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

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0
votes
1answer
114 views

Polynomial that permutes residue classes

Prove that for any integers $d, e > 1$, the polynomial $f$ with integer coefficients permutes the residue classes modulo $p^d$ if and only if it permutes the residue classes modulo $p^e$ where $p$ ...
6
votes
1answer
348 views

Finding an asymptotic for the sum $\sum_{p\leq x}p^m$ [duplicate]

Possible Duplicate: How does $ \sum_{p \leq x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $? What could I use to prove the following conjecture? $ \sum_{p \le x} p^{m} \sim ...
8
votes
1answer
187 views

the constant in the asymptotics of $\sum_{1\le k \le n} \frac{\varphi(k)}{k^2}$

The following thread at math.stackexchange.com proposes a constant term for the asymptotic expansion of $$\sum_{1\le k \le n} \frac{\varphi(k)}{k^2}.$$ I am getting a different term and I would like ...
2
votes
0answers
54 views

Rationality of Polynomial Coefficients. Integral Question.

We are entertaining polynomials with roots, all unique, on the curve $\Upsilon_s = \{{( 1-\cos[\theta])^{-s} \exp(i \theta)} \ | \ \theta \in \mathbb{R} \}$, where $s>0.$ $\Upsilon_s$ looks like a ...
2
votes
1answer
72 views

$ \ \lim_{x\to ∞ }\frac{π(x)} { x^δ} $

Let $\ π(x)$ denote the prime counting function , i.e. the number of primes not exceeding $x$ Then does $$ \ \lim_{x\to ∞ }\frac{π(x)} { x^δ} $$ exist for all real $δ$ $∈ ( 0 , 1 )$
8
votes
1answer
268 views

average order of $\sum\limits_{\substack{1\le k\le n \\ (k,n)=1}} \frac{1}{k}$

Introduce $$\varrho(n) = \sum\limits_{\substack{1\le k\le n \\ (k,n)=1}} \frac{1}{k}.$$ The following thread at math.stackexchange.com proposes to analyse the average order of $\varrho(n)$, i.e. ...
3
votes
0answers
70 views

Schneider's theorem about the transcendence of values of the $j$-function

It is known that the $j$-function takes algebraic values when evaluated at imaginary quadratic integers. This is a result that was proved by Schneider in 1937 apparently. To be precise, Schneider ...
3
votes
2answers
422 views

Euler Maclaurin summation examples?

How does one use Euler Maclaurin to compute asymptotics for sums like $$ \sum_{\substack{n\le x \\ (n,q)=1}} \frac{1}{\sqrt{n}} \quad \text{or} \quad \sum_{\substack{n\le x \\ (n,q)=1}} \frac{\log ...
6
votes
0answers
449 views

Sum of reciprocal of primes in arithmetic progression

In http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf on page 6 (top) the author states that: $$ \sum_{p \le x, \ p \equiv 1 \bmod l} \frac{1}{p} = \frac{\log \log x}{\phi(l)} + O \left ( \frac{\log ...
8
votes
0answers
657 views

Proof of Hardy-Ramanujan inequality in number theory.

On page 3 of http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf the author write that the following inequalities follow from "the Hardy-Ramanujan inequality", but he doesn't point to a proof. The ...
1
vote
2answers
274 views

The product $\sin(\pi s/2)\Gamma(1 - s)$

How to show that $\sin(\pi s/2)\Gamma(1 - s)$ is analytic when $s$ is an even positive integer?
1
vote
2answers
250 views

Is there a way to show that $\sqrt{p_{n}} < n$?

Is there a way to show that $\sqrt{p_{n}} < n$? In this article, I show that $f_{2}(x)=\frac{x}{ln(x)} - \sqrt{x}$ is ascending, for $\forall x\geq e^{2}$. As a result, $\forall n \geq 3$ ...
-1
votes
1answer
104 views

Recursions in decimal expansions of certain integers (OEIS A216407)

http://list.seqfan.eu/pipermail/seqfan/2012-September/010196.html is the link to "The SeqFan Archives" thread "Recursions in decimal expansions" This thread discusses whether there is a ...
9
votes
5answers
991 views

How does one read aloud Vinogradov's notation $\ll$ and $\ll_{\epsilon }$?

