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
56 views

Finite solution of Power Diophantione Equation.

Given an equation $x^2+k=y^3$ where k is a constant and $y=f(x)$,$f(x)$ is differentiable and algebraic. for which- $$\frac{d}{dx}x^{2} \neq\frac{d}{dx} f(x)^3$$ 1. Can I infer that the ...
-3
votes
1answer
41 views

what is the legendre symbol of (18/43) [on hold]

I know how to find legendre symbol and what a legendre symbol is but I am confused in using it for big numbers like 43 can any one help me and give a step by step answer.
0
votes
0answers
16 views

Which theta function is $\theta(x;q) = (x;q)(q/x;q)$?

The physics paper I am reading very non-chalantly defines the theta function as $$ \theta(x;q) = (x;q)(q/x;q) \hspace{0.5in} \tilde{\theta}(x;q) = x^{-1/2}(x;q)(q/x;q) $$ where they are using the ...
2
votes
0answers
29 views

Convergence of series involving Euler's totient function.

I have to show that if $\phi$ is Euler's totient function, then the series $\sum\limits_{n=2}^{\infty} \frac{1}{\phi (n) \log n}$ diverges and $\sum\limits_{n=2}^{\infty} \frac{1}{\phi(n) \log^2 n}$ ...
1
vote
0answers
101 views

Any results for small number Goldbach conjecture research?

It seems to me that most research results on Goldbach conjecture research are for large number. (Example: results of Vinogradov, Terence Tao, Harald Helfgott, etc). My understanding is that those ...
4
votes
0answers
60 views

$d$ and $d+1$ both dividing certain integers

\begin{align} & 1\cdot 72 \\ & 2\cdot 36 \\ & 3\cdot 24 \\ & 4\cdot 18 \\ & 6\cdot 12 \\ & 8\cdot 9 \end{align} When the divisors of a number are listed in this way, let us ...
8
votes
1answer
214 views

When does the “Zetor function” converge?

Let $p_n$ be the n'th non-trivial zero of the Riemann zeta function. We define the Zetor function (acronym of 'zeta' and 'zero') as follows: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}. ...
7
votes
2answers
159 views

Zeta functions in Chebychev's Prime Number theory

In two papers from 1848 and 1850, the Russian mathematician Pafnuty L'vovich Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the ...
2
votes
0answers
169 views

An argument for “Brocard's problem has finite solution”

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem ...
7
votes
1answer
94 views

Is there any relationship between the Riemann z function and strange attractors?

I have this question in mind since the first time I saw a graphical representation of the zeta function (like in the sample below). Just by looking to them I wondered if there is any relationship ...
-2
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0answers
30 views
4
votes
2answers
44 views

A Mertens-like product over primes

MathWorld's page Prime Products gives the 'related result' (7) to Mertens' theorem: $$ \lim_{n\to\infty}\log p_n\prod_{k=1}^n\frac{1}{1+1/p_k}=\frac{\pi^2}{6e^\gamma}. $$ Does this identity have a ...
5
votes
1answer
49 views

Remainders of quadratic trinomial

The problem is to determine, whether there exist a quadratic trinomial $f(x) = ax^2 + bx +c$ with integer coefficients (with $a$ not a multiple of 2014), such that the numbers $\ f(1), \ f(2),\, ...
2
votes
0answers
32 views

On Zero-Free Regions for $\zeta(s)$ and $L(s,\chi)$ with $|t| \le 2$

I'm reading the proof from Hildebrand that for some $c_1 > 0$, the Riemann zeta function $\zeta(s)$ has no zero in the region $\sigma > 1-c_1$, $|t| \le 2$. (Here $s = \sigma + it$ per ...
0
votes
0answers
24 views

Difference of the $2$ sums is $O(x\log(x))$

If $g(x)$ is real-valued on $\{x\in\mathbb R:x\ge1\}$ and satisfies the condition $|g(x)|\le Cx$, with a constant $C$ for all $x\ge1$ then show that; $$\sum\limits_{n\in\mathbb N\atop{n\le ...
4
votes
1answer
56 views

