'm stuck on a line in the proof of Bombieri implies Linnik, where Bombieri: For primitive $\chi$ mod $q$ with $q \leq T$ we define $$N(\alpha, T; \chi)=\#\{\rho=\beta+i\gamma \;:\; \Lambda(\...
0
votes
1answer
153 views

Vanishing of Dirichlet Series

Suppose the function $\sum_{n=1}^{\infty}{a_{n}n^{-s}}$ is $0$ on some open set $U\subset\mathbb{C}$. (Can assume the sum converges absolutely on $U$.) Is it true that $a_{n}=0$ for all $n$? (This ...
4
votes
1answer
134 views

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 $-\frac{\zeta'(s)}{\zeta(...
1
vote
1answer
266 views

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 ...
9
votes
1answer
459 views

Sum of square root of primes

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 $n$ ...
4
votes
1answer
107 views

Trying to understand Theorem 2.27 in a recent paper on the Chebyshev function

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 ...