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
13 views

Motivation of the study of certain mean-square sums

Given a multiplicative function $a(n)$ and a positive real number $\alpha$ in $Q[\sqrt{N}],$ where $N$ is a square free integer. I want to know the motivation of the study of the mean-square value of ...
4
votes
1answer
31 views

$L(1,\chi) = \sum_{n=1}^{\infty}\frac{\chi (n)}{n} > 0$, for $\chi$ be the non-trivial real character

Let q be an odd prime and $\chi$ be the non-trivial real character modulo q. I am trying to prove that $L(1,\chi) = \sum_{n=1}^{\infty}\frac{\chi (n)}{n} > 0$. Note: this question was first asked ...
2
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0answers
50 views

Find motivation for calculating $\int_{2}^{X} A^2(t) A(\alpha t)dt$

I read a thesis of Kong Kar Lun (student of Tsang K.M) about the some mean value theorems for certain errors terms in analytic number theory and in which he gave the asymptotic formulas of the ...
0
votes
0answers
20 views

References for Dirichlet characters and L-functions

I am working on some exercises from my Analytic Number Theory course regarding Dirichlet characters, and I was wondering if someone could provide some references for this. Here's a problem that I'm ...
0
votes
0answers
43 views

Strange equation $1 + 2 + 3 + … = - \frac{1}{12}$ [duplicate]

Let $S = 1 + 2 + 3 + 4 + ...$ $S_1 = 1 - 1 + 1 - 1 + 1 - 1 = 1 - (1 - 1 + 1 - 1 + 1 - ...) = 1 - S_1 \Rightarrow S_1 = 1 - S_1 \Rightarrow 2S_1 = 1 \Rightarrow S_1 = \frac{1}{2}$ $S_2 = 1 - 2 + 3 - ...
7
votes
1answer
76 views

Distribution of random divisor sums modulo n

Let $k$, $n\ge 2$ be positive integers, and choose $\ell$ such that $0\le \ell \le k-1$. For each integer $2\le j \le n$, choose a divisor $d_j$ of $j$, uniformly at random from the divisors of $j$. ...
0
votes
0answers
22 views

finding reference

I want to have a reference in which we have an asumptotic formula for the third moments of the sums of Hecke eigenvalues. If $f$ is a primitive form of an even weight $k$ for the full modular group ...
2
votes
2answers
51 views

Show that $\sum_{n\le x} \mu ^2(n)=\frac{x}{\zeta(2)}+o(\sqrt{x}) \; (x\to \infty)$

Show that $$\sum_{n\le x} \mu ^2(n)=\frac{x}{\zeta(2)}+o(\sqrt{x}) \; (x\to \infty)$$ I've proven so far that $\sum_{n\le x} \mu ^2(n)=\frac{x}{\zeta(2)}+O(\sqrt{x})$. I want to reduce this error ...
3
votes
2answers
51 views

Let $S(x)=\sum_{p\le x,\; q\le x,\; pq\gt x}\frac{1}{pq}$, where p and q are primes. Find the limit of this function.

Let $$S(x)=\sum_{p\le x,\; q\le x,\; pq\gt x}\frac{1}{pq},$$ where $p$ and $q$ denote prime numbers. Show that as $x\to\infty$,$S(x)$ converges to a constant, and find the value of that constant. ...
1
vote
1answer
37 views

For $s > 1,$ we have $\zeta(s) = \prod \left( \dfrac{1}{1-p^{-s}} \right)$

For $s > 1,$ we have $\zeta(s) = \displaystyle \prod \left(\dfrac{1}{1-p^{-s}} \right).$ Fix a positive integer $y,$ let $p_1, \ldots, p_n$ be primes and define $N_y := \{n \in \mathbb{N}: ...
1
vote
1answer
29 views

Example of holomorphic modular form of weight 2?

I read that for holomorphic modular forms of weight 2, $f(q ) = \sum a_n q^n $ Hecke proved $|a_n| < Cn$. Are there any holomorphic modular forms of weight 2? There certainly aren't any for the ...
1
vote
0answers
13 views

Prove that $L(s)$ converges for $s > 0,$ or more generally for all complex $s \in \mathbb{C}$ with $\Re(s) > 0.$ [duplicate]

Let $L(s) = \sum_{n=1}^{\infty} a_n/n^s$ be a Dirichlet series. Suppose that the partial sums of the coefficients $A_n = \sum_{i=1}^n a_i$ are bounded, i.e. there exists a constant $C$ such that ...
1
vote
1answer
46 views

Bounded sum of reciprocals of primes.