How does one read aloud the Vinogradov's notation $\ll$ and $\ll_{\epsilon }$ as in $$f(x)\ll g(x)$$ and $$c\ll_{\epsilon }\left( \prod\limits_{p\mid abc}p\right) ^{1+\epsilon}.$$ Is the first one ...
1
vote
0answers
64 views

modulus of a series

My question is about the possibility of calculating the modulus of the Dirichlet Eta Function for complex numbers with positive real part. This series is uniformly convergent but it is not absolutely ...
4
votes
2answers
625 views

Trends in the distribution of reordered digits of Pi (OEIS A096566)

First let me try to describe in more details below the approach of "reordering" digits of Pi, which is used in OEIS A096566 https://oeis.org/A096566 and what I have done analyzing it so far. I am ...
2
votes
1answer
243 views

Can we give an upper bound for the sum over primes $p_{i}$ of $\sin(p_{i} x)$?

Let $x$ be a positive real number. Consider the sum $\sum \sin(p_i x)$ taken over all primes $p_i$ from 2 till $n$. Call this function $f(n,x)$. Can we give good upper and lower bounds of $f(n,x)$ ...
8
votes
1answer
853 views

how to prove this extended prime number theorem?

A Generalized Prime Number Theorem? Conjecture Let $n$ and $k$ be positive integers with $n - 50 > k^2 > 0$ and $n$ sufficiently large. Then for the odd primes we have, when $p$ is the biggest ...
3
votes
1answer
540 views

Chebyshev's first $\vartheta(x)$ function question

This was an exercise using the first Chebyshev function, $\vartheta(x)= \sum_{p \leq x} \log p.$ The question is simply how to prove (2) below, the rest is my two thoughts on how to proceed. [Edit: ...
3
votes
0answers
397 views

Show that $n \sum\limits_{p \leq n} \frac{\log(p)}{p} = n \log(n) + \mathcal{O}(n)$

Using the fact that $\log(n!) = n \log(n) - n + \mathcal{O}(\log(n))$ I am asked to show that: $$ n \sum_{p \leq n} \frac{\log(p)}{p} = n \log(n) + \mathcal{O}(n) $$ Prior to this result it was ...
4
votes
1answer
122 views

Is the Glaisher–Kinkelin constant transcendental?

As the title says, is it known whether or not the Glaisher constant is a transcendental number?
5
votes
0answers
311 views

Partial summation of a harmonic prime square series (Prime zeta functions)

I am trying to find the following series: $S=\displaystyle\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\dfrac{1}{p_ip_j},A\leq p_1 < p_2 < \dots < p_n \leq B, \lbrace A,B\rbrace \in \mathbb{N}$ ...
23
votes
4answers
1k views

Evaluation of $\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)$?

I would like to evaluate the sum $$ \sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right) $$ Where $\operatorname{Si}$ is the sine integral, defined as: $$\operatorname{Si}(x) := ...
3
votes
3answers
241 views

Trying to prove that $\lim_{n\rightarrow\infty}(\frac{\Gamma '(n+1)}{n!} -\log(n))=0$

In my attempt to prove that $\Gamma'(1)=-\gamma$, I've reduced the problem to proving that $\lim_{n\rightarrow\infty}(\frac{\Gamma '(n+1)}{n!} -\log(n))=0$. Where $\gamma$ is the Euler-Mascheroni ...
5
votes
3answers
257 views

From $\sum_p \frac{\log p}{p^s} = \frac{1}{s-1} + O(1)$ conclude that $\sum_p \frac{1}{p^s} = \log \frac{1}{s-1} + O(1)$

I'm reading a book on analytic number theory. It asks me to prove: $$ \sum_p \frac{\log p}{p^s} = \frac{1}{s-1} + O(1) \tag{A}$$ and conclude, via integration, that: $$ \sum_p \frac{1}{p^s} ...
5
votes
1answer
130 views

