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

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5
votes
1answer
365 views

Residue of Rankin-Selberg L-function for non-trivial nebentypus

Let $f\in S_k(\Gamma_0(N),\chi)$ be a normalized holomorphic newform (i.e. weight $k$, level $N$, nebentypus $\chi$) and write its Fourier expansion as $$ f(z)=\sum_{n\ge 1} ...
15
votes
4answers
737 views

A closed form for the sum $\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$

How can I find a closed form for the following sum? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
4
votes
2answers
244 views

How to proceed doing number theory?

I'm an undergrad majoring in mathematics. Being in first year I'm still exploring new branches of mathematics and till now, It is analysis and Number theory that I've come to have a great interest ...
4
votes
2answers
494 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 ...
0
votes
0answers
32 views

Montgomery&Vaughan's Multiplicative number theory theorem 13.3

I can't understand well the proof of theorem 13.3 There exist a constant $C>0$ s.t. if RH is true, then for every $x\ge 2$ the interval $(x,x+Cx^{1/2}\log x)$ contains at least $x^{1/2}$ prime ...
1
vote
1answer
55 views

the average order of divisor function

In Analytic number theory by Apostol there's a theorem: $$\sum_{n\le x} \sigma(n)= \frac{1}{2} \zeta(2)x^2 + O(x\log x)$$ and then it claims that because we know that $\zeta (2)= \frac{\pi^2}{6} $ ...
1
vote
1answer
41 views

an exercise from apostol analytic number theory

this is the second exercise of chapter 3: if $x\ge 2$ prove that $$\sum_{n\le x} \frac{d(n)}{n}= \frac{1}{2} {\log^2 x} + 2C\log x +O(1)$$ where $C$ is Euler's constant. here's what i've done to ...
1
vote
1answer
34 views

If $\zeta$ is a function of characters what does it mean for it to be regular?

This is from lemma 2.4.1 of Tate's thesis. Lemma 2.4.1: A $\zeta$-function is regular in the "domain" of all quasi-characters of exponent greater than $0$. proof: We must show that for each ...
7
votes
1answer
242 views

Notation in Terry Tao's exposition on the PNT

The exposition I'm talking about can be found here (page 6): http://www.math.ucla.edu/~tao/preprints/Expository/prime.dvi Essentialy, Tao proves the prime number theorem in the elementary way, ...
3
votes
1answer
50 views

Identities for L-series and Euler product

It is a mabe a stupid question for many experts here. There is something wrong in the following reasoning, and now I could not find it. Could someone help me out? Any advice will be highly ...
1
vote
1answer
32 views

Rearrangements of Dirichlet Eta Function

I was wondering if explicit closed forms for rearrangements of $\eta(s)$, for $s$ such that the series is not absolutely convergent, are useful in studying the Dirichlet $\eta$ function. I am asking ...
1
vote
1answer
68 views

Sum of reciprocals of primes for known primes.

I was reading through some old analytic number theory notes earlier and found the interesting fact that even though $\sum\frac{1}{p}$ diverges: $\sum_{\text{known primes}}\frac{1}{p} < 4$. ...
1
vote
1answer
60 views

Congruences of weights of modular forms modulo primes

I'm trying to prove that for two modular forms $f$ and $g$ of weight $k$ and $k'$ respectively, that are congruent modulo a prime $\ell\ge 5$, their weights are congruent modulo $\ell-1$. This is what ...
0
votes
0answers
21 views

Gamma function whose argument is a reciprocal power with integer base and exponent

Consider the analytic continuation of the factorial function $n!$ given by $\Gamma(z)$ (note $n!=\Gamma(n+1)$), and suppose $z=a^{-n}$, where $a,n\in\mathbb{N}$ are positive integers. Are there any ...
7
votes
3answers
578 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 ...
0
votes
0answers
92 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
28
votes
1answer
1k views

Are these zeros equal to the imaginary parts of the Riemann zeta zeros?

Edit 8.8.2013: See this question also. The Fourier cosine transform of an exponential sawtooth wave times $e^{-x/2}$: $$\operatorname{FourierCosineTransform}(\operatorname{SawtoothWave}(e^x)\cdot ...
1
vote
0answers
36 views

Riemann functional equation question?

