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

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3
votes
0answers
44 views

Behaviour of $\zeta(s)$ near $s=1$

I would appreciate if somebody could run this over and see if it works out? any suggestions or pointers would be appreciated. I denote the standard eta function $\eta$ by $\zeta^{*}$. I have not used ...
2
votes
0answers
25 views

Upper bound for $|\zeta'(s)|$ near the line $\sigma=1$, a detailed proof

In page 285 Apostol leaves as a reader's asigment the proof that $|\zeta'(s)|=O(\log^{2}t)$, this is for every $T>0$ there exists a positive constant $K$ (depending on T) such that ...
4
votes
1answer
61 views

On the series $\sum_{n=1}^{\infty} (H_{n}+\exp(H_{n})\log(H_{n}))/n^{s}$, where $H_{n}$ is the $n$th harmonic number

It is known the following (see [1], here is an open access PDF on his homepage): Theorem (Lagarias, 2002). Let $\sigma(n)$ denote the sum of the positive divisors of $n$. Riemann Hypothesis ...
2
votes
2answers
83 views

$\zeta(2n)$ proof [duplicate]

Can anybody pass me on a good source to see the steps in proving, \begin{equation} \zeta(2n) = \frac{(-1)^{k-1}B_2k (2 \pi)^{2k}}{2(2k)!} \end{equation} I know how we start by looking at the product ...
0
votes
1answer
43 views

Why is the Bernoulli Number $B_1$ sometimes $+ \frac{1}{2} $?

By using the recursive formula, \begin{equation} \sum_{i=0}^{n} \binom{n+1}{i} B_i = n+1 \end{equation} we find $B_1$ to be $\frac{1}{2}$ and not $- \frac{1}{2}$. Why is this?
2
votes
1answer
40 views

Character sum of a type of “almost linear” surjective mappings over finite fields

Let $F$ be a finite field of characteristic a prime $p$ with $q$ elements and let $E/F$ be a finite extension of degree $n > 1$ over $F$. Let $\chi$ be the additive canonical character on $F$ ...
10
votes
1answer
126 views

why is $\zeta(1+it) \neq 0$ equivalent to the prime number theorem?

Reading through Titchmarh's book on the Riemann Zeta Function, chapter 3 discusses the Prime Number Theorem. One way to prove this result is to check the zeta function has no zeros on the line $z = 1 ...
1
vote
1answer
39 views

Good introductory book for Probabilistic Number Theory

I have a decent high school knowledge of Elementary Number Theory and it is also a subject I love to study. I have a good background in Real Analysis (not Complex Analysis) and Abstract Algbera. I ...
9
votes
2answers
465 views
+100

Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth are playing the following game. ...
0
votes
0answers
35 views

A residue question in integers

Given $N\in\Bbb N$, is it possible to find $9$ positive integers $A_j,N_i$ with $j\in\{1,2,3\}$, $i\in\{1,2,3,4,5,6\}$ such that following holds? $(1)$ $N\log N < A_j < cN\log N$ at every $j$ ...
4
votes
0answers
124 views

A lower bound for an arithmetic function

Let $N \in \mathbb{N}$ such that $\phi(N) \sim N$, where $\phi$ is the Euler's totient function. Let $A \subset [N] := \{1, 2, \ldots, N\}$. For $n \in \mathbb{N}$ define the function $$ C_A(n) = \#\{ ...
0
votes
1answer
47 views

What is the relative density of the abundant numbers in the positive integers?

The Art and Craft of Problem Solving by Paul Zeitz has the following problem. Now, I have been able to solve parts (a) and (b), part (a) by showing that it can get arbitrarily large, and part (b) by ...
0
votes
0answers
21 views

Hasse-Weil zeta function of projective hypersurfaces

Assume $f$ is a homogeneous integer polynomial in $n\geq 3$ variables such that the hypersurface $f=0$ is irreducible over $\mathbb{Q}$ (but not necessarily over $\overline{\mathbb{Q}}$ so for example ...
2
votes
0answers
29 views

Order of Magnitude

It is well known (see here, for example) that we have $$ \psi\left(\frac{1}{2},T\right)=\sum_{p\leq T}\frac{1}{p}=\log(\log T)+A+O\left(\frac{1}{\log T}\right), $$ where $\psi(\sigma,T)=\sum_{p\leq ...
5
votes
1answer
39 views

What is the Mobius sum $\sum_{n=1}^\infty \frac{(-1)^{n+1}|\mu(n)|}{n^s}$?

It can be observed that, $$A(s) := \sum_{n=1}^\infty \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)}$$ $$B(s) := \sum_{n=1}^\infty \frac{|\mu(n)|}{n^s} = \frac{\zeta(s)}{\zeta(2s)}$$ $$C(s) := ...
1
vote
1answer
35 views

Identity involving the zeta function

This might be very trivial, but a proof I'm reading on the bounds of the zeta-function uses the following fact: If $s=\sigma+it$ is a complex number and if $\sigma\geq 2$, then $|\zeta(s)|\geq ...
2
votes
1answer
54 views

A sum involving twin primes and Prime Number Theorem

This morning I've been watching documentary about asterorids, in a scene an astronomer explains the so called image subtraction process or pixel subtraction, a mathematical model used in computerized ...
1
vote
0answers
25 views

Why are rational numbers required in cusps of congruence subgroups?

