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

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0
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0answers
19 views

Polynomial/ Exponential diophantine equation

I am looking for the reference characterizing all the cases when $$an^2+bn+c=2^m$$ has infinitely many positive integer solutions (m,n). Thanks.
1
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1answer
36 views

Formula for $\sum_{d|n} \frac {\mu(d)}d$

I feel like I've seen a formula somewhere for $\displaystyle \sum_{d|n} \frac {\mu(d)}d$, but I can't remember what it is and can't find it. Does anybody know of a formula?
2
votes
1answer
20 views

Asymptotic formula for sums related to primes

Suppose $0 < \alpha < 1$. What is the asymptotic formula for the sum $$\displaystyle \sum_{p \leq x} \frac{\log p}{p^\alpha}?$$ Thanks for any insights.
1
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0answers
25 views

Asymptotic expression for $3$ term arithmetic progression in the primes

I have found an asymptotic for the following sum using the circle method: \begin{align} R(n)=\sum_{\substack{p_1,p_2,p_3 \le n \\p_1+p_2=2p_3 }} \log (p_1) \log (p_2) \log ...
2
votes
0answers
23 views

Asymptotics of $\sum_{\mathfrak{a}}\frac{n^{k-\epsilon}}{\mathfrak{N}\left(\mathfrak{a}\right)^{r\left(k-\epsilon\right)}}$

In this paper by Brian D. Sittinger, the following claim is made: For an algebraic number field $K$ with norm $\mathfrak{N}$, let $\epsilon=\left[K:\mathbb{Q}\right]^{-1}$. Then, taking the sum over ...
1
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0answers
36 views

Do you know any answer for equation y^2 = x^3 + k? [duplicate]

As you know, the equation y^2 = x^3 + k for k like (4n-1)^3 - 4m^2 that m , n are integers & no prime number that p is congruent to 1 modulo 4 count m, don't have any answer & it's proof is by ...
2
votes
1answer
37 views

Asymptotics of $\sum_{n\leq x}\tau_{k}\left(n\right)$

We define $\tau_{k}\left(n\right)$ to be the number of ordered $k$-tuples of positive integers with product equal to $n$. It is easily shown that this satisfies the recurrence relation ...
2
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0answers
51 views

Any formula for the exact number of primes below a given bound?

Reading The music of the primes, the author relates that Riemann had figured out a formula giving exact number of primes up to a certain bound with no errors. Does such formula really exist? If ...
1
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0answers
38 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 ...
3
votes
1answer
52 views

Moving the integral $Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds$ past Re(s) = 1.

Given the integral $$Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds,$$ I know that the integrand is holomorphic except for simple poles at ...
3
votes
1answer
70 views

What is the inverse of the divisor sum function $\sigma $?

Let $(A, +, *)$ be the commutative ring of arithmetic functions with Dirichlet convolution as the multiplicative operation *. The element $$\sigma(n)=\prod_i \frac{p_i^{k_i+1}-1}{p_i-1}, \text { ...
1
vote
1answer
58 views

On the proof of Fejér-Riesz theorem

I'm having a course about Analytic Number Theory, and I'm having trouble understanding the proof of Fejér-Riesz Theorem: http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf First of all, I didn't ...
1
vote
1answer
156 views
+50

How to calculate this sum like Gauss sum.

I would like to calculate the following sum, which looks like a Gauss sum. Let $n$ be a natural number and let $a,b$ be integers. Denote by $e(x)=e^{2\pi i x/n}$. Consider the sum $$ \sum_{1 \leq j, ...
0
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0answers
38 views

What is $B$ in $\varsigma(s)=e^{A+Bs} \prod_\rho \left(1-\dfrac{s}{\rho}\right) e^{s/\rho}$

$$\varsigma(s)=e^{A+Bs} \prod_\rho \left(1-\dfrac{s}{\rho}\right) e^{s/\rho}$$ (I have proved that $\rho$ is nontrivial zero of the Riemann zeta function in first part of question.) Show that ...
9
votes
3answers
76 views

Is there a monotonic $f$ such that $\sum f(n)$ diverges but $\sum f(p)$ converges?

(where the former summation is over natural numbers $n$ and the latter is over prime numbers $p$, and $f: \mathbb{N} \to \mathbb{R}$ is a monotonic function.) For the class of functions $f_s(n) = ...
3
votes
0answers
108 views

What is the value of $\sum_{p\le x} 1/p^2$?

My question is, what is the value of $$\sum_{p\le x} \frac{1}{p^2}?$$ More generally, what is the value of $$\sum_{p\le x} \frac{1}{p^n}?$$ How can we find it? For $\sum_{p\le x} 1/p$ the idea was ...
11
votes
1answer
151 views

Is $\sum\frac1{p^{1+ 1/p}}$ divergent?

