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

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10 views

How to prove an inequality in number theory without induction?

$\sum_{n=1,(n,m)=1 }^{km} \frac{1}{n} \leq (\gamma + \log(km) + \sum_{p|m}\frac{\log p}{p-1})\prod_{p|m}(1-\frac{1}{p})+\frac{2^{\pi(m)}}{km}$ where $\pi(m)$ is the number of distinct prime ...
0
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0answers
16 views

Hardy- Littlewood Circle Method

I'm currently trying to get to grips with the Hardy Littlewood circle method so I'm working through Vaughan's book. In the past I've been very bad for leaving a point behind if I don't follow it so ...
4
votes
1answer
59 views

Is a strong form of Goldbach conjecture equivalent of Generlized Riemann Hypothesis?

In Andrew Granville's paper: REFINEMENTS OF GOLDBACH’S CONJECTURE, AND THE GENERALIZED RIEMANN HYPOTHESIS He said that: "we show that if a strong form of Goldbach's conjecture is true then every ...
3
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0answers
22 views

Sum of integers and zêta functions

I am working on generalizing some works from the usual rational case to general number fields. That implies some technical changes I am not really at ease with. For instance: $$\sum_{m \leqslant X} m ...
1
vote
1answer
42 views

doesn't exist an $N$ s.t. all $n \ge N$ satisfy an equation.

I came across this problem on my own and i'm asking for any potential techniques/strategies/hints for attacking it. Prove that there does not exist an $N$ such that for every natural number $n ...
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0answers
20 views

$p$-adic Fourier transforms and orthogonality relations

In $\mathbb{C}$, we have the following orthogonality relation $$ \int_{0}^{1} e^{2\pi i (m-n)x} dx = \begin{cases} 1 & \mbox{ if } m = n;\\ 0 & \mbox{ otherwise.} \end{cases} $$ Do we have ...
2
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0answers
71 views

What are some practical attempts to disprove Riemann Hypothesis?

Most people believe Riemann Hypothesis is true. Since RH has not been proved yet, so it is not completely insane to disprove RH. Among the ways to disprove RH, straightforward ways, such as: try to ...
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0answers
24 views

Estimate of the log derivative of zeta function in the classic zero-free region

We know that the Riemann zeta function $\zeta$ has no zeros in the region $\{\beta+it:\beta>1-\frac{c}{\log(2+|t|)}\}$, where $c>0$ is an absolute constant. This is known as the classical ...
3
votes
1answer
200 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 ...
2
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1answer
85 views

How to find the bound of this sum?

Let $t>0,a(t)=\arg(\Gamma(1/4+it))$,$\kappa(n)=\frac{1}{2}x\pi n^2$,we need to calculate the bound,$A(x)$, of the following finite sum: $$ S(x)=\sum_{1\le n\le ...
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0answers
8 views

Factorization of Dirichlet characters

A Dirichlet character $\chi$ of modulus $q$ is called primitive if it cannot be factored as $\chi=\chi'\chi_0^{''}$, where $\chi_0^{''}$ is a principal character and $\chi'$ is a character of modulus ...
26
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4answers
4k views

Riemann zeta function at odd positive integers

Starting with the famous Basel problem, Euler evaluated the Riemann zeta function for all even positive integers and the result is a compact expression involving Bernoulli numbers. However, the ...
1
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2answers
53 views

What are the applications of Sigma Function?

I read about the Sigma Function today.It tells that- The $\sigma(n)$ is the sum of all the positive divisors of $n$. But I had no idea how they can be useful.What are the practical applications ...
1
vote
3answers
36 views

Why does Dirichlet Series of Mangoldt Function has simple pole of order 1 at s = 1

Could someone explain why $\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s} = -\frac{\zeta'(s)}{\zeta(s)}$ has a first order pole at $s=1$ with residue 1? That's what I found from Apostol's Introduction to ...
4
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1answer
58 views

My attempt to follow Tatuzawa and Iseki strategy to get a bound for $\int_2^x \frac{dt}{\log t}-\pi(x)$, where $\pi(x)$ is the prime counting function

I don't know if this exercise is in the literature, where $Li(x)=\int_2^x\frac{dt}{\log t}$ is the logarithmic integral and $\pi(x)$ is the prime counting function Question. Compute a good bound ...
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1answer
35 views

Finding an upper bound for a sum over primes

Fix $X>\geq 1$ a real number and let $1\leq y<X.$ Is there a positive constant $B$ such that $$\prod_{y<p\leq X} \left(1+\frac{3}{p}+ \sum_{\nu \geq 2} \frac{(\nu+1)^2}{p^{\nu}}\right)\leq ...
5
votes
2answers
89 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 ...
0
votes
1answer
13 views

Is Hurwitz contour integration the same as the Fourier series of the Hurwitz Zeta function?

