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

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5
votes
2answers
168 views

To estimate $\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$

How may we estimate $$\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$$ where for every positive integer $m$ , $d(m)$ denotes the number of positive divisors of $m$ ?
6
votes
1answer
162 views

Finding near-integers in a range

I have a (transcendental) constant $\alpha$ and a fixed parameter $\varepsilon>0.$ I'd like to find all positive integers $n<\ell$ for which $\|n\alpha\|<\varepsilon,$ where $\|x\|$ is the ...
3
votes
1answer
539 views

Definition of nebentypus in $L$-functions.

In Iwaniec and Kowalski, the term nebentypus is mentioned several times in the book. Every time it seems to just refer to a character $\chi$. Since I don't see the authors defining nebentypus, can ...
1
vote
1answer
83 views

Iwaniec Kowalski Notation

On page 532 of the book analytic number theory by Iwaniec and Kowalski, the following notation is used: $C^{~\infty}$ and $\tau(n,\chi)$. Could anyone tell me what these represent? (the former is ...
6
votes
2answers
248 views

Prime power Gauss sums are zero

Fix an odd prime $p$. Then for a positive integer $a$, I can look at the quadratic Legendre symbol Gauss sum $$ G_p(a) = \sum_{n \,\bmod\, p} \left( \frac{n}{p} \right) e^{2 \pi i a n / p}$$ where ...
1
vote
2answers
68 views

showing that $\log(N) \leq \prod_{n \leq N} {(1-p^{-1})^{-1}}$

i can't see that $H_n \leq \prod_{n \leq N}{(1-p^{-1})^{-1}}$ and i can't see that $\log(N) \leq \prod_{n \leq N} {(1-p^{-1})^{-1}}$ p is prime and $H_n$ is harmonic series
5
votes
1answer
567 views

Clarkson's Proof of the Divergence of Reciprocal of Primes

In Tom Apostol's book, he credits the proof of the divergence of the sum of reciprocal of primes to Clarkson. To begin, we assume $\{p_n\}$ is an enumeration of the primes and ...
7
votes
5answers
522 views

Proving $\sqrt{2}\in\mathbb{Q_7}$?

Why does Hensel's lemma imply that $\sqrt{2}\in\mathbb{Q_7}$? I understand Hensel's lemma, namely: Let $f(x)$ be a polynomial with integer coefficients, and let $m$, $k$ be positive integers ...
3
votes
0answers
146 views

Existence of zeros of Mellin transform and properties of function to be transformed

Mellin transform of function $f(x)$ defined for $x\geqslant 0$ is given by $$ f^\ast(z) =\int\limits_0^\infty x^{z} f(x) \frac{dx}{x}. $$ I consider only exponentially decreasing (there exist such ...
2
votes
1answer
147 views

Stable points and the fundamental domain of the modular group

Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq 1/2\}$ its fundamental domain. How ...
8
votes
2answers
541 views

Approximation of $\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$ [duplicate]

I am reading about the Riemann hypothesis, and the article mentioned the Li function: $$\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$$ They said that this function can be approximated: ...
0
votes
0answers
89 views

By establishing a recurrence relation and using induction, or other-wise, show that this sequence is 3-adically Cauchy?

this is a question from a book I'm struggling with, please could you provide a clear proof Consider the sequence of rational numbers $a_1 = 1+3,a_2 = 1+\frac{3}{1+3},a_3= 1 + \cfrac{3}{1 ...
2
votes
1answer
320 views

For what primes $p$ does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically?

this is a question from a book I'm struggling with, please could you provide a clear proof For what primes p does the series $1!+2!+3!+4!+ \cdots $ converge $p$-adically? kind thanks
0
votes
1answer
80 views

For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically

For which primes p does the series $\sum_{i=0}^\infty (\frac{10}{11})^i$ converge p-adically and, when it does, to what limit?
4
votes
1answer
69 views

$a_{[n/1]}+a_{[n/2]}+…+a_{[n/n]}=1$

The sequence $a_n$ satisfy $$a_{[n/1]}+a_{[n/2]}+...+a_{[n/n]}=1,$$ for all $n \in \Bbb N$. (the subscript $[n/k]$ is the integer part of $n/k$) $Proof:$for any $k>0$,$$\lim_{n \rightarrow ...
4
votes
2answers
997 views

Using sum of logarithms of primes to prove the number of primes up to $n$ is $O(n/\log n)$

I need to show that the number of primes up to $n$ (i.e. $\pi(n)$) is $O(n/\log n)$. In the previous exercise of this question I proved that ${\displaystyle \sum_{i=1}^{\pi(n)}\log p_{i}} \leq Cn$ for ...
2
votes
1answer
261 views

Explicit Formula for $\zeta(s)$

In the explicit expression for $$\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2}) $$ $ x^\rho$ denotes $x^{\mathrm{Re} \rho}$. I wanted to ...
5
votes
1answer
158 views

Understanding a very elementary property of factorials

I've seen this stated in a few places. If $$\vartheta(x) = \sum_{p\le{x}} \log (p) \qquad \psi(x) = \sum_{m=1}^{\infty}\vartheta\left(\sqrt[m]{x}\right)$$ Then $$\log(x!) = \sum_{m=1}^{\infty} ...
2
votes
1answer
58 views

How to show $\sum_{d>x,P^+(d)\le y} \mu(d)/d = O (\log{y}\cdot e^{-\log{x}/2\log{y}})$?