Is $\displaystyle\sum\frac1{p^{1+ 1/p}}$ divergent? How can we prove that it is divergent or convergent in analytic number theory? I know what bound of the n-th prime number is, and that its order is ...
3
votes
1answer
60 views

Hardy-Ramanujan theorem's “purely elementary reasoning”

I'm reading through The normal number of prime factors of a number $n$. I'm confused by a remark on the second page: let $f(n)$ represent the number of distinct prime factors of $n$. Then we can ...
2
votes
1answer
34 views

How can one show that $\prod_{n<p\leq2n}p\leq C(2n,n)$?

I am trying to rove that $\prod_{n<p\leq2n}p \leq C(2n,n) \leq 2^{2n}$, where $C(2n,n)= \frac{2n!}{n! n!}$ and $p$ is prime. I can prove the second part by induction, but first part induction ...
1
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0answers
31 views

Is the integral $\int_1^{\infty} {A(t)}{t^{-s-1}}dt$ a holomorphic function of $s$?

My question is whether, for $Re(s)=\sigma > 3/4$, $$s\int_1^{\infty} \dfrac{A(t)}{t^{s+1}}dt$$ is holomorphic, where $A(x)=O(x^{3/4})$. Under absolute value, it is easy to see that the integral ...
1
vote
1answer
37 views

Number of the positive integers up to $\sqrt x $ generated by the prime factors of $x$.

Let $x$ be a natural number and $F_x$ be the set of distinct prime factors of $x$. One more let $\langle F_x \cup \{ 1 \} \rangle$ be multiplicative semigroup generated by the set. Then, the problem ...
2
votes
0answers
46 views

Find the integral values for which $\left(\pi(x+y)\right)^2=4\pi(x)\pi(y)$

Let $\pi(x)$ be the prime counting function. Find all integral values of $x,y$ such that, $$\left(\pi(x+y)\right)^2=4\pi(x)\pi(y)$$ I have no idea as to where to begin with. I think that probably ...
1
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0answers
26 views

Non-trivial odd characters mod m

I am stuck with this problem of marcus: I proved it when the charater is even. But I cannot prove the given formula when the character is odd. Please help.
1
vote
1answer
43 views

Can we define Mobius function for any real number and any complex number ?

All: To me, Mobius function is a bit mysterious. I just want to know if we can define Mobius function for any real number or any complex number ? Can anyone point out any resource on this ? Thank ...
3
votes
1answer
81 views

Absolutely convergent function

I am trying to show that if $\displaystyle\sum_{n\le x}f(n)=Cx+O(x^{3/4})$, where $f$ is non-negative multiplicative function and $C$ is a positive constant, then ...
0
votes
0answers
37 views

Using Perron's formula for asymptotic behaviors

I happen to read this post about trying to get the formula of $\sum_{n=1}^N n^m$ for Perron's formula. The general Perron's formula is $$\sum'_{n\le x} a(n)=\frac{1}{2\pi i}\int_{\text{Re ...
4
votes
1answer
63 views

Riemann Hypothesis, is this statement equivalent to Mertens function statement?

All: I saw one form of Riemann Hypothesis, it says: $$ \lim ∑(μ(n))/n^σ $$ Converges for all σ > ½ Is this statement same as the order of Mertens function is less than square root of n ?
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0answers
15 views

Estimates for a Mertens-type Product.

The first corollary of Theorem 8 of this paper by Rosser and Schoenfeld states that $$\prod_{p\leq x}\left(\frac{p}{p-1}\right)<e^{\gamma}(\log x)\left(1+\frac{1}{\log^2 x}\right)$$ for all $x\geq ...
1
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0answers
56 views

Riemann's explicit formula for $\pi(x)$

Riemann's explicit formula $J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2,$ where ...
0
votes
1answer
41 views

existance of decimal expansion

Can every real number be written in decimal expansion? I mean, can every real number $a$ be expressed as follows: $$\text{For }\, a \in \mathbb {R}^{+},\quad ...
1
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0answers
42 views

On the prime number theorem in shorts intervals

In 1988 Heath-Brown (" The number of primes in a short interval ", J. reine angew. Math. 389, 22-63) proved this theorem: Let $\varepsilon\left(x\right)\leq\frac{1}{12}$ be a non-negative function ...
3
votes
1answer
91 views

Cramer and Riemann Conjecture Implication

Cramer's conjecture gives $$p_{n+1}-p_n= O(\log^2 p_n)$$ while Riemann Hypothesis yields just $$p_{n+1}-p_n= O(\sqrt p_n\log^2 p_n).$$ Does Cramer conjecture on prime gaps imply Riemann Hypothesis ...
2
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0answers
21 views

parity problems for sieve methods, is it only for Selberg Sieve or for all sieve methods?