Whittaker and Watson show a derivation of the Hurwitz representation of the Hurwitz Zeta function as a trigonometrical series. This represenation is achived by doing a countour integration. The ...
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0answers
41 views

Some inequalities with Mangoldt function

Let $\Lambda$ be the Mangoldt function defined by $$-\dfrac{\zeta'(s)}{\zeta(s)}=\sum_{n=1}^{+\infty}\frac{\Lambda(n)}{n^s}$$ then $$\Lambda(n)=\left\{% \begin{array}{cc} \log p & ...
2
votes
1answer
37 views

Is a modular form on $\text{SL}_2(\mathbb{Z})$ also a modular form on congruence subgroups?

Is a modular form $f$ of weight $k$ with respect to $\text{SL}_2(\mathbb{Z})$ always a modular form to a congruence subgroup $\Gamma$ (for example $\Gamma_1(N)$)? If the transformation law ...
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1answer
36 views

Understanding the proof of $\phi(n)=\sum_{d\mid n}\mu (d) \left(\frac{n}{d}\right)$.

A proof for the identity $\phi(n)=\sum_{d\mid n}\mu (d) \left(\frac{n}{d}\right)$ several times in this website. After studying the book Apostle's Analytic Number Theory and failed to understand I ...
8
votes
3answers
281 views

Explicit formula for floor(x)?

In number theory we have so-called explicit formula's in terms of the Riemann zeta zero's. For instance to count the sum of the logarithms of the primes below some given integer. (second Chebyshev ...
0
votes
3answers
380 views

Properties of Mertens Function

I am astounded by how little information about Mertens function M(n) (partial sums of the Möbius function) is on the Internet. Thus, I would be thankful if someone could clear up some of my confusion. ...
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0answers
9 views

What's the function that it is neccesary to show being bounded locally integrable in the Wiener-Ikehara Theorem?

When I am reading in (this video from an official channel in You$\color{red}{\text{Tube}})$ mathscienciechannel, that has the most high quality, in my attempt to understand the facts that currently I ...
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0answers
16 views

An understandable example of the Hadamard three-lines theorem

I would like to know Question. A detailed example, proving your statements and giving the specific function and the strip, of the Hadamard three-lines theorem. If it is possible explain how ...
0
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0answers
40 views

Characters with values on the $p$-adic complex field $\mathbb{C}_p$?

Characters $\psi : G \to \mathbb{C}$ from abelian groups $G$ to the complex field $\mathbb{C}$ are well-known and appear all over. Is there an analogue for the $p$-adic complex numbers $\mathbb{C}_p$, ...
-2
votes
1answer
25 views

Effective upper bound for a sum over prime numbers

Fix $y$ a positive real number. Is there an effective bound for the following sum i.e a positive constant B such that $$\sum_{p>y}\sum_{\nu \geq 4} \frac{1}{p^{9\nu/32}} \leq B.$$ Many thanks.
2
votes
1answer
44 views

Generalised Gauss sums

Let $\chi$ be a non-trivial Dirichlet character modulo an odd prime $p$ and let $f(x) \in \mathbb{Z}[x]$ be a polynomial. We define the generalised Gauss sum $$ G(\chi, f):=\sum_{y \in \mathbb{F}_p^*} ...
5
votes
0answers
189 views

Show that : $2, 5, 13, 17, 29, 421, 401, 53, 281,…,\rightarrow \infty$? $a_{n+1}=\operatorname{ GPF}(qa_n+p)$

I denote by $\operatorname{ GPF}(n)$ the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...
3
votes
0answers
85 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
20 views

On computations around $\sum_{n=1}^N\frac{n\Lambda(n)}{n+N}$, where $\Lambda(n)$ is von Mangoldt function

By specialization with $F(x)=\frac{1}{1+x}$ in Apostol's Theorem 4.17 (Apostol, Introduction to Analytic Number Theory (Springer)), for intergers $N\geq 1$ one has $$\frac{\log ...
16
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0answers
309 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
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0answers
12 views

Bound on smallest $n$ for consistency of a system of equations?

Given small $\epsilon>0$ how small should $n\in\Bbb N$ be such that if $a,b,c,d,q,r,u,v,x,y,m,m'\in\Bbb N$ with $gcd(a,b)=gcd(a,x)=gcd(b,y)=1$ the following relations can hold with constraints ...
5
votes
1answer
67 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 ...
2
votes
0answers
31 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 ...
0
votes
0answers
20 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 $p_i^{e_i}$) ...
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0answers
56 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 ...
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0answers
38 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 ...
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0answers
33 views
1
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1answer
23 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
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 ...
0
votes
0answers
12 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
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0answers
22 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. $$ ...
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0answers
24 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)) ...
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1answer
35 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
108 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
95 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
88 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 ...
7
votes
1answer
147 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
832 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 ...