I can't see why $$\sum_{d>x,P^+(d)\le y} \mu(d)/d = O (\log{y}\cdot e^{-\log{x}/2\log{y}})\;,$$ where $P^+(d)$ is the greatest prime factor of $d$. Can anyone give a hint? Thanks.
4
votes
1answer
104 views

evaluate $\phi(50!)$

I want to evaluate $\phi(50!)$, where $\phi$ is the Euler totient function, so i take the factorization in primes of $50!$ $$2^{47}\times 3^{22}\times 5^{12}\times 7^8\times 11^4\times 13^3\times ...
2
votes
1answer
355 views

A convergence problem: splitting a double sum

I have been facing some difficulties with the following question. For an absolutely convergent series $\sum_m a_m$, and the Möbius function $\mu(n)$, $x=(x_1,x_2)\in \mathbb{R}^2$, and $\alpha ...
9
votes
3answers
940 views

A cohomological statement equivalent to the Riemann Hypothesis

Is there a possibility for looking for a theory of cohomology and an equivalent cohomological statement for Riemann hypothesis over $\mathbb{Z}$?
8
votes
2answers
333 views

Bounds on a sum of gcd's

Does there exist a positive real number $C$ and a positive integer $M$ such that for all $n > M$ we have: $$\sum_{i=1}^n\sum_{j=1}^n\gcd (i, j)\ge Cn^2 \log n$$ This originally appeared as an ...
0
votes
1answer
82 views

What's the name of this class of transcendental numbers?

I'm considering the set $$\left\{\sin(k)\mid k\in\Bbb Z\backslash \left\{0\right\}\right\}.$$ All of its members are transcendental numbers, but together they don't form the complete set of all ...
5
votes
2answers
346 views

Sum of squares of sum of squares function $r_2(n)$

Let $r_2(n)$ denote the number of representations of $n$ as a sum of two squares. What is known about the sum of squares of this function, $\sum_{i=1}^n r_2(i)^2$ In particular is anything ...
8
votes
1answer
949 views

how to understand $\log\zeta(s)$ (Riemann zeta function)?

I know that if a function $f$ is analytic and has no zeros in a simple connected region, then we can define $\log{f}$ making it analytic in that region. Let's assume $Re(s)>1$. Is $\zeta(s)$ ...
0
votes
3answers
380 views

Properties of Mertens Function

I am astounded by how little information about Mertens function M(n) (partial sums of the Möbius function) is on the Internet. Thus, I would be thankful if someone could clear up some of my confusion. ...
7
votes
5answers
219 views

$(1+\frac{1}{n\log n})^n-1=O(\frac{1}{n})$.

When I solved a problem, I could solve it if I assumed that $(1+\frac{1}{n\log n})^n-1=O(\frac{1}{n})$ I tried to prove it, but I failed. Actually, I don't convince if it is true. Is it correct? If ...
3
votes
1answer
230 views

Is this the way to estimate the amount of lucky twins?

To estimate the amount of prime twins between $3$ and $x$ we just take $x \prod_{p}(1-2/p)$ where $p$ runs over the primes between $3$ and $\sqrt x$. Lucky numbers are similar to prime numbers. Does ...
3
votes
1answer
162 views

A strange quantum potential: $V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$

So I have a strange quantum potential I have been playing with: $$V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$$ where $\mu$ is the Möbius function. This is what ...
4
votes
2answers
194 views

How to show $e^{2 \pi i \theta}$ is not algebraic.

I was wondering if someone could possibly help me figure out how to show $e^{2 \pi i \theta}$ is not algebraic when $\theta$ is irrational. Thanks!
2
votes
1answer
82 views

What error bound would an epsilon closer to the Riemann hypothesis give?

$s=1$ line gives: $$\psi(x) = x(1+o(1))$$ classical zero free region gives: $$\psi(x) = x + O(x e^{-c \sqrt{\log x}})$$ for some positive constant $\delta$ RH gives: $$\psi(x) = x + ...
1
vote
1answer
179 views

Applications of prime-number theorem in algebraic number theory?

