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

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0
votes
0answers
12 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
17 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 $\sum\limits_{n=2}^{\infty} \frac{1}{\phi(n) \log^2 n}$ ...
-2
votes
0answers
30 views

a question in De koninck and luca's analytic number theory [on hold]

what is your idea,can you introduce a book for these kind of problems?
3
votes
0answers
58 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 ...
2
votes
0answers
29 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
23 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 ...
5
votes
1answer
47 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),\, ...
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 ...
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) = ...
7
votes
1answer
91 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 ...
1
vote
1answer
30 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
27 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
71 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
48 views

Multiplicity one theorem for GL(n) and SL(n)

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...
0
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0answers
54 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 ...
-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 ...
4
votes
2answers
48 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 ...
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) = ...
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 ...
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}} ...
2
votes
1answer
59 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) ...
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 ...
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 ...
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 ...
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 ...
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 ...
1
vote
0answers
18 views

Positive proportion sets of numbers not divisible by twin primes.

Is it possible to explicitly construct a set of integers $S$ which contains a positive proportion of the positive integers and every integer in $S$ is not divisible by any prime $p$ in the set of ...
2
votes
0answers
30 views

Generalization of Dirichlet convolution

The Wikipedia page on the Mobius inversion formula gives the following formula in passing: if $$G(x)=\sum_{k=1}^x \alpha(x)F(x/k)$$ for some arithmetic function $\alpha(n)$ possessing a Dirichlet ...
1
vote
1answer
54 views

Wrong proof of the functional equation for $ \zeta (s) $ but why is the result correct?

If I introduce the function $ f(x)= |x|^{s-1} $ inside Poisson summatory formula and use the fact that $$ \sum_{n=-\infty}^{\infty}|n|^{s-1}=2\zeta (1-s) $$ If I combine this expression in the ...
7
votes
2answers
165 views

What percentage of numbers is divisible by the set of twin primes?

What percentage of numbers is divisible by the set of twin primes $\{3,5,7,11,13,17,19,29,31\dots\}$ as $N\rightarrow \infty?$ Clarification Taking the first twin prime and creating a set out of its ...
-2
votes
1answer
32 views

Least quadratic Non residue [closed]

What are all results known yet using without using riemann hypothesis on the bounds on Least quadratic non residue .
2
votes
0answers
42 views

Titchmarsh S function

SO it is known that Titchmarsh S function $$ S(T)= \pi^{-1} arg\quad \zeta\bigg(\frac{1}{2}+iT\bigg)$$ under the assumption of *riemann hypothesis * gives $$ S(T)=O(\frac{\log T}{\log \log T})$$ can ...
1
vote
1answer
39 views

Group of Dirichlet Characters Modulo $q$ is Isomorphic to $(\mathbb{Z} / q\mathbb{Z})^*$

I'm currently reading a book on analytic number theory, and shortly after defining Dirichlet characters, the author stated that one can prove that for a given $q\in\mathbb{N}$, the group of Dirichlet ...
2
votes
0answers
37 views

Subset of numbers analogous to primitive polynomials over finite fields

It is well known that many problems in number theory have an analogue on the ring of polynomials over finite fields and vice versa, the primes in $\mathbb{F}_q[x]$ being the irreducible polynomials. ...
2
votes
1answer
99 views

Asymptotics for square-free numbers in an arithmetic progression

Set $$Q(s,\chi)=\sum_{n=1}^{\infty}\frac{\mu(n)^2\chi(n)}{n^s},\quad (s=\sigma+i\tau),$$ where $\chi$ is a character $\mod q$, Show that $Q(s,\chi)=L(s,\chi)H(s,\chi)$ where $H(s,\chi)$ is a ...
1
vote
0answers
45 views

Comparative prime number theory with a small tweak

Fix $a, k \in \mathbb{N}$ relatively prime. For $x \in \mathbb{R}$ recall the function $$ \pi(x; k, a) = \sum_{\substack{p \leq x \\ p \equiv a \pmod{k}} } 1 $$ where $p$ denotes the primes. ...
2
votes
1answer
26 views

Logarithm of the n'th prime.

Let $P_n$ denote the n'th prime number. How could we conclude the following from the prime number theorem? $$ \log(P_n)=\log n + \log\log n + o(1) $$ Maybe by showing that $P_n=An\log n $ for a ...
1
vote
2answers
18 views

On estimating a prime related Diophantine equation related to a partition .

A friend gave me the following algebraic combinatorics question which I couldn't solve Let $p$ be a prime number and $f(p)$ the smallest integer for which there exists a partition of the set $\{2,3, ...
4
votes
2answers
51 views

Question from the proof of the Prime Number Theorem

My question is pretty trivial, but I just wanted to ask about something I can't see at all. In the proof of the PNT supplied in these notes, it is asserted that when $|t| \ge 2$ ...
2
votes
1answer
29 views

A question about step in the proof of Selberg's formula

Recently I've found the following paper, discussing and proving Selberg's symmetry formula: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/Balady.pdf My question concerns proofs of ...