About Mertens' first theorem

Mertens first theorem states that $ \sum_{ p \le x } \frac{\log p}{p} = \log x + R $ with $| R | \le 2$ . Is it correct that the limit $ \lim_{x \to \infty} \sum_{ p \le x } \frac{\log p}{p} - \log x ...
3
votes
1answer
46 views

Why is $\sum\left(\left\lfloor\frac{x}{p}\right\rfloor+\left\lfloor\frac{x}{p^2}\right\rfloor+\dots\right)\log p=\sum\frac{x}{p}\log p+O(x)$?

Why is $\sum\limits_{\substack{p:\text{prime}\\p\le x\\}}\left(\left\lfloor\frac{x}{p}\right\rfloor+\left\lfloor\frac{x}{p^2}\right\rfloor+\dots\right)\log ...
3
votes
0answers
114 views

Conjecture concerning sums of reciprocals of largest prime factors

Let $x$ be an integer, $r(x)$ the reciprocal of the largest prime factor of $x$. Let $f(n) = \sum_{k=1}^{n-1} r(k) r(n-k)$ for which $k$ and $(n-k)$ are coprime. For $n = 3 \dots 10$, $f(n) = ...
2
votes
0answers
127 views

Entropy Rate of a sequence of Random Variables with Distributions related to Primes

Let us consider a stochastic process $\mathcal{X}=\{X_i\}_{i \in \mathbb{N} }$ where $X_i$'s are independent and $X_i$ is distributed as $$X_i=p_k \ \mbox{w. p.}\frac{p_k}{\sum_{l=1}^{i}p_l},\ 1\leq ...
1
vote
0answers
27 views

How to prove that $ \int_{2}^{x} \frac{dt}{(\log(t))^{k}} = O \Big{(} \frac{x}{(\log(x))^{k}} \Big{)} $ as $x \to \infty$? [duplicate]

For a homework exercise, we are asked to prove that $$ \int_{2}^{x} \frac{dt}{(\log(t))^{k}} = O \Big{(} \frac{x}{(\log(x))^{k}} \Big{)} \quad \text{, as } x \to \infty . $$ The following hint is ...
1
vote
1answer
31 views

Show that $\int_{-T}^T |\zeta(\frac{1}{2} + it)|^4 \, dt \sim T \log(T)^4 $

I have been reading about "mean value theorems in number theory" such as $$\int_{-T}^T |\zeta(\frac{1}{2} + it)|^4 \, dt \sim T \log(T)^4 $$ How to prove such a result? One source says it is ...
3
votes
1answer
29 views

Asymptotic for primitive sums of two squares

A positive integer $n$ can be written primitively as the sum of two squares, meaning $n = x^2 + y^2$ with $\gcd(x,y)=1,$ precisely when $n$ is not divisible by $4$ or by any prime $q \equiv 3 \pmod ...
2
votes
2answers
36 views

$\int_2^x\frac{dt}{\log^kt}=O\left(\frac{x}{\log^kx}\right)$

I seek to prove the identity $$\int_2^x\frac{dt}{\log^kt}=O\left(\frac{x}{\log^kx}\right)$$ I was given the following hint: Split the integral into $\int_2^{f(x)}+\int_{f(x)}^x$ for a ...
1
vote
0answers
72 views

Simple Zero of the Riemann Zeta Function

Let $s=σ+it$. Assume that $ζ(s)-1/(s-1)$ has an analytic continuation to the half plane $σ>0$. Show that if $s = 1 + it$, with $t≠0$, and $ζ(s) = 0$ then $s$ is at most a simple zero of $ζ$. I ...
1
vote
0answers
52 views
-1
votes
0answers
25 views

Application of Cauchy-Schwartz to an exponential sum involving von Mangoldt function

