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

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2
votes
0answers
67 views
+100

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 ...
2
votes
0answers
18 views

Eisenstein series of weight $2$ for $\Gamma_0(N)$ : where am I wrong?

Let $A_{N,2}$ be the set of triples $(\psi,\varphi,t)$ such that $\psi$ and $\varphi$ are primitive Dirichlet characters modulo $u$ and $v$ with $(\psi\varphi)(-1)=1$, and $t$ is an integer such ...
8
votes
3answers
336 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
45 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
74 views

Evaluating an integral using Gamma function [on hold]

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
163 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 ...
3
votes
0answers
26 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 ...
4
votes
3answers
60 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
59 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 ...
3
votes
0answers
150 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 ...
0
votes
0answers
30 views

Argument for finite solution of power Diophantione Equation.

Assume the equation $4x^3=y^2+3$ has infinite positive integer solution. If $x,y$ has general solution then it is clear that for any $x$(rational, integer), there is a $y$. It can be said there is a ...
1
vote
1answer
55 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
263 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
204 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
1answer
85 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
1answer
41 views

Dirichlet Convolution Associativity

I am unsure of the proof of associativity. So far I have: \begin{align} [f\ast (g\ast h)](n)&=\sum\limits_{ab=n}f(a)[g\ast h](b)\notag\\ ...
2
votes
0answers
28 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
51 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 ...
1
vote
0answers
13 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 ...
7
votes
2answers
161 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
26 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
31 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
30 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
35 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. ...
1
vote
0answers
40 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
21 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 ...
14
votes
3answers
800 views

How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?

I saw reference to this result of Chebyshev's: $$\sum_{p\leq n} \frac{\log p}{p} \sim \log n \text{ as }n \to \infty,$$ and its relation to the Prime Number Theorem. I'm looking into an ...
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
47 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
2answers
97 views

Growth rate of product of smallest prime factors

For $n\in \mathbb{N}$, let $p(n)$ denote the smallest prime dividing $n$. Then consider the function $f:\mathbb{N}\rightarrow \mathbb{N}$ defined by $f(n)= \prod_{k=1}^{n}p(k)$. Question: What is ...
2
votes
1answer
27 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 ...
1
vote
0answers
33 views

Rate of Convergence of $A_{s,k}=\prod_{p}\left(1-p^{-1}\right)^{s-k+1}\sum_{m=0}^{k-1}{s\choose m}\left(1-p^{-1}\right)^{k-1-m}p^{-m}$

I'd like to know how fast the infinite product $$A_{s,k}=\prod_{p}\left(1-p^{-1}\right)^{s-k+1}\sum_{m=0}^{k-1}{s\choose m}\left(1-p^{-1}\right)^{k-1-m}p^{-m}$$ converges, where the product is taken ...
0
votes
1answer
23 views

Vanishing property of logarithmic derivative of zeta function

I was trying to derive the explicit formula for the integrated Chebyshev $\psi$ function, $\psi_1$ defined as \begin{equation}\psi_1(x)=\int_1^x\psi(y)dy\end{equation} But I have stumbled upon one ...
1
vote
2answers
64 views

Number of distinct prime divisors of an integer $n$ is $O(\log n/\log\log n)$

I strongly believe that the claim is true; but I'm neither a mathematician nor speaking French and hope that somebody can confirm it, since related questions (here, here and here) either don't have an ...
1
vote
1answer
37 views

Is there numerical evidence supporting the predicted density of the primes of the form $x^2+1$?

A famous conjecture (due I think to Hardy and Littlewood) states that if $P(x)$ denotes the number of primes of the form $n^2+1$ less than or equal to $x$, then $$P(x)\sim \frac{C\sqrt x}{\log x}$$ ...
0
votes
0answers
55 views

Another question/observation about Mersenne numbers and Euler's totient function

This is a follow up to this question Upper bound for Euler's totient function on composite Mersenne numbers and an ongoing project with lots of questions related to Mersenne numbers. I'm sorry if ...
0
votes
0answers
18 views

The estimation of $\sum^{K_0+K}_{k=K_0+1} \min\left\{ U, \frac{1}{\left< \alpha k + \beta \right>} \right\}$

I have some difficulty with understanding the proof of the following theorem: Suppose $\alpha$ is a real number which has the form $\alpha = \frac{h}{q} + \frac{\theta}{q^2}$, $(q,h)=1$, $q \geq ...
1
vote
1answer
84 views

Question concerning the Dirichlet density of a subset of the set of primes

I have the following question: I am reading Serre's book "A Course in Arithmetic" (see http://www.math.purdue.edu/~lipman/MA598/Serre-Course%20in%20Arithmetic.pdf). On page 75, it is stated that the ...
0
votes
0answers
18 views

How to estimate $\zeta(s)/\zeta(ks)$ in zero-free regions?

