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

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2
votes
0answers
41 views

derivative of riemann zeta function

I try to find a represantation for the derivative of the riemann zeta function. I do have for $\Re(s)>0$ and $s \neq 1$ $\zeta(s)=\dfrac{1}{s-1}+1-s\int_{1}^{\infty} \dfrac{x-\lfloor x \rfloor}{x^{...
5
votes
1answer
71 views

Average prime value in n factorial.

I was wondering about the (weighted) average prime value in the factorisation of $n!$. $\\$ If we call $f(n)$ the average prime value in $n!$, then $f$ seems to increase rather linear. Is there a ...
1
vote
0answers
70 views

Is this a new twin prime sieve method? Any information or comments is very appreciated.

I'm studying the twin prime numbers. Instead of sieving prime numbers, I found this method to sieve $\{x: x \neq \pm 1 \text{( mod $p$)}, x \in \mathbb{N}, p \le p_i\}$, so that $(x-1,x+1)$ will be ...
0
votes
0answers
25 views

Why is conductor of a Dirichlet character the product of conductors of other Dirichlet characters?

Let $n=\prod_{i=1}^np_i^{e_i}$ with $p_i$ different prime numbers and $e_i$ positive integers. Given a Dirichlet character $\chi$ modulo $n$ we can define the characters $\chi_i$ (modulo $p_i^{e_i}$) ...
3
votes
0answers
87 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, ...
1
vote
1answer
26 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) = -\frac{B_k}{...
0
votes
1answer
25 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 $M(x)=\...
0
votes
0answers
21 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} e^{su}\mathcal{L}f(z)...
1
vote
0answers
37 views

Can you share some information to help study this unified sieve function for prime, twin prime and Goldbach sums of $2n$?

Let $p_i$ be the $i^{th}$ prime number. For Goldbach sums of $2n$, let $p_i$ be the largest prime less than $\sqrt{2n}$, define $$ P(p_i,n,x)=\sum_{p\le{p_i}}\frac{c_p}{p}\left(1+2\sum_{k=1}^{p-1}(1-\...
0
votes
0answers
27 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
41 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 ...
0
votes
0answers
24 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. $$ P(p_i,x)=\...
2
votes
1answer
99 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 ...
4
votes
3answers
62 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 ...
17
votes
0answers
329 views

Simplify the sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes $\le N^2$

I was looking for a closed form but it seemed too difficult. Now I'm seeking help to simplify this sum. The 50 bounty points or more will be awarded for any meaningful simplification of this sum. I ...
1
vote
0answers
15 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
23 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 : https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%...
1
vote
0answers
54 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 $...
2
votes
1answer
95 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 "...
7
votes
1answer
156 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. $$ P(N,x)=\sum_{n=2}^{N}\frac{1}{n}\...
0
votes
0answers
24 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 x=\...
2
votes
0answers
93 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 } \...
1
vote
0answers
43 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
56 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
31 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
37 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, Wilson’...
1
vote
0answers
35 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
38 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 $\chi$...
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 ...
2
votes
1answer
25 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} \pi^{\frac{s}2}\int_0^\...
4
votes
1answer
52 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 ...
5
votes
2answers
101 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 ...
4
votes
1answer
96 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{ prime}...
0
votes
0answers
35 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 $\gamma+\sum_{p}\left(...
3
votes
1answer
229 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....
1
vote
1answer
52 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 $f=1*g$ (i.e.,$f(n)=\sum_{d\...
3
votes
1answer
52 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 ...
4
votes
2answers
96 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 ...
5
votes
2answers
97 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
48 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 ...
2
votes
1answer
40 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 ...
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 $...
1
vote
0answers
30 views

What's about $ \sum_{n=1}^{\infty} \frac{ \mu\left( \sigma (n)\right)}{n^3} ,$ where $\mu(n)$ is Möbius function and $\sigma(n)=\sum_{d\mid n}d$?

Let $ \mu (n)$ the Möbius function and $ \sigma (n)$ the sum of divisors function, then the arithmetical function $g(n)= \frac{ \mu\left( \sigma (n)\right)}{n^3} $ isn't multiplicative since $gcd(2,...
1
vote
1answer
29 views

On the Density of Deficient Odd Numbers and Abundant Integers

Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) < 2x$, then $x$ is said to be deficient, while if $\sigma(x) > 2x$, $x$ is said to be abundant. (Of course, when $\sigma(x) ...
2
votes
1answer
67 views

On the proof of “The infinite series $\sum_{n=1}^{\infty} p_n^{-1}$ diverges”.

The following text is from the book Introduction to Analytic Number Theory by T. M. Apostol : Theorem 1.13 $ \ $ The infinite series $\sum_{n=1}^\infty 1/p_n$ diverges. Proof. The following ...
0
votes
1answer
44 views

Estimates of $\Omega_{\text{av}}(n)$

Ramanujan proved that the average number of distinct divisors of $x$ for $x$ on $[1,n], ~\omega_{\text{av}}(n),$ and the average number of divisors including repetitions, $\Omega_{\text{av}}(n),$ are ...
2
votes
0answers
58 views

Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...
6
votes
1answer
71 views

Combinations of four consecutive primes in the form $10n+1,10n+3,10n+7,10n+9$

Here $n$ is some natural number. For example, among the primes $< 1000$ I found four such combinations: \begin{array}( 11 & 13 & 17 & 19 \\ 101 & 103 & 107 & 109 \\ 191 &...
0
votes
0answers
49 views

Which values of $n$ is this inequality related to prime numbers true for?

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the $i$-...
0
votes
1answer
125 views

Does the Riemann Hypothesis for finite fields imply the original RH?

Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ where $p$ is a prime. The zeta function, $\zeta(E, s)$ for $E$ is defined as $\zeta(E,s) = \dfrac{(1-\alpha p^{-s})(1-\beta p^{-s})}{(...