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

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8
votes
2answers
315 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
163 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
174 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
127 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
204 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
525 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
116 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
502 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
195 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
631 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
178 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
276 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 ...
7
votes
1answer
1k views

Why the Riemann hypothesis doesn't imply Goldbach?

I'm interested in number theory, and everyone seems to be saying that "It's all about the Riemann hypothesis (RH)". I started to agree with this, but my question is: Why then doesn't RH imply the ...
8
votes
2answers
254 views

Structure of the group of arithmetic functions

This question was originally posted in Elements of finite order in the group of arithmetic functions under Dirichlet convolution. and it goes as follows: Let G be the group consisting of all ...
4
votes
3answers
481 views

Number of representable as sum of 2 squares

How to find asymptotically (or some reasonable bound, at least $ o(n) $) number of numbers, representable as a sum of squares of 2 numbers? (in case of bound I am interested in both: lower and upper ...
1
vote
2answers
180 views

Sum of Stieltjes constants

Does anyone know of any papers or resources dealing with the following question: For which values of $s=\sigma+it$ does the following sum of Stieltjes constants hold, ...
3
votes
1answer
179 views

Vonmangoldt sums

The dirichlet series for the Vonmangoldt function, $\Lambda(n)$, which is equal to zero when $n$ is not a prime a power, and $\ln(p)$ when it is a prime power say, $n=p^j$, is ...
7
votes
1answer
241 views

Prime number sum

Let $p$ denote a prime, and let $\{x\}$ denote the fractional part of $x$. Suppose that the following statement is true for all non-integer real numbers $x$: $$\lim_{n\to\infty}\frac{\sum_{p\leq n}^\ ...
4
votes
1answer
905 views

What are dirichlet characters?

What are dirichlet characters? I don't really understand the definitions given by wikipedia or wolframalpha, are they defined to be partially multiplitictive just because that gives them, an euler ...
2
votes
1answer
109 views

Dirichlets theorem on primes

Is there a proof of dirichlets theorem that does not require complex analysis?
4
votes
1answer
334 views

Von mangoldt function dirichlet series

The dirichlet series for the Vonmangoldt function, $\Lambda(n)$, which is equal to zero when $n$ is not a prime a power, and $ln(p)$ when it is a prime power say, $n=p^j$, is ...
7
votes
2answers
972 views

Derivative of the Riemann zeta function for $Re(s)>0$.

The Riemann zeta function can be analytically continued to $Re(s)>0$ by the infinite sum $$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}.$$ Can we differentiate this with ...
9
votes
2answers
447 views

Prove that $\sum_{n=1}^\infty \mu(n) \log n/n =1$.

One can show that the Prime Number Theorem is equivalent to the statement $$ A(x):= \sum_{n \leq x} \frac{\mu(n)}{n}=o(1),\qquad \qquad (1)$$ i.e. that $A(x) \to 0$ as $x \to \infty$. Given that the ...
7
votes
1answer
237 views

Von Mangoldt function estimate

In many treatments of Vinogradov's three prime theorem, one considers the summation $S(\alpha) = \sum_{k \leq N}\Lambda(k)e^{2\pi i\alpha k}$ in place of $T(\alpha) = \sum_{p \leq N}e^{2\pi i\alpha ...
2
votes
1answer
147 views

Proving $\sum\limits_{n=1}^\infty\frac{|\mu(n)|}{n^s}=\frac{\zeta(s)}{\zeta(2s)}.$

I am trying to show that $$D(|\mu|,s)=\sum_{n=1}^\infty\frac{|\mu(n)|}{n^s}=\frac{\zeta(s)}{\zeta(2s)}.$$ By a previous exercise I know that $D(\lambda,s)=\zeta(2s)/\zeta(s)$, where $\lambda$ is ...
4
votes
2answers
560 views

Bounds on a sum involving the Möbius function

In Apostol's Analytic Number Theory, Apostol defines $$A(x):= \sum_{n \leq x} \frac{\mu(n)}{n}$$ and proves that $A(x)=o(1)$ implies the Prime Number Theorem, by showing that ...
2
votes
1answer
98 views

Why does a certain equality in Iwaniec's Topics in classical automorphic forms hold?

I have been reading the book Topics in classical automorphic forms for a while, where I encountered a formula in the middle of Sec 8.6 (p143) $$F(z;\beta,\gamma)=\frac ...
1
vote
3answers
100 views

Questions about assigning a probability to a randomly chosen large integer $n$ being prime

I heard this question a few days ago, so reciting from memory: If I were to randomly choose an arbitrarily large positive integer $n$, could I write a function that determines the likelihood of it ...
3
votes
0answers
202 views

existence of closed forms of certain Dirichlet series

In the following post we find an interesting calculation of $$ \sum_{n\ge 1} \frac{\mu(n)^2}{n\varphi(n)}.$$ I had been trying to do this calculation myself, setting the slightly more ambitious goal ...
0
votes
1answer
116 views