Let $f(x_1, ..., x_n)$ be a polynomial in $\mathbb{Z}[x_1, x_2, ..., x_n]$. Let $\Lambda$ denote the Von Mangoldt function. Suppose I have an exponential sum of the form $$ S(\alpha) = \sum_{1 \leq ...
0
votes
0answers
27 views

Show that $Q(x)-\frac{6x}{\pi^2}=\Omega_{\pm}(x^{1/4})$

Let $Q(x)$ denote the number of square-free numbers not exceeding $x$. Show that $$Q(x)-\frac{6x}{\pi^2}=\Omega_{\pm}(x^{1/4}).$$
1
vote
2answers
33 views

non analytic functions

Find two functions, each of which is nowhere analytic, but whose sum is an entire function. I can give examples of functions that are analytic nowhere, but can't find two that add to an entire ...
3
votes
1answer
69 views

Distribution of composite numbers

This question is moved from mathoverflow, there are several excellent answers at mathoverflow which improve my question greatly. For more information, please see the original question posted on ...
1
vote
1answer
25 views

Lower and upper bounds for $\tau(n)$

How to prove the following statement: If $n$ is the product of k powers of primes, i.e. $n=\prod\limits^{k}_{i=1}p_i^{\alpha_i}$ then $\omega (n) = k$ and $\Omega=\sum\limits_{i=1}^{k}\alpha_i$ $$ ...
3
votes
2answers
46 views

How to prove this asymptotic formula?

How to prove this asymptotic formula? $$ \prod\limits_{p\leq x}\left(1+\frac{1}{p}\right) \sim \frac{6 e^C}{\pi^2}\log x $$ Where we multiply over all primes less than or equal to x. I have little ...
12
votes
2answers
587 views

On prime factors of $n^2+1$

It is a well-known conjecture that there are infinitely many primes of the form $n^2+1$. However, there are weaker results that one can prove. For example, There are infinitely many positive ...
2
votes
1answer
110 views

Show that $1/\zeta(2k) = \sum_{m \le K} \mu (m)/m^{2k} + O(1/K)$

Show that $1/\zeta(2k) = \sum_{m \le K} \mu (m)/m^{2k} + O(1/K)$. I have already proved that $1/\zeta(s) = \sum_{m=1}^{\infty} \mu (m)/m^s$. But how do I show that if $k\ge 1$, $1/\zeta (2k) = ...
4
votes
2answers
49 views

Error term of a Tauberian theorem and lattice points in circles

Suppose $\{a_n\}$ is a sequence of non-negative real numbers, $a_n = O(n^M)$ for a positive number $M$ and it's Dirichlet series $L(s)=\sum \frac{a_n}{n^s}$ has an analytic continuation to a ...
1
vote
1answer
45 views

A question on the Lagrange Inversion Formula

I have to use the L.I.F. for \begin{align*} s\left(x,y\right)=\frac{1}{2}\left(1-x-y-\sqrt{1-2x-2y-2xy+x^2+y^2}\right) \end{align*} to obtain that \begin{align*} s\left(x,y\right) = ...
3
votes
1answer
47 views

$\sum_{n=0}^\infty z^n = \prod_{m=0}^\infty \left(1+z^{2^m}\right)$

When reading Iwaniec and Kowalski's Analytic Number Theory, I came across the following "identity" on page 11 (the Amazon link has a free book preview which includes page 11): $$\sum_{n=0}^\infty z^n ...
8
votes
2answers
170 views

Minimizing over partitions $f(\lambda) = \sum \limits_{i = 1}^N |\lambda_i|^4/(\sum \limits_{i = 1}^N |\lambda_i|^2)^2$

I'm trying to characterize the behavior of the the quantity: $$A = \frac{\sum \limits_{i = 1}^N x_i^4}{(\sum \limits_{i = 1}^N x_i^2)^2},$$ subject to the constraints that $$ \sum \limits_{i = 1}^N ...
5
votes
1answer
150 views

