Questions on the use of the methods of real/complex analysis in the study of number theory.
1
vote
0answers
51 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 ...
3
votes
2answers
308 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
143 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 $f(n,x)$ ...
6
votes
1answer
192 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 ...
2
votes
1answer
251 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
109 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
85 views
Is the Glaisher–Kinkelin constant transcendental?
As the title says, is it known whether or not the Glaisher constant is a transcendental number?
2
votes
0answers
108 views
Partial summation of a harmonic prime square series (Prime zeta functions)
I am trying to find the following series:
$S=\displaystyle\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\dfrac{1}{p_ip_j},A\leq p_1 < p_2 < \dots < p_n \leq B, \lbrace A,B\rbrace \in \mathbb{N}$
...
17
votes
4answers
573 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:
$$\operatorname{Si}(x) := ...
2
votes
3answers
132 views
Trying to prove that $\lim_{n\rightarrow\infty}(\frac{\Gamma '(n+1)}{n!} -\log(n))=0$
In my attempt to prove that $\Gamma'(1)=-\gamma$, I've reduced the problem to proving that $\lim_{n\rightarrow\infty}(\frac{\Gamma '(n+1)}{n!} -\log(n))=0$.
Where $\gamma$ is the Euler-Mascheroni ...
5
votes
3answers
183 views
From $\sum_p \frac{\log p}{p^s} = \frac{1}{s-1} + O(1)$ conclude that $\sum_p \frac{1}{p^s} = \log \frac{1}{s-1} + O(1)$
I'm reading a book on analytic number theory. It asks me to prove:
$$ \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
99 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
71 views
relationship between Connes trace formula and Weil's trace formula
Connes trace formula $$ Tr{U(h)}=2h(1)log\Lambda + \sum_{v} \int d^{*}x \frac{h(u^{-1})}{|1-u|} $$
Weil's trace $$ \int_{C}h(u)|u|d^{*}u- ...
8
votes
2answers
179 views
Concrete Example of the Birch and Swinnerton-Dyer Conjecture
The Setup
Consider an elliptic curve $E$ in Weierstrass form
$y^2=x^3+ax+b$
with $a,b \in \mathbb{Z}$. As usual, we let $\Delta_E$ be the discriminant of the polynomial, and we set
$N_p := $ ...
7
votes
1answer
311 views
Effective Upper Bound for the Number of Prime Divisors
Let $\omega(n) = \sum_{p \mid n} 1$. Robin proves for $n > 2$,
\begin{align}
\omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}}.
\end{align}
Is there a similar ...
2
votes
1answer
271 views
What is the Birch and Swinnerton-Dyer Conjecture?
This is probably a really silly question, but I was wondering if someone could explain the Birch and Swinnerton-Dyer conjecture to me in a simple way. I've read a lot about it, but cannot understand ...
3
votes
1answer
179 views
Calculating a summation of a $\theta$ function
Let $ \theta_z(t) = \sum \limits_{m,n\in\mathbb{z}}e^{-\pi Q_z(m,n)t}$ where $Q_z(m,n)=y^{-1}|mz+n|^2$.
I need to prove that $\theta_z(t)=t^{-1}\theta_z(t^{-1})$.
Now, looking that up I know that ...
5
votes
0answers
538 views
one to one mapping between the floor function and the Riemann prime counting function
We have the following 'transform' of a real valued, piecewise continuous function $f(x)$ :
$$T[f(x)]=1+\sum_{n=1}^{\infty}\int_{\mathbb{R}^{n}_{+}}f\left(\frac{x}{\Lambda _{n}} \right ...
1
vote
1answer
95 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
154 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 $s$:
$$-\frac{\Pi(-s)}{2\pi i}\int_{|z|=\epsilon}(-2\pi in ...
5
votes
0answers
302 views
Residue of Rankin-Selberg L-function for non-trivial nebentypus
Let $f\in S_k(\Gamma_0(N),\chi)$ be a normalized holomorphic newform (i.e. weight $k$, level $N$, nebentypus $\chi$) and write its Fourier expansion as
$$ f(z)=\sum_{n\ge 1} ...
4
votes
1answer
131 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
$$
M(x) = \frac{1}{2 \pi i} ...
1
vote
1answer
130 views
Two Representations of $\log \zeta$
I was looking for representations of $\log \zeta$ and found these two:
$ \displaystyle \log\zeta(s)=\color{red}{s}\sum_{n>0} \frac{P(ns)}{n\color{red}{s}}$ from here [$\color{red}{s}$ inserted ...
0
votes
1answer
60 views
Question regarding the function $R_X(t)=\frac{1}{\pi} \sum_{p\leq x} \frac{\sin(t\log p)}{\sqrt{p}}$
I want to show that the expected value $\mathbb{E}_{\omega ,T}(R_x(t)^{2k})$ behaves asymptotically as:
$$\frac{(2k)!}{k!\cdot 2^k} \left(\frac{\log(\log T)}{2\pi^2}\right)^k$$
for $T^\epsilon < ...
1
vote
2answers
191 views
Asymptotic behaviour of $\sum_{p\leq x} \frac{1}{p^2}$
As the title suggests, I want to find the asymptotic behaviour of this sum as $x\rightarrow \infty$, I tried by summation by parts but didn't succeed I also tried using the asymptotic behvaiour of the ...
7
votes
2answers
170 views
Generalized PNT in limit as numbers get large
If $\pi_k(n)$ is the cardinality of numbers with k prime factors (repetitions included) less than or equal n, the generalized Prime Number Theorem (GPNT) is:
$$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln ...