How can one adapt Apostol's proof that the bounded sum of the reciprocals of the first primes is $$\log\log x + C + O(1/ \log x) $$ to conclude the same about $$ \sum\limits 1/(p+1) $$ ? I just need a ...
3
votes
1answer
35 views

Why is Newman's Analytic Theorem neccessary

In a proof of the prime number theorem along the lines of Newman's, we establish that $-\frac {\zeta'(s)}{\zeta(s)}-\frac 1{s-1}$ possesses an analytic continuation to $\Re(s)\ge 1$ and that ...
4
votes
1answer
40 views

An asymptotic formula for the bounded sum of primes.

How to prove the following asymptotic formula?: $$ \sum\limits_{p\leq x} p \sim \frac{x^2}{2 \log x} $$ I'm stuck and I don't know where to start. I've been suggested the use of $\pi (x) = ...
2
votes
2answers
30 views

Estimate of the bounded sums of the tau function logarithms

Is the following estimate correct? Let $\tau (n)$ be the function that counts how many divisors of n are there. Then: $$ \sum\limits_{n\leq x} \log(\tau(n))=\log 2 \log\log x + O(1) $$ I've been ...
1
vote
0answers
164 views

“Necessary” condition for Power Diophantine Equation.

Motivation: Brocard’s problem $n!+1$ being a perfect square Observations: Given a power Diophantine equation of $k$ variables with a “general solution” (provides infinite integer solutions) to ...
0
votes
1answer
51 views

On the summatory function of $\Lambda(n)/n$

In this paper is written that the prime number theorem in the form $\psi(x) = ( 1 + o(1) ) x$ is elementary equivalent to $$\sum_{n \le x } \frac{\Lambda(n)}{n} = \log x - \gamma + o(1) $$ I started ...
0
votes
1answer
24 views

want a better bound the expression

Consider the expression $$ f_n(x)=\sum_{d|n,1<d\leq x} \Lambda(d)\left(\frac{1}{\log d}-\frac{1}{\log x}\right) $$ I've got $f_n(x)=\operatorname{O}\left(\frac{ x}{(\log x)^2}\right)$, but I've ...
3
votes
1answer
26 views

$\forall \theta>1/2$ $\exists x_0$ such that $\forall x\geq x_0 \,\{n\in\mathbb{N}\colon n\in[x,x+x^\theta],\ n\text{ is sq.free}\} \neq \emptyset$

I proved that $\left|\{n\in\mathbb{N}\colon n\leq x,\ n\text{ is squarefree}\}\right|=\frac{x}{\zeta(2)} + O(\sqrt{x})$. How does one use this result to prove that: $\forall \theta>1/2$ $\exists$ ...
0
votes
1answer
33 views

$\sum_{p|q_n} (\log p)^\alpha\sim n(\log n)^\alpha, n\to\infty $, where $q_n:=\prod_{j\leq n}p_j$

Let $\alpha \in (0,1)$ and let $p_j$ be the sequence of primes and let $q_n:=\prod_{j\leq n}p_j$. Prove that $$ \sum_{p|q_n} (\log p)^\alpha \sim n(\log n)^\alpha \qquad n\to\infty ...
1
vote
1answer
41 views

Jacobi Identity guide

Can any one guide me how can I prove this identity. $$\prod_{n=1}^{\infty}(1-q^j)^3=\sum_{n=0}^{\infty}(-1)^n(2n+1)q^{(n^2+n)/2}$$
1
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0answers
25 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
38 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 $\sum\limits_{n=2}^{\infty} \frac{1}{\phi(n) \log^2 n}$ ...
4
votes
0answers
67 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
37 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
30 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 ...
6
votes
1answer
68 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
50 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
66 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
55 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
137 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
112 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
33 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
38 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
42 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
1answer
83 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
70 views

Multiplicity one theorem for GL(n) and SL(n) [closed]

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...
0
votes
0answers
62 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 ...
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
38 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
83 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
32 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
50 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
57 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
52 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
50 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
116 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
182 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
101 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}} ...