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

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2
votes
1answer
40 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
26 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
52 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
71 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
69 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
34 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
66 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
39 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
40 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
8 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
312 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
57 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
81 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
975 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
132 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
48 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
61 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
33 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
64 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
193 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
125 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
48 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
49 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
145 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
84 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
249 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
39 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
81 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
92 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 ...
1
vote
1answer
38 views

what is the value of Chebyshev function at non-integer value?

What is the value of Chebyshev $\psi(x)$ function at non-integer values ? For example, what is the value of $\psi(3.56)$? I have seen, in same place, it seems that $$\psi(3.56)=\psi(3)$$ And in ...
4
votes
2answers
203 views

Does the function change the sign infinitely many times?

Let $0<\epsilon<\frac{1}{2}$; $\mu$ denotes the Moebius function. Consider the sums $$ \sum_{k=1}^n\frac{\mu(k)}{k^\epsilon}. $$ Do they change the sign infinitely many times as $n\to\infty$? ...
2
votes
1answer
33 views

On Abel summation $\sum_{e<n\leq x}\left(\mu(n)\cdot\int_2^n\frac{ds}{\log s}\right)$, where $\mu(n)$ is the Möbius function

By Abel's identity for $Li (x)=\int_2^x\frac{ds}{\log s}$, $a(n)=\mu(n)$ the Möbius function and $[y=e,x]$ (see Theorem 4.2, page 77 of [1]) and an application of Fundamental Calculus Theorem we ...
1
vote
0answers
62 views

On the sequence $n^{H_{n}lcm(1,2,\cdots,n)}$, where $H_n=1+1/2+\cdots+1/n$ is the nth harmonic number and Prime Number Theorem

For an integer $\geq 1$, if $H_n=1+\frac{1}{2}+\cdots+\frac{1}{n}$ the nth harmonic number then $lcm(1,2,\cdots,n)\cdot H_n$ is an integer and Definition. For an integer $n\geq 1$, the general ...
0
votes
1answer
85 views

Does Riemann Hypothesis imply strong Goldbach Conjecture? [duplicate]

In Andrew Granville's 2007 paper: "REFINEMENTS OF GOLDBACH’S CONJECTURE, AND THE GENERALIZED RIEMANN HYPOTHESIS" He said: "an averaged strong form of Goldbach's conjecture is equivalent to the ...
1
vote
0answers
43 views

Conditions of Euler Product

We know that if the infinite sum of a multiplicative function is absolute convergent, then the sum can be expressed as infinite product and the infinite product is absolutely convergent. Does there ...
3
votes
1answer
51 views

Question about complex analysis in proof in Ingham

This is a detail from a proof in Ingham's Distribution of Prime Numbers, p. 91-92. He forms a Dirichlet integral and assumes for contradiction that the numerator $c(x)\geq 0.$ Then he bounds $f(s)$ in ...
0
votes
1answer
49 views

is there a Globally convergent series for Riemann Xi function?

According to Wikipedia, there is a global convergent series for Riemann Zeta function: https://en.wikipedia.org/wiki/Riemann_zeta_function#Globally_convergent_series Is there a similar global ...
1
vote
0answers
36 views

modern proof of the conditional three prime theorem by Hardy and Littlewood

Hardy and Littlewood proved the three prime theorem under the GRH(generalized Riemann hypothesis) in an old paper: Some problems of `Partitio numerorum'; III: On the expression of a number as a sum of ...
0
votes
0answers
33 views

A combinatorics question arising from taking powers of a sum

Let $N \in \mathbb{N}$ and let $f$ be an arithmetic function. For a positive integer $k \geq N$ consider the sum $$ S_k(N) = \left( \sum_{n=1}^N \mathbf{1}_{f(n) = 0} \right)^k = \sum_{1 \leq n_1, ...
11
votes
1answer
155 views

Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges

I am going through A. J. Hildebrand's lecture notes on Introduction to Analytic Number Theory. I'm currently stuck at the exercises at the end of Chapter 3 (Distribution of Primes I - Elementary ...
3
votes
1answer
72 views

Looking an asymptotic for $\sum_{k\leq x}\Lambda(k)e^k$, where $\Lambda (n)$ is the von Mangoldt function

Using Abel's identity (see Theorem 4.2 in page 77 of [1]) and Prime Number Theorem (Theorem 4.4 in page 75) I compute $$\frac{1}{x}\sum_{k\leq x}\Lambda(k)e^k\sim 1\cdot e^x-\frac{1}{x}\int_1^x ...
3
votes
1answer
60 views

Numbers as sum of two relatively prime composite numbers

It is not hard to prove by analytical method the existence of a positive integer $n$ such that for all integers $m > n$ the following assertion is true: There exist two positive integers $a$ and ...
3
votes
0answers
66 views

What are the (possible) fundamental reasons for all zeros of zeta function to be on the critical line?

What are the (possible) fundamental reasons for all zeros of zeta function to be on the critical line ? Are there particular mechanism to make this happen ? Because of Levinson and Corney's work, we ...
18
votes
0answers
190 views

Sums of the form $\sum_{d|n} x^d$

Let $$S(x,n) = \sum_{d|n} x^d, n \in \Bbb N$$ Do these sums appear in the literature? What are they called if they do and what is known about them?
1
vote
0answers
39 views

How to derive a formula related to the Gauss sum

Let $\chi$ be a Dirichlet character modulo $m$ induced by $\chi'$ modulo $m'$. We define $$ \tau(\chi):= \sum_{a(mod \ m)} \chi(a) e(a/m). $$ Could someone please show me how to derive the formula: ...