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

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3
votes
1answer
95 views

Prove a complex number is real

Let $z$ be a complex such that $|z-1| =1$, and consider the complex numbers $v$ and $w$ such that: $w = z^2 -z$ and $3\arg(v) = 2\arg(w)$, where arg is the argument of a complex number. Show that $$ ...
1
vote
0answers
36 views

A shortcut for analytic continuation?

Let $P(x)$ be a nonconstant integer polynomial with nonnegative coëfficiënts such that the equation $y= P(y)$ has only one real solution $q$. Let $x_1=P(0)$ and $x_n = P(x_{n-1})$. $$f(z) = ...
1
vote
0answers
13 views

Estimates related to a sum over primes from a fixed, sparse set

Let $E$ be a fixed infinite sequence of primes such that $\sum_{p \in E} \frac{1}{p} = \infty$. Assume that $\sigma > 1$ depends on some parameter $x \rightarrow \infty$ in such a way that $\sigma ...
0
votes
0answers
15 views

Relation between Bombieri theorem and p-adic squares

Koblitz states in his book on p-adic numbers on page 84: Suppose that $\alpha \in \mathbb Q$ is such that $1 + \alpha$ is the square of a nonzero rational number $a/b$. Let $S$ be the set of all ...
0
votes
0answers
17 views

Additive character sum over additive subgroups of finite fields, with special monomial arguments

Let $F$ be a finite field with $q = p^n$ elements, let $\psi$ be a non-trivial additive character of $F$, let $m$ be an integer coprime to $q-1$, and let $K$ be a large subspace of $F$, say $K$ is the ...
3
votes
0answers
35 views

Additive character sum over intersection of additive and multiplicative subgroups of finite fields

Let $H$ be a multiplicative subgroup of the finite field $\mathbb{F}_q$ with $q$ elements, say $H$ is the subgroup of $d$-th powers, $d \mid q-1$. Let $L$ be a subspace of $\mathbb{F}_q$ over some ...
2
votes
1answer
82 views

Are there any intuitive reasons for Goldbach conjecture to be true?

One thing puzzled me is that, despite its simple form, I have not seen any intuitive reasons for Goldbach conjecture to be true. Typical heuristic reason is based on probability arguments. Such ...
1
vote
0answers
19 views

Does Linnik's approximation to Goldbach's problem also work for the power of 3, 5, 7, etc ?

Linnik proved in 1951 the existence of a constant K such that every sufficiently large even number is the sum of two primes and at most K powers of 2. Roger Heath-Brown and Jan-Christoph ...
3
votes
0answers
42 views

Average Order of $\frac{1}{\mathrm{rad}(n)}$

Again a question about $\mathrm{rad}(n).$ Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing $n$. Or equivalently, ...
0
votes
0answers
97 views

How to use an inverse Mellin transform to get to the $\mathrm{core}(n)$?

From this question here: Moreover, if multiplicative function $\mathrm{core}(n)$ is defined to map positive integers "$n$" to square-free numbers by reducing the exponents in the prime power ...
2
votes
0answers
13 views

Construct an example of a Dirichlet series with specific abscissa of convergence and absolute convergence

I want, for each $\alpha \in [0,1]$ to construct an example of a Dirichlet series for which $\sigma _ 1 = \alpha$ and $\sigma _0 = 1$ where $\sigma _0$ is the abscissa of convergence and $\sigma _1$ ...
0
votes
0answers
45 views

Integrating a series expansion of $\{x\}\lfloor x\rfloor$ coming from Fourier series of sawtooth function.

Originally, I ran into this problem on some basic math puzzle site (where it was asked to find the definite integral for some specific integer values), but I decided to try and find the indefinite ...
1
vote
0answers
34 views

Locating a pdf of certain paper

One of the references in the book Opera de Cribro by Friedlander and Iwaniec is: J. Friedlander and H. Iwaniec, A polynomial divisor problem, J. Reine Angew. Math. 601 (2006), 109-137 Does anybody ...
3
votes
1answer
89 views

Twin prime conjecture implies $\limsup_{n\to\infty}\frac{\sigma(n)\pi(n)}{n^2}\left(\pi(\log n)-\frac{\pi_2(\log n)}{2C_2}\right)=e^{\gamma}$?