Understanding a very elementary property of factorials

I've seen this stated in a few places. If $$\vartheta(x) = \sum_{p\le{x}} \log (p) \qquad \psi(x) = \sum_{m=1}^{\infty}\vartheta\left(\sqrt[m]{x}\right)$$ Then $$\log(x!) = \sum_{m=1}^{\infty} ...
2
votes
1answer
76 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 ...
8
votes
3answers
340 views

Counting the Number of Integral Solutions to $x^2+dy^2 = n$

It is a well known result that the number of integer solutions $(x,y), x>0, y\ge 0$ to $x^2+y^2 = n$ is $\sum_{d|n}\chi(d)$, where $\chi$ is the nontrivial Dirichlet character modulo $4$ such that ...
2
votes
1answer
60 views

Jacobi Identities

Can anyone guide me how can I prove these two identities? a)$$\prod_{n=1}^{\infty}\frac{1-q^{2n}}{1-q^{2n-1}}=\sum^{\infty}_{n=1}q^{n(n+1)/2}$$ b) ...
1
vote
3answers
89 views

Evaluating an integral using Gamma function [closed]

For $r \in (0,2)$, I would like to evaluate the integral $$\frac{2}{r} \int_0^{\infty} \frac{\sin(u)}{u^r} du.$$ The answer should be $$\frac{\pi \cdot \mathrm{cosec}{\frac{r\pi}{2}} ...
3
votes
1answer
168 views

Zeros of alternating zeta function

Is there an intuitive explanation for why all the known non-trivial solutions of $$\frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots=0$$ have a real part of $\frac{1}{2}$?
31
votes
2answers
3k views

Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. The following are exluded: Books by mathematical ...
4
votes
0answers
33 views

Squarefree products of a class of primes

Numbers which are the sum of two squares are the product of a square and a collection of distinct primes which are 1 or 2 mod 4. Landau proved that there are $\sim kx/\sqrt{\log x}$ such numbers up ...
5
votes
3answers
72 views

Show that $\sum\limits_{p \leqslant x}1/p = \frac{\pi(x)}{x} + \int_2^x \frac{\pi(u)}{u^2} du.$

Show that $$\displaystyle\sum\limits_{p \leqslant x}1/p = \dfrac{\pi(x)}{x} + \int_2^x \dfrac{\pi(u)}{u^2} du.$$ In the equation above, $\pi(x)$ denotes the prime counting function. To get ...
0
votes
1answer
61 views

How to introduce an integer function into $\zeta$ function instead of $n$

I have a problem that I am struggling with since long and probably it is simple but I can not get through. So your help would be very welcome. Known that Riemann $\zeta$ function is defined as sum ...
1
vote
1answer
60 views

Show that $\limsup_{x \to \infty} \frac{\pi(x)}{x/ \log x} \geqslant 1. $

Show that $$\displaystyle\limsup_{x \to \infty} \dfrac{\pi(x)}{x/ \log x} \geqslant 1. $$ I've seen $\displaystyle\lim_{x \to \infty}$ operator, but I haven't seen $\displaystyle\limsup_{x \to ...
9
votes
3answers
272 views

For what $t$ does $\lim\limits_{n \to \infty} \frac{1}{n^t} \sum\limits_{k=1}^n \text{prime}(k)$ converge?

The average of all primes is $$\lim\limits_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \text{prime}(k) ,$$ which diverges. What is the smallest $r$ such that for $t>r$, $$\lim_{n \to \infty} ...
5
votes
1answer
208 views

Heuristic explanation for oscillatory behavior of first $n$ primes' multiples

Let $A$ be the set of all multiples of the first $n$ primes. The asymptotic density of $A$ should be given by $\mu=1-\prod_{i=1}^n(1-1/p_i)$. Letting $a_k$ be the $k$th element of $A$, the function ...