It is said that sieve methods have parity problems. Terence Tao gave this "rough" statement of the problem: "Parity problem. If A is a set whose elements are all products of an odd number of primes ...
3
votes
0answers
76 views

What will be the consequences if second Hardy-Littlewood conjecture turns out to be true? [migrated]

It is generally believed that the Second Hardy-Littlewood Conjecture is false. But it has not been proved (or disproved) yet. My question is, What would be the consequences if Second ...
1
vote
1answer
68 views

How to compute $\lim_{s \to 1} (s-1) \frac{\zeta'(s)}{\zeta(s)} $ ?

I wish to verify the conditions of a certain theorem to prove that the integral $$\int_{1}^{\infty} \frac{\psi(x) - x}{x^2} dx $$ converges. (Where $\psi(x) = \sum_{n\leq x} \Lambda (n) $, and ...
9
votes
2answers
230 views

What are some equivalent statements of (strong) Goldbach Conjecture?

What are some equivalent statements of (strong) Goldbach Conjecture ? We all know that Riemann Hypothesis has some interesting equivalent statements. My favorites are involved with Mertens ...
2
votes
2answers
164 views

Riemann Hypothesis and Prime Count

Let $\pi(a)$ be the number of primes below $a>0$. The prime number theorem states $\pi(a)\sim\frac{a}{\ln a}$. My question is trivial. Is $$\frac{a}{\ln a}\leq\pi(a)\leq\frac{a}{\ln a}+c\sqrt{a}\ln ...
2
votes
2answers
72 views

Chebyshev's first function prime count

How is Chebyshev's first function $$\vartheta(N)=\sum_{p\leq N}\log p$$ useful in counting primes? Can it alone be used to analytically derive the prime number theorem?
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votes
2answers
130 views

Importance of the zero free region of Riemann zeta function

I have heard that for improving the error term in the Prime Number Theorem, we need better and better estimates on the zero free region. I have also heard that the best possible error term comes from ...
1
vote
0answers
42 views

Summing the reciprocal of $\phi(n)^2$

I am currently reading Vaughan's book 'The Hardy-Littlewood method' and am working on exercise 3.3 (page 37). I have tried working through a special case but that didn't help either. More concretely, ...
2
votes
1answer
20 views

How to find all Dirichlet characters

I want to know all the Dirichlet characters modulo $m$. I know that the number of such characters are $\phi(m)$. But how do find each and every character. for small moduli I could do it using some ...
2
votes
3answers
69 views

Arithmetic functions of particular type

Any there any natural functions real valued single variable that: changes (increases) values only at primes but otherwise stay constant (like a non periodic increasing staircase)? whose increase in ...
2
votes
0answers
64 views

Prove that $\sum\limits_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$. [closed]

Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that $$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$ ...
0
votes
1answer
45 views

Looking for methods for approximating an iterative equation regarding primes

In a previous question, I was looking for an equation for counting the number of the number of integers between $1$ and $x$ that have a prime factor besides $2$ or $3$. There were 2 iterative ...
5
votes
0answers
43 views

Kloosterman sum and multiples of 16

A Kloosterman sum is defined as $$K(a,b;m)=\sum_{0\leq x \leq m-1}_{\gcd(x,m)=1} e^{2\pi \mathcal{i} (ax+bx^*)/m}$$ where $a,b,m \in \mathbb{N}$ and $x^*$ is the inverse of $x$ modulo $m$. How can ...
1
vote
0answers
23 views

GCD of Arguments of Kloosterman Sum

A Kloosterman sum is defined as $$K(a,b;m)=\sum_{0\leq x \leq m-1}_{\gcd(x,m)=1} e^{2\pi \mathcal{i} (ax+bx^*)/m}$$ where $a,b,m \in \mathbb{N}$ and $x^*$ is the inverse of $x$ modulo $m$. It ...
2
votes
0answers
37 views

Riemann Zeta Function and Laurent Expansion

In the wikipedia page "1+2+3+4+..." http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF (and specifically in the section "Zeta Function Regularization")it is stated without reference ...
1
vote
1answer
141 views

Abscissa of convergence for a Dirichlet series

Let $\alpha \in \mathbb{Z}$ and $f(n) = n^{i \alpha n}$. What is the abscissa of convergence, $\sigma_c$, for the associated Dirichlet series, $\sum_{n=1}^{\infty} \frac{f(n)}{n^s}$? Since $|f(n)| = ...
0
votes
2answers
88 views

Prove that as $x\to\infty $, $\sum\limits_{p \leq x} \frac{1}{p \log \log p} \approx \log \log \log x$

Prove that as $x\to\infty$, $$\sum_{p \leq x} \frac{1}{p \log \log p} \approx \log \log \log x$$ Here sum is taken over primes.I tried to use the partial summation formula but could not ...
0
votes
1answer
46 views

Convergence of a certain series of Primes

This is a problem from Alan Baker's Comprehensive Course in Number Theory. We have to show that $\displaystyle \sum\limits_{p} \frac{1}{p (\log\log p)^{\delta}}$ converges for all $\delta >1$.Here ...