Let $\sigma(n)$ the sum of positive divisor function, $\pi(x)$ is the prime counting function, $\pi_2(x)$ is the twin prime counting function (we will assume that Twin prime conjecture holds), $C_2$ ...
2
votes
1answer
42 views

Estimates for a product involving primes

Is $$\prod_{p\leq z}\bigg(1 + \frac{p}{(p-1)}\frac{\log z}{\log p}\bigg) = O(z)?$$ where $p$ ranges over the primes less than $z$. I believe I can show that it is $O(z\log \log z)$.
10
votes
2answers
224 views

What is your idea about this conjecture?

I conjecture that in a consecutive sequence of $n$ natural numbers all greater than $n$, there exists at least one number which is not divisible by any prime number less than or equal to $n/2$. Can ...
2
votes
1answer
37 views

A more general Kloosterman-type sum

Let $\mathbb{F}_q$ be a finite field and let $a,b \in \mathbb{F}_q$ not both zero. Let $\psi$ be the canonical additive character on $\mathbb{F}_q$. The classical Kloosterman sum is given by $$ K(a,b) ...
1
vote
0answers
20 views

What would be a good source to learn the Deuring-Heilbronn phenomenon?

I was trying to read from the book "Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis" by Montgomery. However, I found the text highly condense and difficult to read, ...
1
vote
1answer
47 views

Could someone explain how the Gram series relates to Riemann's function?

I was reading an article on the distribution primes which mentions the following equation for Riemann's function $R(x)$: $$R(x) = \sum_{n=1}^{\infty}\frac{\mu(n)}{n}\text{li}(x^{1/n}) = 1 + ...
5
votes
1answer
66 views

Number of solutions to $x_1x_2+x_3x_4 = 1$ (mod $n$)

Show that the number of solutions $N$ to $x_1x_2+x_3x_4 = 1 \pmod n$ is $$ N=n^2\phi(n)\prod_{p|n} \left( 1+\frac{1}{p} \right). $$ Only thing I know how to start the problem is to consider: $$ N = ...
2
votes
1answer
68 views

Primes and arithmetic progressions

In a book on complex analysis, the authors prove: Given finitely many (non-trivial) arithmetic progressions of natural numbers $$a_1, a_1+d_1, a_1+2d_1, \cdots $$ $$a_2, a_2+d_2, a_2+2d_2, ...
0
votes
0answers
31 views

Equidistribution mod $1$ of sequence of rational numbers from interval $(0,1]$.

Define a sequence of rational numbers $\frac{a}{b}$ (gcd$(a,b)=1$) from interval $(0,1]$ as follows: $\frac{a_1}{b_1}$ comes before $\frac{a_2}{b_2}$ if $b_1 < b_2$ and $\frac{a_1}{b}$ before ...
0
votes
0answers
57 views

Proof of Prime Number Theorem

I am looking for a detailed proof of the Prime Number Theorem using analytic methods (that is, using $\zeta(s)$). What is a good reference to read?
3
votes
1answer
36 views

The sum $\sum_{n\leq x}\sum_{\substack{1\leq k\leq n \\ gdc(k,n)=1}}cos^2\pi \frac{k}{n}$ diverges as $x$, when $x$ tends to infitity

I want to know if it is possible find an easy proof (this is without an use of an strong result) of Question. Prove that the following sum diverges as $x\to\infty$ $$\sum_{n\leq ...
4
votes
1answer
39 views

Integers with a divisor in a given interval

Please bear with me, I have a notation question. In Kevin Ford's paper with the above title, the following statement occurs in Theorem T1, p. 369: If $2 ≤ y ≤ z ≤ x$, then $$H(x, y, z) = x\left(1 + ...
0
votes
0answers
7 views

Analogue of the Shifted Convolution Problem for $\lambda_{Sym^r}f$

Given a primitive form $f$ for the full modular group $SL_2(Z)$ of an even weight $k.$ Philippe Michel on his paper 'On the Shifted Convolution Problem' ...
10
votes
2answers
258 views

Motivation on how does complex analysis come to play in number theory?

