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

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14
votes
0answers
217 views
+50

Find a closed form formula for $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes.

I meant by "closed form formula" a formulate that doesn't have summation or has very few terms. Maybe there's a better term for this meaning. I found this function that has very interesting property ...
1
vote
0answers
25 views
1
vote
1answer
21 views

Hurwitz Zeta in terms of Bernoulli polynomials.

@Raymond Manzoni showed nicely in this post how the Riemann zeta function is related to the Bernoulli numbers using the Euler-Maclaurin sum. The result is : \begin{eqnarray} \zeta(1-k) = ...
0
votes
1answer
19 views

On a bound about $\sum_{n\leq n}\sqrt{\frac{x}{n}} \left[\sqrt{\frac{n}{x}} M \left(\frac{x}{n} \right) \right] $

From the fact that $f(x)= \left[f( x) \right]+ \left\{ f(x) \right\} $, where $ \left\{ x \right\} $ is the fractional part function, one can write by a direct substitution for the function ...
0
votes
0answers
6 views

Factoring the conductor of a Dirichlet character and factoring generalized Gauss sums

Let $m=m_1m_2\cdots m_r$ with $m_i$ positive integers with $\gcd(m_i,m_j)=1$ for $i\neq j$. Given a Dirichlet character $\chi$ modulo $m$ we can define the characters $\chi_i$ (modulo $m_i$) by ...
0
votes
0answers
16 views

On the Laplace transform $\int_0^\infty e^{-sx}d \left( \ \int_2^{e^{1+x}}\frac{dt}{\log t}\right) $

I've read the basics about Laplace transform, and I know that since for $\Re s>1$, $\frac{e^x}{1+x}$ has exponential order, then $$F(s)=\int_0^\infty e^{-sx}\frac{e^{1+x}}{1+x}dx$$ is well defined, ...
0
votes
0answers
9 views

How to prove the Mellin's inverse formula?

Let $f:[0,+\infty)\to \mathbf{C}$ be a continuous, compactly supported function. Establish the Mellin's inverse formula $$f(u)=\frac{1}{2\pi i}\lim_{T\to \infty}\int_{c-iT}^{c+iT} ...
0
votes
0answers
18 views

What is support and spectrum of this nonnegative trigonometric function (or Finite Fourier Sum )?

This is a follow up of another question. The zeros of the following cosine sum shows the prime distribution, and the gap between the zeros can help to study the gap between prime numbers. $$ ...
0
votes
0answers
19 views

Average order of divisor function ; Theorem in Apostol

In Introduction to Analytic number theory by Apostol, a theorem states that: For all x $\geq$ 1, we have $$\sum_{n\le x} \sigma(n)= \frac{1}{2} \zeta(2)x^2 + O(x\log x)$$ The definition of O(g(x)) ...
1
vote
1answer
31 views

Hecke $L$-function of cusp form is entire

Let $f=\sum a(n)q^n\in S_k(N,\chi)$ be a cusp form of integral weight. Can someone give me the proof of the fact that : the Hecke $L$-function $L(f,s)=\sum\frac{a(n)}{n^s}$ is entire. I searched in ...
5
votes
1answer
103 views

Prime Zeta Function proof help: Why are these expressions not equal?

I was trying to create a formula for the Prime Zeta function and I partially succeeded except for one frustrating error. I was only able to formulate an approximation. Consider the following sum: ...
2
votes
1answer
91 views

Derive zeta values of even integers from the Euler-Maclaurin formula.

Euler showed: \begin{equation} B_{2 k} = (-1)^{k+1} \frac{2 \, (2 \, k)!}{ (2 \, \pi)^{2 k}} \zeta(2 k) \end{equation} for $k=1,2, \cdots$. We could from here find $\zeta(2k)$ in terms of the ...
2
votes
2answers
80 views

Definition of Dirac Delta function on the surface of a unit sphere

I am looking for a definition of a Dirac Delta function which is defined on the 2D unit sphere surface in 3D. In other words, I am looking for a function which is zero everywhere on the 2D spherical ...
6
votes
1answer
133 views

Why can this cosine sum function show all primes less than $N^2$?

I constructed this cosine sum that puts all primes within N on line y=1, and its zeros show the sieve by primes less than N. For $x<N^2$, they are all primes. $$ ...
1
vote
2answers
821 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 ...
4
votes
3answers
44 views

$\sum_{p} \chi(p)/p$ is conditionally convergent for non-principal character

Let $\chi$ is a non-principal character. Show that the sum $\sum_{p}\frac{\chi(p)}{p}$ is conditionally convergent. Then show that the product $\prod_{p}(1-\frac{\chi(p)}{p})^{-1}$ is conditionally ...
1
vote
1answer
42 views

Difference between Eulers product and Zeta Function at a finite values

So a very important formula proven by Euler is that is equal to Of course these formulas give you the same value when they reach infinity, but my question is that say $s=1$. What would be the ...
4
votes
0answers
78 views

Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
2
votes
1answer
84 views

Prove that $\prod\limits_{2 < p \leq y}\left(1-\frac{2}{p}\right)\sim\frac{D}{\log ^2 y}$ [duplicate]

I'm writing my bachelor thesis about Brun's sieve method and his theorem. In one proof I found this statement without further explanation. It is important to show that the product doesn't converge ...
1
vote
0answers
12 views

Diamond-Steinig identity

I'm self-studying analytic number theory in tao's blog. There is an exercise that I can't solve. Let ${k \geq 1}$. Show that ${\Lambda_{2k} + \Lambda_k * \Lambda_k}$ can be expressed as a linear ...
0
votes
0answers
18 views

What is right hand side limit of Dirichlet eta function of -1?