Riemann zeta function $\zeta(s)$ $(s=\sigma+i\tau)$, the best zero-free region to known to date, namely $$\sigma \ge 1-c(\log \tau)^{-2/3}(\log\log \tau)^{-1/3} \quad (\tau \ge 3).$$ and we have ...
5
votes
1answer
58 views

Why does Titchmarsh say that we can move the derivative under $\frac{2}{\pi}\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cosh(\alpha t) \, dt$

If we define the Riemann-Xi function as $$ \Xi(t) = \xi(\frac{1}{2} + it)$$ where $$\xi(s) = \frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s),$$ then according to Titchmarsh in his ...
9
votes
2answers
121 views

Evaluating $\sum_{\gcd\left(m,n\right)=1}\frac{1}{m^2n^2}$

I was wondering how one would evaluate the sum $$\sum_{\gcd\left(m,n\right)=1}\frac{1}{m^2n^2}.$$ The first thought that came to mind to to try something like this: ...
2
votes
1answer
51 views

Asymptotic for $\sum a_nb_n$ if asymptotic for $\sum a_n, \sum b_n$ are known

Let us assume that $a_n>0$ and $b_n>0$ for each n. Also let $$ \sum_{n\leq x} a_n \sim f(x) $$ and $$ \sum_{n\leq x} b_n \sim g(x) $$. What can we say about the asymptotic on $\sum_{n \leq x} ...
3
votes
2answers
201 views

Apostol - Analytic Number Theory, Chapter 3 problem 4a

The problem comes from "Introduction to Analytic Number Theory" by Tom M. Apostol, Chapter 3, Problem 4a: Question: Prove $\sum_{n \le x} \mu(n)[ \frac xn]^2 = \frac{x^2}{\zeta(2)} + O(x \log(x))$ ...
0
votes
0answers
27 views

Comparing a primorial $p\#$ to Dusart's upper bound for the $n$th prime

The number of elements of a reduced residue system modulo a primorial $p$ is $\varphi(p\#)$ I thought that it would be interesting to compare each primorial $p_i\#$ to the Dusart's estimate for the ...
1
vote
1answer
36 views

Chebyshev's theorem on the distribution of primes

I a lecture V. Arnold says that Chebyshev had proved that the limit $$\lim_{n\to \infty}\frac{\pi(n)}{n/\mathrm{log}(n)}$$ if exists is equal to one. Where I can find the proof? Thanks!
0
votes
1answer
29 views

Writing a Gauss sum as a sum over divisors

Let $\chi$ be a Dirichlet character modulo $q$ induced by a primitive character $\chi^*$ modulo $d$ for some divisor $d$ of $q$. Let $n$ be a positive integer, and consider the generalised Gauss sum ...
1
vote
0answers
33 views

Reasoning about Pierre Dusart's estimate of the $n$th prime

Cited here, Pierre Dusart established the following lower bound for $p_n$: $$p_n > n(\ln n + \ln \ln n - 1)$$ Using a spreadsheet and plugging in different values of $n$, I noticed that for an ...
3
votes
1answer
35 views

What is the best estimate known for the upper bound for the difference between consecutive primes?

Bertrand's Postulate gives us that: $$p_n < p_{n+1} < 2p_n$$ So that: $$p_{n+1} - p_n < p_n$$ In this answer, this paper is cited which says in Prop 6.8 that: For $x \ge 396738$ ...
0
votes
2answers
44 views

Properties of $\underset{k\geq1}{\sum}\frac{1}{\left(2k-1\right)^{s}}$

Is this function $$\underset{k\geq1}{\sum}\frac{1}{\left(2k-1\right)^{s}},\,Re(s)>1$$ well known? In particular I'm interessed about analytic continuation and its zeros and poles. Have this ...