I am not sure if this is a appropriate question. If it isn't, let me know and I'll delete it. $\textbf{Background}$ I am an undergraduate student and I'm very interested in number theory. I've tried ...
4
votes
1answer
50 views

Riemann zeta function and the volume of the unit $n$-ball

The volume of a unit $n$-dimensional ball (in Euclidean space) is $$V_n = \frac{\pi^{n/2}}{\frac{n}{2}\Gamma(\frac{n}{2})}$$ The completed Riemann zeta function, or Riemann xi function, is $$\xi(s) ...
0
votes
0answers
18 views

Importance of the real-rooted asympototics of $f_n(z)$ that uniformly converges to Riemann $\Xi(z)$ function

We are learning Riemann $\Xi(z)$ and Riemann $\zeta(s)$ functions. This question is related to an earlier one. (1) Suppose that a family of functions, $f_n(z)$, uniformly converges to Riemann ...
7
votes
1answer
78 views

Evaluating an integral - is it a two dimensional beta function? This arises from a variant of Goldbach's conjecture.

Let $\gamma>0$. I would like a nice way to prove that $$\int_{\begin{array}{c} 0\leq s,t\leq1\\ s+t\leq1 ...
13
votes
3answers
956 views

Are the nontrivial zeroes of the Riemann zeta function countable?

It is known that the set of non trivial zeros is an infinite set. But is it known if it is a countable, or uncountable infinite set?
1
vote
0answers
125 views

Assuming the Riemann hypothesis, does this integral give the Riemann zeta zeros when increasing Working Precision in Mathematica?

It is probably well known that the Riemann zeta zeros satisfy the following equation: $$\frac{\arg \left(\zeta \left(\rho _n+\frac{1}{1000000000000000}\right)\right)}{\pi }+\frac{\vartheta ...
0
votes
1answer
33 views

Number of pairwise non-isomorphic spanning trees of the wheel $W_n$, with restrictions

I recently encountered this problem. Frankly I'm stuck; would be nice for some help. Here it is: Let $N,k$ be positive integers. By $p_k(N)$ we denote the number of integer partitions of $N$ with ...
1
vote
1answer
44 views

A combinatorial sum and identity involving Stirling numbers of the second kind

Let $n, k \geq 1$. Let $a(j), 1\leq j \leq k$, be a sequence of real numbers. Consider the sum $$ \sum_{j=1}^k j! S(k, j) {n \choose j} a(j), $$ where $S(k,j)$ are Stirling numbers of the second kind. ...
0
votes
1answer
54 views

Analysis Texts Advice

I am after a nice analysis textbook which predominately covers inequalities and asymptotics. Something with some number theory, such as a very elementary analytic number theory text would be great!
5
votes
1answer
56 views

An asymptotic behavior of $\operatorname{Li}_{-n}(a)$ for $n\to\infty$

Suppose $a,b\in(0,1)$. I'm interested in comparison of an asymptotic behavior of $\operatorname{Li}_{-n}(a)$ and $\operatorname{Li}_{-n}(b)$ for $n\to\infty$. Such functions exhibit approximately ...
0
votes
1answer
30 views

Multiplicative function and Euler product

Theorem 1 Let $G \subset \mathbb{C}$ be an area and $\sum_{n=1}^{\infty}a_n e^{-\lambda _ns} $ and $\sum_{n=1}^{\infty}b_n e^{-\lambda _ns} $ two Dirichlet-series that converge on $G$ and represents ...
1
vote
0answers
51 views

Estimates of the sum involving both the Mobius function and Mertens function.