While we consider the action of congruence subgroups on $\mathbb{H}$ (the upper half plane), we compactify using an additional point at infinity, that is fine. But why do we add even all rational ...
2
votes
0answers
35 views

Online lecture Videos on Algebraic or Analytic Number Theory or Sieve Theory [closed]

I found some lectures on youtube but I need something which starts from the basics.Any help will be truly appreciated.
1
vote
1answer
29 views

Why only congruence subgroups for modular forms?

When we define modular forms, why do we restrict ourselves to congrumence subgroups? Why not any subgroup of finite index? Or, even more generally why not any subgroup? Is it just a matter of ...
1
vote
1answer
48 views

The Dirichlet series $\sum_{n=1}^{\infty}\frac{rad(n)}{n^s}$

Following the example to compute $\zeta (s)\sum_{n=1}^{\infty}\frac{\phi(n)}{n^s}=\zeta (s-1)$, converges absolutely if $\sigma>2$, where $\phi(n)$ is the Euler's totient function and $s=\sigma + ...
2
votes
1answer
33 views

Understanding Wright's proof of Landau's theorem

I'm reading Wright's A simple proof of a theorem of Landau in which the core argument is a proof by induction and I find myself stuck on a major point. I must be misunderstanding notation or something ...
2
votes
1answer
46 views

A question from Titchmarsh's zeta function book.

On page 30, he writes that $\xi(0)=-\zeta(0)=1/2$, but on page 16 he writes that: $\xi(s)=1/2 s(s-1)\pi^{-1/2s}\Gamma(1/2s)\zeta(s)$ in eq.(2.1.12); so if I plug into this equation $s=0$ then I get ...
1
vote
3answers
57 views

Estimating the sum of reciprocals of products of two primes

It's rather well-known that $$ \sum_{p \leq X} \frac{1}{p} \sim \log \log X,$$ where this is a sum over the positive integer primes. Can we efficiently estimate the sum $$ \sum_{p,q \leq X} ...
0
votes
1answer
59 views

Out of all the proofs of the PNT, which one is the most accessible?

I have been studying the continuation of the Riemann zeta function $\zeta(s)$ for the past while. I can prove that all the zeroes must lie in the critical strip.I am currently in the process of using ...
3
votes
0answers
47 views

Linear independence of primitive Dirichlet characters and convolution

This is not an exercise but merely a question I have. Fix $N \in \mathbb{N}$ and suppose there exist some values $a_k \in \mathbb{C}$, for $k \in \mathbb{Z}_N$, such that $$ \sum_{k \in \mathbb{Z}_N} ...
8
votes
1answer
86 views

Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?

Khinchin showed that given the simple continued fraction of a real number, $$r = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1} {\ddots}}}$$ then it is almost always true that the partial quotients $a_i$ ...
9
votes
0answers
170 views

Maximum integer not in $\{ ax+by : \gcd(a,b) = 1 \land a,b \ge 0 \}$

Ryan asked about a variation of the coin problem, which was whether for any coprime natural numbers $x,y$ every sufficiently large natural number is $ax+by$ for some coprime natural numbers $a,b$. ...
2
votes
0answers
52 views

Finite Ramanujan expansions over a finite field

I'm wondering if we could have an analogy in finite fields. After all, the Discrete Fourier Transform (DFT) has been generalized to finite fields as well (with essentially identical properties as in ...
0
votes
0answers
41 views

Behaviour of the sum of divisors function via logarithmic means versus an elementary problem equivalent to the Riemann Hypothesis due to Lagarias

It is known the following (see [1], here is an open access in his homepage www.math.lsa.umich.edu/~lagarias/doc/elementaryrh.pdf): Theorem (Lagarias, 2002). Let $\sigma(n)$ denote the sum of the ...
1
vote
0answers
27 views

Erdős' papers on Analytic Number Theőry

My adviser has often mentioned that Paul Erdős' works on Analytic Number Theory contain a myriad of techniques that any number theorist must know. What are some of his papers in Analytic Number Theory ...
2
votes
1answer
52 views

An identity involving Gauss sums and convolution

For a Dirichlet character $\chi$ modulo $N$, the Gauss sum attached to $\chi$ is given by $$G_\chi(m) = \sum_{k \in \mathbb{Z}_N} \chi(k) e^{2\pi i mk/N}.$$ Suppose one has an $N$-periodic function ...
7
votes
0answers
127 views

Is the Euler function $\phi$ constant in arbitrarily large intervals?

Is it true that for every $k \in \mathbb{N}$ there exists a natural number $x$ such that $\phi(x)=\phi(x+1)=\cdots=\phi(x+k)$, where $\phi$ is the Euler's totient function? I thought about this ...
1
vote
1answer
45 views

Is there any PDE that applies specifically to Number Theory?

Given the advanced results obtained by analytic means in Number Theory, it puzzles me why I don’t recall ever seeing a partial differential equation used to good effect in Number Theory. Is there ...
0
votes
0answers
35 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
86 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
55 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
70 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 ...
1
vote
1answer
27 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
72 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
47 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
16 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
45 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
76 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
33 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
42 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
55 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, ...
14
votes
1answer
195 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
32 views

What is general Riemann's Hypothesis? [duplicate]

What makes it so important in analytic number theory?