Dirichlet arithmetic progression theorem, or more generally, Chabotarev density theorem, has applications to algebraic number theory, especially in class-field theory. Since we might think of the ...
8
votes
2answers
317 views

Intuitive explanation with rigorous details why zeta has infinitely many zeros?

I have seen a proof outline that $\zeta$ has infinitely many zeros on the critical line here but what I really want is: Simplest possible (least "magic") argument that explains why zeta has ...
10
votes
5answers
1k views

Why does zeta have infinitely many zeros in the critical strip?

I want a simple proof that $\zeta$ has infinitely many zeros in the critical strip. The function $$\xi(s) = \frac{1}{2} s (s-1) \pi^{\tfrac{s}{2}} \Gamma(\tfrac{s}{2})\zeta(s)$$ has exactly the ...
37
votes
2answers
4k views

Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. The following are exluded: Books by mathematical ...
1
vote
1answer
164 views

Don't understand a bound on Dirichlet L function for principal character

$s= \sigma + it$ is any complex number with real part $> 0$. This came up because $L(s,\chi) = \zeta(s)\prod_{p | q} (1-p^{-s})$ and I have a bound for zeta I want to change to a bound for $L$ ...
4
votes
1answer
178 views

The Riemann zeta function: Didn't Dirichlet get there first?

I took some notes on the historical use of zeta functions in number theory here Why the zeta function? but from looking up the dates I didn't realize something: I remember reading that its not called ...
4
votes
1answer
129 views

How to derive the class number formula?

What's a simple way to derive the class number formula, if it is simpler I only need it for quadratic fields: $$ \lim_{s\to 1} (s-1)\zeta_K(s)=\frac{2^{r_1}\cdot(2\pi)^{r_2}\cdot h_K\cdot ...
9
votes
1answer
281 views

Is there a $k$ such that $a_n=\frac{n^k!}{(n^k!!)^2}$ converges?

Lately I have been playing around with the sequence $$a_n(k) := \frac{n^k!}{(n^k!!)^2}.$$ For $k=1$, it does not look much like it converges. I don't know $k=2$ it converges, but it doesn't really ...
5
votes
1answer
210 views

What is the analytic continuation of a multifactorial?

The $\Gamma$ function is the analytic continuation of the factorial function. Is there a similar analog for multifactorials? I am particularly interested in the double factorial. All Google has ...
19
votes
0answers
532 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = ...
3
votes
0answers
77 views

A proof concering $\Re(\log\zeta(\sigma+it))$

I have been trying to prove that $$\Re(\log\zeta(\sigma+it))=\sum_{n=2}^\infty\frac{\Lambda(n)}{n^\sigma\log n}\cos(t\log n),$$ but now I've given up, so I looked up the answer in the back of the ...
2
votes
1answer
49 views

A number-theoretical estimation-inequality

I have some trouble understanding the following number-theoretical estimation: $$\sum_{k\le \sqrt{n}} (1-k^2/n)^{1+o_n(1)}=n^{1/2+o(1)} \ (n\to\infty),$$ where $o_n(1)$ denotes a $o(1)$ function ...
5
votes
1answer
118 views

Infinitely many primes of the form $p = a + qb$?

Is there a proved result that establishes the status of the following. Are there infinitely many primes in the progression $a + qb$ where $(a,b) = 1$, not both odd, and $q$ ranges over all ...
6
votes
1answer
517 views

Generating function for the divisor function

Earlier today on MathWorld (see eq. 17) I ran across the following expression, which gives a generating function for the divisor function $\sigma_k(n)$: $$\sum_{n=1}^{\infty} \sigma_k (n) x^n = ...
1
vote
1answer
197 views

Dirichlet Characters modulo $260$

I want to count the number of Dirichlet characters with given properties: Number of Dirichlet characters modulo $260$ Number of quadratic Dirichlet characters modulo $260$ Number of primitive ...
5
votes
1answer
641 views

Möbius function sum

If gcd(a,b)=1, $1\leq b\leq a$, and $\mu(k)$ is the Möbius function, what is $$\sum_{k=0}^\infty\frac{\mu(ak+b)}{(ak+b)^s}$$ Can it be expressed in terms of other functions? Can I get it in the form ...
5
votes
1answer
183 views

When a number is a square in the p-adic rationals - proof question (Quadratic Residues)

I'm a little stuck with the proof of a theorem I'm trying to understand. The theorem is as follows: "For odd prime $p$, suppose for $\alpha \in Q_{p}$ (the p-adic rationals) that $|\alpha|_p=1$. Then ...
0
votes
1answer
282 views

Evaluating the sum of $\omega(n)$ in an arithmetic progression [closed]

Let $\omega(k)$ count how many distinct prime factors k has, Then I can prove that for any coprime integers $a,b$ $$\lim_{n\to\infty}\frac{\sum_{k=2}^n\omega(ak+b)}{\sum_{k=2}^n\omega(k)}=1$$ Does ...