I want to ask on the estimates of the sum $$ \sum_{n=1}^{\infty} \mu(n)M\Big(\frac{x}{n}\Big)=\frac{1}{2\pi i }\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{x^s}{s\zeta(s)^2}ds.$$ But it is little known ...
1
vote
1answer
150 views

Argument of the Riemann zeta function on Re(s)=1

I refer to the lovely answer to this question. Is there an exact formula for the argument of the Riemann zeta function? Specifically, I would like to know the arguments along the line Re$(s)=1$. Some ...
4
votes
0answers
108 views

Arithmetic Derivative

In Calculus, whenever we see a constant and want to take the derivative of it, it always is $0$. However in Number Theory, we have something called the arithmetic derivative in which we can ...
1
vote
0answers
47 views

Show that $| f (0)| ≤ \sqrt{6}$

Suppose $f$ is $C$-analytic in $|z| ≤ 1$, $f ≪ 2$ for $|z| = 1$, Im $z ≥ 0$ and $f ≪ 3$ for $|z| = 1$, Im $z ≤ 0$. Show then that $| f (0)| ≤ \sqrt{6}$. I know to consider $f(z)f(-z)$ but not sure ...
0
votes
1answer
37 views

Proving that if a function lies within an integral domain, it satisfies the remainder and the root factor theorem but not

Remainder theorem states that for a c in F and f(x) in F(x). When we divide f(x) by x - c then the remainder is f(c). the Root Factor Theorem states that for c in F is a root of f(x) in F(x) if ...
0
votes
0answers
33 views

A question about selberg asymptotic formula

All: I tried to verify the following formula *Selberg: $\sum_{p \leq x}(\log p)^2+ \sum_{pq \leq x}\log p\log q = 2x\log x + O(x)$ I did some simple calculations, but I could not verify the ...
0
votes
0answers
51 views

How to compute similarity between two numbers

is there some metric for similarity between two numbers which have range 0~1 I hope that sim(1,1) = 1 sim(1,2) = 0.5 ... ... sim(50,47) = 0.78 sim(100,99) = 0.99 something like that.. if two number ...
0
votes
0answers
33 views

Approximating the integral $\int_1^x (t-[t])f'(t)dt$

In applications of Euler's summation formula to find the asymptotic behaviour of something like $\sum_{n\leq x}f(n)$, one typically gets integrals of the form $$\int_1^x (t-[t])f'(t)dt$$ where $[t]$ ...
3
votes
1answer
82 views

An understandable explanation of Euler Maclaurin formula for $\sum_{k=1}^n\log^2 k$, and related questions

Reference [3] for example, provide us the Euler Maclaurin formula, that with $m=1$ and $f(x)=\log^2 x$, defined for $x>0$, gives $$\sum_{k=1}^n\log^2 k=\sum_{k=2}^n\log^2 k=\int_1^n\log^2 ...
14
votes
3answers
242 views

Series of the totient function

Good morning, I wonder if : $$\sum_{n} \frac{(-1)^n}{\varphi (n)}$$ converges or not. where $\varphi (n)$ is the Euler function. Do you have any idea ?
2
votes
0answers
32 views

Asymptotic local limit theorem and applications in analytic number theory

I'm wondering if one could get similar results to the classical local limit theorem if one assumes that conditions, such as independence and identicallity of distribution of the random variables ...
1
vote
1answer
68 views

A very silly question about the Erdos-Kac theorem

Let $m$ be a positive integer and let $\omega(m)$ be the number of distinct prime factors of $m$. The Erdos-Kac theorem, see here, is the following: For all $x \in \mathbb{R}$, $$ \lim_{n \to ...
1
vote
0answers
91 views

When does linear combination of real-rooted entire functions of genus 0 or 1 remain real-rooted?

In our search of a family of entire functions to approximate Riemann $\Xi(z)$ function, we encounter the following family of functions: $$f_m(z,n,b)=\sum_{k=1}^m (-1)^k u_k(z,n,b)\tag{1}$$ where ...