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

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0
votes
0answers
24 views

Finding references about modular forms and symmetric power $L$-functions

I want to write an introduction to my thesis which is about modular forms and symmetric power $L$-functions. Could you give me good references to these two topics as simple as possible to get an ...
5
votes
2answers
72 views

$L$-function, easiest way to see the following sum?

What is the easiest way to see that$$\sum_{(m, n) \in \mathbb{Z}^2 \setminus \{0, 0\}} (m^2 + n^2)^{-s} = 4\zeta(s)L(s, \chi)?$$Here $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to ...
1
vote
0answers
41 views

Dirichlet characters with values in a finite field

Although the classical Dirichlet characters are complex valued, it seems to me rather useful that the characters attain values in a finite field; thus homomorphisms from $\mathbb{Z}_N^*$ to ...
2
votes
1answer
66 views

How to prove that $\frac 12+ \frac 13+\dots + \frac 1n < \log n < 1 + \frac 12+ \dots + \frac {1}{n-1} $?

If $n \in \mathbb N$ and $n \geq 2$, then we have $\frac 12+ \frac 13+\dots + \frac 1n < \log n < 1 + \frac 12+ \dots + \frac {1}{n-1} $. My try : Once if we can prove that for all $k \in ...
0
votes
0answers
20 views

Smooth interpolation for complex variable function. [on hold]

Is there any smooth interpolation function $T(z)$ that could smoothly connect two complex variable rational polynomial function $H_1(z)$ and $H_2(z)$, for example $$ H(z) = \begin{cases} ...
1
vote
1answer
24 views

Zeros of Dirichlet L-functions on the line $\Re(s)=1$ in proof of Dirichlet's theorem

In the proof of Dirichlet's theorem, we show that $L(s,\chi_0)$ has a simple pole at $s=1$ where $\chi_0$ is the principal character and that $L(1,\chi)\neq 0$ otherwise. Therefore the logarithmic ...
6
votes
1answer
67 views

Is $\pi(n)$ a Rational Function?

Are there some two-variable polynomials $P(n,\log n)$ and $Q(n,\log n)$ which we have the bellow equation for prime counting function $\pi(n)$ for $n \in \mathbb{n}$? $$\pi(n) = \Bigl{\lfloor} ...
4
votes
1answer
43 views

Prime Zeta Function proof help: Why are these expressions not equal?

I was trying to create a formula for the Prime Zeta function and I partially succeeded except for one frustrating error. I was only able to formulate an approximation. Consider the following sum: ...
0
votes
1answer
15 views

Definition of “Contractive Invariant Plane”

Can someone please explain the definition of a contractive invarient Plane found in: the paper It is nearly at the very beginning of the Introduction. By contractive do they mean a contractive map? ...
0
votes
0answers
37 views

What is the Fourier transform of Riemann Zeta function?

All: Is there an explicit form of Fourier Transform of Riemann Zeta function ? Also, is there an discrete Fourier Transform (DFT) of Riemann Zeta function ? I remembered I had seen something like ...
0
votes
1answer
71 views

Is the Riemann zeta function $\zeta(s)$ exactly $\pi(x)$?

Let $\pi(x)$ denote the number of primes less than or equal to a certain x value. The prime number theorem says that $x/\log x$ (or more accurately $x/(\log x-1)$) has been the most popular method ...
2
votes
1answer
44 views

Does $\zeta(s)^2 \pm \zeta(1-s)^2$ have roots at the $\rho$s?

Maybe a strange (or stupid) question, but does $$\zeta(s)^2 \pm \zeta(1-s)^2$$ also have roots equal to the non-trivial zeros ($\rho$) ? At first sight you would expect so, however when I tried to ...
1
vote
1answer
31 views

Expected value of discrete functions. [closed]

I am doing some research in number theory(High school-so nothing advanced). During this I came across this post. I have not done much statistics. So could someone explain to me why if $\displaystyle ...
6
votes
1answer
38 views

counting function of system of equations and Circle method

I came up with the follwing question while looking on Davenport's book: Analytical Methods for Diophantine equations and Inequalities. When introducing the Circle method gives an example on how to ...
2
votes
1answer
52 views

How to get sine term in Analytical continuation of $\zeta(s)$

I am able to prove the symmetric functional equation that Riemann gives in his paper, using Poisson Summation and properties of $\theta(x)$. The functional equation is given like so, ...
13
votes
1answer
166 views

Sum of Reciprocals of Primes in Imaginary Quadratic Field Diverges (2014 Miklós Schweitzer)

Problem 5 of the 2014 Miklós Schweitzer states: Let $\alpha$ be a non-real algebraic integer of degree two, and let $P$ be the set of irreducible elements of the ring $\mathbb{Z}[\alpha]$. Prove that ...
0
votes
0answers
31 views

What is general Riemann's Hypothesis? [duplicate]

What makes it so important in analytic number theory?
4
votes
1answer
36 views

Divide a square into different parts

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with geometry, which perhaps yields the shortest, simplest proofs, but other ...
1
vote
1answer
44 views

Growth of the zeta function on the line $Re(s)=\frac{1}{2}$

I've seen that on the line $Re(s)=\frac{1}{2}$, $\zeta(s)=O(t^{\frac{1}{4}})$ where, as usual, $s=\sigma+it$. My teacher has told me that this can be derived directly from the functional equation of ...
3
votes
1answer
23 views

How do I estimate the error term when computing the number of integers which have the fractional part of their square roots in a given interval?

I'm trying to find the number of integers $n \leq N$ such that fractional part of $\sqrt n\in (\alpha,\beta]$ where $(\alpha,\beta]\subseteq(0,1]$. The approximate number is of course ...
1
vote
1answer
58 views

Proving the functional equation $\theta (x) = x^{-\frac{1}{2}} \theta (x^{-1})$ from the Poisson summation formula

We have the relationship $\theta (x) = x^{-\frac{1}{2}} \theta (x^{-1})$ Now I know one uses the Poisson summation formula to prove this. The Poisson summation formula comes from Fourier Transform ...
0
votes
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
votes
0answers
46 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
19 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
55 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
19 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
47 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
49 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
27 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
42 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
29 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
27 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
152 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
32 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
vote
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
37 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
66 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
36 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
28 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
63 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 ...