Möbius Inversion Clarification

I'm teaching myself Möbius inversion. From Wikipedia it appears if $F$ and $G$ are complex-valued then $G(x)=\sum\limits_{1 \le n \le x} F(x/n)$ implies $F(x)=\sum\limits_{1 \le n \le x} \mu(n) ...
1
vote
0answers
101 views

Relationship between Connes trace formula and Weil's trace formula

Connes trace formula $$ \mathrm{Tr}\,{U(h)}=2h(1)\ln\Lambda + \sum_{v} \int d^{*}x \frac{h(u^{-1})}{|1-u|} $$ Weil's trace $$ \int_{C}h(u)|u|d^{*}u- ...
9
votes
2answers
426 views

Concrete Example of the Birch and Swinnerton-Dyer Conjecture

The Setup Consider an elliptic curve $E$ in Weierstrass form $y^2=x^3+ax+b$ with $a,b \in \mathbb{Z}$. As usual, we let $\Delta_E$ be the discriminant of the polynomial, and we set $N_p := $ ...
8
votes
1answer
1k views

Effective Upper Bound for the Number of Prime Divisors

Let $\omega(n) = \sum_{p \mid n} 1$. Robin proves for $n > 2$, \begin{align} \omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}}. \end{align} Is there a similar ...
4
votes
1answer
1k views

What is the Birch and Swinnerton-Dyer Conjecture?

This is probably a really silly question, but I was wondering if someone could explain the Birch and Swinnerton-Dyer conjecture to me in a simple way. I've read a lot about it, but cannot understand ...
3
votes
1answer
499 views

Calculating a summation of a $\theta$ function

Let $ \theta_z(t) = \sum \limits_{m,n\in\mathbb{z}}e^{-\pi Q_z(m,n)t}$ where $Q_z(m,n)=y^{-1}|mz+n|^2$. I need to prove that $\theta_z(t)=t^{-1}\theta_z(t^{-1})$. Now, looking that up I know that ...
5
votes
0answers
604 views

one to one mapping between the floor function and the Riemann prime counting function

We have the following 'transform' of a real valued, piecewise continuous function $f(x)$ : $$T[f(x)]=1+\sum_{n=1}^{\infty}\int_{\mathbb{R}^{n}_{+}}f\left(\frac{x}{\Lambda _{n}} \right ...
1
vote
1answer
144 views

Eisenstein series solution

Denote the function $$ \Psi (x,y)= y^{1/2+ik}+ \sum_{g\in SL(2,\Bbb Z)} \frac{y^{1/2+ik}} {|c_{g}z+d_{g}|^{1/2+ik}}\tag{1}$$ My question is if I can write the wave function in terms of the Eisenstein ...
2
votes
1answer
243 views

Strange application of Cauchy's Integral Theorem

According to my book, Riemann's Zeta Function, Cauchy's Integral Formula is applicable to the following integral for all negative values of $s$: $$-\frac{\Pi(-s)}{2\pi i}\int_{|z|=\epsilon}(-2\pi in ...
7
votes
2answers
440 views

Residue of Rankin-Selberg L-function for non-trivial nebentypus

Let $f\in S_k(\Gamma_0(N),\chi)$ be a normalized holomorphic newform (i.e. weight $k$, level $N$, nebentypus $\chi$) and write its Fourier expansion as $$ f(z)=\sum_{n\ge 1} ...
7
votes
1answer
312 views

Is $M(x)=O(x^σ)$ possible with $σ≤1$ even if the Riemann hypothesis is false?

