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

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7
votes
1answer
214 views

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 ...
3
votes
1answer
163 views

Is there an explicit formula connected to $(\log\zeta(s))^2$?

Riemann used $\log\zeta(s)$ and, essentially, Perron's formula to find the explicit formula for his prime counting function, $\Pi(n)$: $li(x)-\displaystyle\sum_{\rho}li(x^\rho)-\log 2-\int_{x}^{\...
1
vote
2answers
852 views

Abscissa of Convergence (and of Absolute Convergence) of the Derivative of a Dirichlet Series

Given the series: $$F(s) = \sum f(n) n^{-s}$$ with abscissa of convergence $\sigma_c$. It's derivative would be: $$F'(s) = - \sum_{n = 1}^\infty \frac{f(n) \log(n)}{n^s}$$ Aopstol, "Intro to ...
8
votes
4answers
644 views

The asymptotic expansion for the weighted sum of divisors $\sum_{n\leq x} \frac{d(n)}{n}$

I am trying to solve a problem about the divisor function. Let us call $d(n)$ the classical divisor function, i.e. $d(n)=\sum_{d|n}$ is the number of divisors of the integer $n$. It is well known that ...
6
votes
1answer
438 views

A general explicit formula for the generalized divisor summatory function?

Mertens function has, by residues, an explicit formula of $M(x)=\displaystyle\sum_{\rho}\frac{x^\rho}{\rho\zeta'(\rho)}-2+\sum_{n=1}^\infty\frac{(-1)^{2 n}(2\pi)^{2n}}{(2n)! n \zeta(2n+1)x^{2n}}$ ...
7
votes
0answers
114 views

Weierstrass product expression for Klein's j-invariant

The first sentence of @ccorn's answer to a previous question of mine was: “Because of the modular symmetries of $j(\tau)$, the zeros of $j(\tau)$ are precisely the $\operatorname{SL}(2,\mathbb{...
4
votes
2answers
520 views

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, ...
2
votes
1answer
96 views

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 $$ \sum_{p\le x}\frac{\sqrt{\log p}}{p} $$ ?
3
votes
0answers
789 views

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 ...
7
votes
1answer
504 views

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 ...
5
votes
1answer
321 views

Applying Möbius Inversion to $\Pi(x)$ and $\pi(x)$

I would appreciate help as to how to apply the Möbius inversion theorem to prime counting $\Pi (x)$ and $\pi (x)$, where: $$\Pi(x) := \sum_{n = 1}^\infty \frac{1}{n} \pi(x^{1/n})$$ and $\pi (x) = \...
3
votes
1answer
193 views

Convergence of the Zeta and Phi functions

I want to show that the following functions (in the picture) are absolutely and locally uniformly convergent if real part of complex number $s$ is bigger than 1. Absolute part for zeta function is ...
5
votes
1answer
120 views

Help using the inclusion-exclusion principle to prove $\sum_{n|d}\mu(d)\frac{n}{d}=\varphi(n)$

$(1)$Using the Dirichlet convolution theorem, $$\sum_{n|d}\mu(d)\frac{n}{d}=\sum_{ab=n}\mu(a)b$$ Let $n=p_1\cdot p_2\cdots p_n\cdot k$, where $k$'s prime factorisation contains $p_k$ only if $ k\in\{...
6
votes
2answers
98 views

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 ...
2
votes
0answers
157 views

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 ...
3
votes
0answers
80 views

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 ...
1
vote
1answer
104 views

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 $\textrm{ord}(n) \sim \log(n)$.
2
votes
0answers
45 views

Apostol ANT chapter 13 Question 9

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 ...
1
vote
2answers
102 views

Dirichlet series for prime sequence

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 ...
1
vote
2answers
67 views

At what rate are composites removed in a set after each prime multiple is cancelled out?

I was looking at sieves today, mainly sieving for primes and I noticed a pattern type thing. As I crossed out primes in a small set, the number of composites that were crossed out decreased. I haven't ...
5
votes
2answers
177 views

Prove that $\sum_{n\leq x}d^2(n)=O(x\log^3 x)$

Prove that $$\sum_{n\leq x}d^2(n)=O(x\log^3 x),\tag1$$ where $d(n)$ is the divisor function: $d(n)=\sigma_0(n)=\sum_{a\mid n}1.$ I can prove that $$\sum_{n\leq x}d(n)=x\log x+(2\gamma -1)x+O(\...
1
vote
1answer
474 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
votes
1answer
75 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
434 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
votes
0answers
84 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
votes
1answer
252 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
votes
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
vote
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
votes
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
vote
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
votes
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
votes
0answers
273 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
votes
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
votes
2answers
450 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
votes
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
votes
0answers
189 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
vote
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
votes
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 ...
19
votes
1answer
449 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: ...
11
votes
2answers
776 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 $...
0
votes
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
votes
1answer
75 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
votes
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
votes
0answers
156 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}-...
1
vote
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
votes
2answers
132 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
votes
0answers
388 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
218 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
votes
1answer
126 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 ...
5
votes
0answers
132 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 ...