What is right hand side limit of Dirichlet eta function of -1 ? Left hand side ia 1/4, right hand side doesn't exist as I understand, motivation : ...
0
votes
0answers
13 views

Can you get the average order of $ \left( 1+|\mu(n)| \right)^{M(n)} $, where $\mu(n)$ and $M(n)$ are the Möbius and Mertens functions, respectively

When yesterday I was interested in do a little study about the arithmetic function $$f(n)=\left( 1+|\mu(n)| \right)^{M(n)},$$ defined for integers $n\geq 1$, which $\mu(n)$ is the Möbius function and ...
0
votes
0answers
20 views

On a closed-form from particular values of the Riemann zeta function and divisor functions

I am looking if I can get a closed-form for an infinite series, but I don't know for what it is possible, without finish my computations (see my Question, below). From Applications (8.1 Infinite ...
0
votes
0answers
23 views

On the zeroes of sine complex function, and a search for a special sequence, following Riemann's approach

If there are no mistakes from the Fourier expansion series for the fractional part function we can write, using a substituion, that for $1<x<e^2$ with uniform convergence $$\frac{1}{2}\log ...
2
votes
0answers
90 views

Estimating a contour integral which includes the riemann zeta function

I would like to understand the following paper, it is about the Erdös-Kac theorem http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa417.pdf (Site 75). My problem is to estimate $$I_2 := \dfrac{1}{2 \pi i } ...
12
votes
2answers
1k views

Is there an explicit irrational number which is not known to be either algebraic or transcendental?

There are many numbers which are not able to be classified as being rational, algebraic irrational, or transcendental. Is there an explicit number which is known to be irrational but not known to be ...
1
vote
0answers
42 views

What's about $\sum_{n=1}^\infty e^{-p_n u}$, where $p_n$ is the nth-prime number?

I am assuming that the following function, for which I am asking as reference request, should be known in the literature, since Glaisher studied the Prime Zeta Function, and my computation is the ...
1
vote
0answers
52 views

Convergence of the Euler product

Suppose that the Riemann Hypothesis is true. It is well known that then the Dirichlet series $$\sum_{n=1}^\infty\frac{\mu(n)}{n^s}$$ converges in the half-plane ${\rm {Re}}\, s>\frac{1}{2}$. Does ...
0
votes
0answers
25 views

I am looking a comparison of this computation and Riemann's approach for $lcm(1,2\ldots,x)$

Looking a comparison with a reasoning due to Riemann, I ask to me about the behaviour as $x\to\infty$ of the following arithmetical function $$ \left( \prod_{n\leq x}n^{-\mu(n)}\right)\cdot \left( ...
2
votes
0answers
32 views

Does Wilson’s Theorem characterize the gamma function?

Wilson’s Theorem can be stated as follows: n is a prime if, and only if, n is an integer > 1 such that n divides (n - 1)! + 1. However, Γ(n) = (n - 1)! (for any positive integer n) Therefore, ...
2
votes
1answer
21 views

Show that a periodic, completely multiplicative arithmetic function is a Dirichlet character to some module $q$

Show that if $f$ is a periodic, completely multiplicative arithmetic function, then $f$ is a Dirichlet character to some modulus $q$. A Dirichlet character modulo $q$ is an arithmetic function ...
1
vote
0answers
31 views

Products of $k^{\mu(k)}$, where $\mu(n)$ is Möbius function, and the Prime Number Theorem

We can write $$e^{-\Lambda(n)}=\prod_{d\mid n}d^{\mu(d)},$$ where $\mu(n)$ is the Möbius function and thus $\Lambda(n)$ is von Mangoldt's function. Then taking the product from $1$ to $N$ we've for ...
2
votes
1answer
23 views

A change of variables in Riemann's proof of the functional equation of $\zeta(s)$

In Riemann's functional equation proof it says: From $$\Gamma(s)=2\int_0^\infty e^{-x^{2}}x^{2s-1}dx,$$ after variable substitution, we get $$\Gamma\left(\frac{s}2\right)=n^{s} ...
3
votes
2answers
88 views

Prove that the value of the constant $C$ must be $1$

After proving the prime number theorem in class, our professor directs us to a remark by Lagrange that for large values of $x$, $\pi(x)$ is approximately equal to $$ \frac{x}{\log x - B}. $$ (This is ...
4
votes
1answer
44 views