The wiki page on Mertens conjecture and the Connection to the Riemann hypothesis says Using the Mellin inversion theorem we now can express $M$ in terms of 1/ζ as $$ M(x) = \frac{1}{2 \pi i} ...
1
vote
1answer
174 views

Two Representations of $\log \zeta$

I was looking for representations of $\log \zeta$ and found these two: $ \displaystyle \log\zeta(s)=\color{red}{s}\sum_{n>0} \frac{P(ns)}{n\color{red}{s}}$ from here [$\color{red}{s}$ inserted ...
0
votes
1answer
69 views

Question regarding the function $R_X(t)=\frac{1}{\pi} \sum_{p\leq x} \frac{\sin(t\log p)}{\sqrt{p}}$

I want to show that the expected value $\mathbb{E}_{\omega ,T}(R_x(t)^{2k})$ behaves asymptotically as: $$\frac{(2k)!}{k!\cdot 2^k} \left(\frac{\log(\log T)}{2\pi^2}\right)^k$$ for $T^\epsilon < ...
2
votes
2answers
325 views

Asymptotic behaviour of $\sum_{p\leq x} \frac{1}{p^2}$

As the title suggests, I want to find the asymptotic behaviour of this sum as $x\rightarrow \infty$, I tried by summation by parts but didn't succeed I also tried using the asymptotic behvaiour of the ...
8
votes
2answers
253 views

Generalized PNT in limit as numbers get large

If $\pi_k(n)$ is the cardinality of numbers with k prime factors (repetitions included) less than or equal n, the generalized Prime Number Theorem (GPNT) is: $$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln ...
3
votes
1answer
675 views

Change of order of summation.

I feel like an idiot for asking this, so bear my stupidity. I have the sum $\sum_{n\leq N} \sum_{p | n ; \ p \ prime} 1$, and I want to change the order of summation of these two sums I think it ...
3
votes
1answer
99 views

Possible errors in my professor's notes, Abel summation

In my professor's notes he has written this: $$\int_1^N \frac{\{t\} - \frac{1}{2}}{t}dt = \int_1^N\frac{1}{t}d \left(\int_1^t B(y)dy \right) = \int_1^t B(y)dy|_1^N + \int_1^N \frac{\int_1^t ...
5
votes
1answer
259 views

Asymptotics for sums of the form $\sum \limits_{\substack{1\leq k\leq n \\ (n,k)=1}}f(k)$

How can we find an asymptotic formula for $$\sum_{\substack{1\leq k\leq n \\ (n,k)=1}}f(k)?$$ Here $f$ is some function and $(n,k)$ is the gcd of $k$ and $n$. I am particularily interested in the case ...
6
votes
1answer
290 views

Questions about the L-function for Eisenstein Series

$E_a(z,s)$ denotes the Eisenstein series expanded at the cusp $a$. For each cusp $a=\frac{u}{w}$ of $\Gamma_0(N)$, we define the Eisenstein series $$ ...
7
votes
1answer
470 views

Evaluating a series with the Möbius function and greatest common divisor.

Problem: Let $\gcd(a,b,c,d)$ refer to the largest integer $r$ such that $r$ divides each of $a,b,c,d$. Evaluate the series ...
5
votes
2answers
201 views

Bounds for $\zeta$ function on the $1$-line

I was going over my notes from a class on analytical number theory and we use a bound for the $\zeta$ function on the $1$ line as $\vert \zeta(1+it) \vert \leq \log(\vert t \vert) + \mathcal{O}(1)$ ...
5
votes
2answers
2k views

Main differences between analytic number theory and algebraic number theory

What are some of the big differences between analytic number theory and algebraic number theory? Well, maybe I saw too much of the similarities between those two subjects, while I don't see too much ...
5
votes
2answers
161 views

“Convergent” Integral in Davenport's Multiplicative Number Theory

I am currently learning analytic number theory using Davenport's Multiplicative Number Theory book, and at some point I believe something silly is happening. I have great faith that I am wrong AND ...
11
votes
4answers
629 views

Constructing arithmetic progressions

It is known that in the sequence of primes there exists arithmetic progressions of primes of arbitrary length. This was proved by Ben Green and Terence Tao in 2006. However the proof given is a ...
2
votes
1answer
178 views

A prime conjecture

Let $n_k$ for $k=1,2,...,i$ be a finite sequence of positive integers, with $i>1$ and $n_1=0$. If there is a prime p such that for every positive integer m, one or more integers in {${(m+n_k)|1\leq ...