Polynomial that permutes residue classes

Prove that for any integers $d, e > 1$, the polynomial $f$ with integer coefficients permutes the residue classes modulo $p^d$ if and only if it permutes the residue classes modulo $p^e$ where $p$ ...
6
votes
1answer
358 views

Finding an asymptotic for the sum $\sum_{p\leq x}p^m$ [duplicate]

Possible Duplicate: How does $ \sum_{p \leq x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $? What could I use to prove the following conjecture? $ \sum_{p \le x} p^{m} \sim ...
8
votes
1answer
192 views

the constant in the asymptotics of $\sum_{1\le k \le n} \frac{\varphi(k)}{k^2}$

The following thread at math.stackexchange.com proposes a constant term for the asymptotic expansion of $$\sum_{1\le k \le n} \frac{\varphi(k)}{k^2}.$$ I am getting a different term and I would like ...
2
votes
0answers
54 views

Rationality of Polynomial Coefficients. Integral Question.

We are entertaining polynomials with roots, all unique, on the curve $\Upsilon_s = \{{( 1-\cos[\theta])^{-s} \exp(i \theta)} \ | \ \theta \in \mathbb{R} \}$, where $s>0.$ $\Upsilon_s$ looks like a ...
2
votes
1answer
75 views

$ \ \lim_{x\to ∞ }\frac{π(x)} { x^δ} $

Let $\ π(x)$ denote the prime counting function , i.e. the number of primes not exceeding $x$ Then does $$ \ \lim_{x\to ∞ }\frac{π(x)} { x^δ} $$ exist for all real $δ$ $∈ ( 0 , 1 )$
8
votes
1answer
272 views

average order of $\sum\limits_{\substack{1\le k\le n \\ (k,n)=1}} \frac{1}{k}$

Introduce $$\varrho(n) = \sum\limits_{\substack{1\le k\le n \\ (k,n)=1}} \frac{1}{k}.$$ The following thread at math.stackexchange.com proposes to analyse the average order of $\varrho(n)$, i.e. ...
3
votes
0answers
72 views

Schneider's theorem about the transcendence of values of the $j$-function

It is known that the $j$-function takes algebraic values when evaluated at imaginary quadratic integers. This is a result that was proved by Schneider in 1937 apparently. To be precise, Schneider ...
3
votes
2answers
432 views

Euler Maclaurin summation examples?

How does one use Euler Maclaurin to compute asymptotics for sums like $$ \sum_{\substack{n\le x \\ (n,q)=1}} \frac{1}{\sqrt{n}} \quad \text{or} \quad \sum_{\substack{n\le x \\ (n,q)=1}} \frac{\log ...
6
votes
0answers
469 views

Sum of reciprocal of primes in arithmetic progression

In http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf on page 6 (top) the author states that: $$ \sum_{p \le x, \ p \equiv 1 \bmod l} \frac{1}{p} = \frac{\log \log x}{\phi(l)} + O \left ( \frac{\log ...
9
votes
0answers
682 views

Proof of Hardy-Ramanujan inequality in number theory.

On page 3 of http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf the author write that the following inequalities follow from "the Hardy-Ramanujan inequality", but he doesn't point to a proof. The ...
1
vote
2answers
275 views

The product $\sin(\pi s/2)\Gamma(1 - s)$

How to show that $\sin(\pi s/2)\Gamma(1 - s)$ is analytic when $s$ is an even positive integer?
1
vote
2answers
255 views

Is there a way to show that $\sqrt{p_{n}} < n$?

Is there a way to show that $\sqrt{p_{n}} < n$? In this article, I show that $f_{2}(x)=\frac{x}{ln(x)} - \sqrt{x}$ is ascending, for $\forall x\geq e^{2}$. As a result, $\forall n \geq 3$ ...
-1
votes
1answer
105 views

Recursions in decimal expansions of certain integers (OEIS A216407)

http://list.seqfan.eu/pipermail/seqfan/2012-September/010196.html is the link to "The SeqFan Archives" thread "Recursions in decimal expansions" This thread discusses whether there is a ...
9
votes
5answers
1k views

How does one read aloud Vinogradov's notation $\ll$ and $\ll_{\epsilon }$?

How does one read aloud the Vinogradov's notation $\ll$ and $\ll_{\epsilon }$ as in $$f(x)\ll g(x)$$ and $$c\ll_{\epsilon }\left( \prod\limits_{p\mid abc}p\right) ^{1+\epsilon}.$$ Is the first one ...
1
vote
0answers
64 views

modulus of a series

My question is about the possibility of calculating the modulus of the Dirichlet Eta Function for complex numbers with positive real part. This series is uniformly convergent but it is not absolutely ...
4
votes
2answers
648 views

Trends in the distribution of reordered digits of Pi (OEIS A096566)

First let me try to describe in more details below the approach of "reordering" digits of Pi, which is used in OEIS A096566 https://oeis.org/A096566 and what I have done analyzing it so far. I am ...