Large gap between two consecutive square-free numbers

Let $q_n$ denote the $n$-th square-free number. By Chinese remainder theorem (see this post), it is not difficult to show that there is arbitrarily large gap between two consecutive square-free ...
3
votes
1answer
120 views

Estimate of the derivative

Show that if $f(x)=x^2+O(x)$, and $f$ is differentiable with non-decreasing derivative $f'(x)$, then $f'(x)=2x+O(\sqrt{x})$. I know that if $f'$ is not non-decreasing, then the statement is not true. ...
3
votes
1answer
69 views

Sum of products of $(1 - 1/p)$

Let $\pi(n)$ denote the number of primes not greater than $n$, and $p_k$ the $k$th prime, so that $p_{\pi(n)}$ denotes the largest prime not greater than $n$. I'm interested in the value of the ...
1
vote
1answer
43 views

Wintner's mean value theorem

This is an exercise (exercise 2.22 p80) from A.J. Hildebrand's Introduction to analytic number theory (an online lecture notes). Let $g$ be an arithmetic function, and let $f=1*g$ ...
4
votes
1answer
78 views

Euler Product formula for Riemann zeta function proof

In class we introduced Reimann Zeta function $$ \zeta (x)=\sum_{n=1}^{+\infty} \frac{1}{n^x} $$ And we proved its domain was $D=(1,+\infty)$ Now Euler proved that $$ \zeta(x)=\prod_{p\text{ ...
0
votes
0answers
20 views

Relation between Meissel–Mertens constant and Euler–Mascheroni constant

From the Wikipedia page, the Meissel–Mertens constant $M$ is defined as the limit: $$M:=\lim_{n\to\infty}\left(\sum_{p\leq n}\frac{1}{p}-\log\log n\right).$$ Why is it equal to ...
2
votes
1answer
37 views

Asymptotic estimate for the sum $\sum_{n\leq x} 2^{\omega(n)}$

How to find an estimate for the sum $\sum_{n\leq x} 2^{\omega(n)}$, where $\omega(n)$ is the number of distinct prime factors of $n$. Since $2^{\omega(n)}$ is multiplicative, computing its value at ...
5
votes
2answers
87 views

Proving that $\pi(2x) < 2 \pi(x) $

In our analytic number theory class we were given the following problem as homework: prove rigorously that for large $x$ the number of primes in $(1,x]$ exceeds that in $(x,2x]$. In class we proved ...
1
vote
1answer
65 views

Analytic Number Theory: Problem in Bertrand’s postulate

I am trying to learn Bertrand’s postulate. I can not understand two steps Why $\displaystyle\sum_{n \leq x}\log n=\sum_{e \leq x} \psi\left(\frac{x}{e}\right)$, where ...
1
vote
1answer
45 views

Asymptotic estimate of the sum $\sum_{n\leq x}1/\phi^2(n)$

How to show that we have the following estimate: $$\sum_{n\leq x}\frac{1}{\phi^2(n)}=c+O(\frac{1}{x}),$$ where $\phi$ is the Euler's totient function and $c$ is a constant. I tried to use the ...
19
votes
0answers
525 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = ...
2
votes
1answer
36 views

Number of subsets $S$ of $[n]$ such that $\gcd(S)$ is coprime to $m$

Fix positive integers $m,n$. Is there a way to count the number of non-empty subsets $S$ of $[n] = \{1, \ldots, n\}$ such that $\gcd(S)$ is coprime to $m$? Can we come up with an expression for such a ...
10
votes
5answers
1k views

Why does zeta have infinitely many zeros in the critical strip?

I want a simple proof that $\zeta$ has infinitely many zeros in the critical strip. The function $$\xi(s) = \frac{1}{2} s (s-1) \pi^{\tfrac{s}{2}} \Gamma(\tfrac{s}{2})\zeta(s)$$ has exactly the ...
10
votes
1answer
109 views

Equality involving Hasse zeta function of commutative ring finitely generated over $\mathbb{Z}$

Let $\mathbb{F}_q$ be a finite field consisting of $q$ elements. Imitating Riemann's zeta function$$\zeta(s) = \sum_{n = 1}^\infty {1\over{n^s}},$$define$$\zeta_{\mathbb{F}_q[t]}(s) = \sum_f ...
1
vote
1answer
36 views

A multiplicative function satisfying $\lim_{p^m\to\infty} f(p^m)=0$ implies $\lim_{n\to\infty} f(n)=0$

Let $f$ be a multiplicative function satisfying $\lim_{p^m\to \infty} f(p^m)=0$. Show that $\lim_{n\to\infty} f(n)=0$. By unique factorization, we can write $n=\prod_{i=1}^k p_i^{\alpha_i}$, where ...
4
votes
2answers
223 views

Divide a square into different parts

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with geometry, which perhaps yields the shortest, simplest proofs, but other ...