2
votes
1answer
421 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
79 views
Possible errors in my professor's notes, Abel summation
In my professor's notes he has written this:
$$\int_1^N \frac{\{t\} - \frac{1}{2}}{t}dt = \int_1^N\frac{1}{t}d \left(\int_1^t B(y)dy \right) = \int_1^t B(y)dy|_1^N + \int_1^N \frac{\int_1^t ...
4
votes
1answer
186 views
Asymptotics for sums of the form $\sum \limits_{\substack{1\leq k\leq n \\ (n,k)=1}}f(k)$
How can we find an asymptotic formula for
$$\sum_{\substack{1\leq k\leq n \\ (n,k)=1}}f(k)?$$
Here $f$ is some function and $(n,k)$ is the gcd of $k$ and $n$. I am particularily interested in the case
...
4
votes
0answers
133 views
Questions about the L-function for Eisenstein Series
$E_a(z,s)$ denotes the Eisenstein series expanded at the cusp $a$. For each cusp $a=\frac{u}{w}$ of $\Gamma_0(N)$, we define the Eisenstein series
$$
...
4
votes
0answers
217 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 ...
4
votes
2answers
119 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 $\vert \zeta(1+it) \vert \leq \log(\vert t \vert) + \mathcal{O}(1)$ ...
2
votes
2answers
378 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 similarties between those two subjects, while I don't see too much ...
5
votes
2answers
121 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 ...
8
votes
4answers
211 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
131 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 ...
6
votes
1answer
172 views
Euler's summation by parts formula
I'm beginning analytic number theory and I see this formula in Apostol's book : If $f$ has a continuous derivative $f'$ on the interval $[y,x]$, where $0 < y < x$, then
$$
\sum_{y < n \le x} ...
3
votes
1answer
83 views
Exclusive prime factors
Let $S$ be an finite or infinite subset of the primes. Let $f(x)=1$ if $x$ has no factors in $S$. If not, $f(x)=0$. Is there a way to calculate the limit $\displaystyle\sum_{n=1}^{x} f(n)/x$, as $x$ ...
10
votes
5answers
544 views
A good reference to begin analytic number theory
I know a little bit about basic number theory, much about algebra/analysis, I've read most of Niven & Zuckerman's "Introduction to the theory of numbers" (first 5 chapters), but nothing about ...
5
votes
0answers
104 views
Dedekind sums + strange integral
I need to write $\displaystyle \int_0^1 f(x)\,dx$, where $f(x) = \# \Sigma_{pq} \cap (x, x+1)$ for $x \in [0, 1]$, where $$\Sigma_{pq} = \left \{ \frac{k}{p} + \frac{l}{q}:\ 1 \leq k \leq p-1, 1 \leq ...
6
votes
1answer
99 views
Limit of ratios of numbers with $m$ factors and primes
This is my first question.
Let $a_1, a_2,\ldots, a_k$ be natural numbers $\leq n$ with $m$ prime factors.
Let $p_1, p_2, \ldots, p_r$ be the prime numbers $\leq n$.
Let $$C_{m,n} = ...
2
votes
1answer
98 views
Some asymptotics for zeta function.
I need to use the functional equation for $\zeta(s)$ and Stirling's formula, to show that for $s=\sigma +it$ , with $\sigma <0$:
$$ |\zeta(s)| << \left(\frac{t}{2\pi}\right)^{1/2-\sigma}$$
...
3
votes
3answers
275 views
Other functional equations for $\zeta(s)$?
For the Riemann zeta function, we know of the standard functional equation that relates $\zeta(s)$ and $\zeta(1-s)$. I wanted to know whether there are functional equations that relates $\zeta(s)$ ...
8
votes
1answer
111 views
Comparing average values of an arithmetic function
Suppose $f(n)$ is a positive real-valued arithmetic function such that
$$
\frac1n\sum_{k=1}^nf(k)\sim g(n)
$$
for $g(x)$ a monotonic increasing function. What can be said about the asymptotic behavior ...
3
votes
1answer
217 views
Is there a simple way to prove Bertrand's postulate from the prime number theorem?
Is there a simple way to prove Bertrand's postulate from the prime number theorem?
The prime number theorem immediately implies Bertrand's postulate for sufficiently large $n$, but it fails to ...
11
votes
1answer
215 views
What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?
We have the following evaluations:
$$\begin{aligned}
&\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\
&\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = ...
5
votes
0answers
106 views
Limit inferior of the quotient of two consecutive primes
I have recently read an article about the prime number theorem, in which Mathematicians Erdos and Selberg had claimed that proving $\lim \frac{p_n}{p_{n+1}}=1$, where $p_k$ is the $k$th prime, is a ...
0
votes
0answers
86 views
$L(1+it,\chi)\neq 0 $ whenever $t \neq 0 \in \mathbb{R}$
I understand that the proof of the assertion in the title uses the same method which proves that zeta function satisfies $\zeta(1+it)\neq 0$, where the above $L$ is Dirichlet L-function.
I.e, you ...
5
votes
1answer
187 views
Always a prime between $x$ and $x+cf(x)$
What is the asymptotically slowest growing function $f(x)$, such that there exists constants $a$ and $b$, such that for all $x>a$, there is always a prime between $x$ and $x+bf(x)$?
$f(x)=x$ ...
6
votes
2answers
202 views
Question regarding Von-Mangoldt function.
Let $\psi(x) := \sum_{n\leq x} \Lambda(n)$ where $\Lambda(n)$ is the Von-Mangoldt function.
I want to show that if $$ \lim_{x \rightarrow \infty} \frac{\psi(x)}{x} =1 $$ then also $$\lim_{x\rightarrow ...