I was looking through the derivation of the Riemann functional equation, and I understand how to obtain the result $$ \pi^{-\frac s2} \Gamma (\frac s2) \zeta(s) = \pi^{-\frac{1-s}{2}} ...
0
votes
1answer
12 views

Given $d$, for how many $m$'s is $d$ a quadratic residue mod $m$?

Let $d$ be a fixed, square-free integer, and let $M$ be some very large number. I would like to count the numbers $m \leq M$ such that $m \perp d$ and $d$ is a quadratic residue modulo $m$. Call this ...
4
votes
1answer
72 views

Analytic Continuation of Zeta Function using Bernoulli Numbers

In my complex analysis textbook by Stein and Shakarchi, as an exercise, I am supposed to extend $\zeta(s)$ to the entire complex plane using Bernoulli numbers, but I am stuck. I can prove that $$ ...
1
vote
1answer
34 views

Transformation property for classical Siegel modular forms of weight 2

Let $\mathbb{H}_g = \{ \tau \in GL_g(\mathbb{C}) | \; {^t\tau} = \tau, Im(\tau) >0\}$ be the Siegel upper half space. There are Eisenstein series $$ E_{2k}(\tau) := \sum_{\gamma\in (P_0\cap ...
9
votes
3answers
542 views

Equivalence to the prime number theorem

I was just reading this question and answer: How will this equation imply PNT and it raised a whole new question: Given that $\sum_{n\le x} \Lambda(n)=x+o(x)$, prove that $$\sum_{n\le x} ...
0
votes
0answers
34 views

Riemann Zeta Function at Real Values of the variable s

My question is: Is the Riemann Zeta function for real values of $s$ $( s = \sigma + 0\,i)$ a monotone function of $\sigma \,$?
2
votes
0answers
39 views

Tau Summatory Function

It is well known that the divisor summatory function can be calculated in $O(x^{1/2})$ via $$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^{\lfloor \sqrt{x}\rfloor} \lfloor\frac{x}{k}\rfloor - \lfloor ...
15
votes
1answer
226 views

Are there infinite many $n\in\mathbb N$ such that $\pi(n)=\sum_{p\leq\sqrt n}p$?

Are there infinite many $n\in\mathbb N$ such that $$\pi(n)=\sum_{p\leq\sqrt n}p,\tag{1}$$ where $\pi(n)$ is the Prime-counting_function? For example, ...
2
votes
0answers
75 views

Is this the chord G Major I am hearing as base tones from interference of zeta zeros times eigenvalues of the von Mangoldt function matrix?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: ...
1
vote
0answers
43 views

Wolstenholme Number

Does Wolstenholme Numbers have perfect squares other than 1 and 49? The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749 seems to be a complicated problem
22
votes
1answer
2k 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. Here is my list: The Riemann Hypothesis: A Resource ...
1
vote
1answer
32 views

Twin primes : prove the convergence of $ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} (\log(p)+\frac{1}{p})²$

let $T$ be the twin primes set : $p \in T $ if and only if $ p$ and $p+2$ are primes. Can you help me establish the convergence of : $$ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} ...
5
votes
2answers
169 views

least common multiple $\lim\sqrt[n]{[1,2,\dotsc,n]}=e$

The least common multiple of $1,2,\dotsc,n$ is $[1,2,\dotsc,n]$, then $$\lim_{n\to\infty}\sqrt[n]{[1,2,\dotsc,n]}=e$$ we can show this by prime number theorem, but I don't know how to start I ...
2
votes
0answers
31 views

Growth rate of arithmetical function

I'm interested in how one would estimate the growth rate of $$f(n)=\sum_{k\le n}\mu^2(k)\log(k)$$ I.e. sum of logarithms of square free integers. I can think of some trivial methods in my head ...
0
votes
0answers
24 views

A question on the big-O value of the complex integral especially in the number theory

My question is quite simple and elementary. Let $A(x)=\sum_{1}^{x}a(n)$ and $\alpha(s)=\sum_{1}^{\infty}a(n)n^{-s}$. Then, as we know, $$ A(x)= ...
1
vote
1answer
46 views

Problem from Apostol's analytic number theory book

Im trying to solve the exercise 13.2 in Apostol's analytic number theory book: Let $A(x)=\sum_{n\leq x}a(n)$, where $a(n)$ is zero unless $n=p^k$ for some prime $p$, in that case $a(n)=1/k$. Prove ...
1
vote
1answer
33 views

Positive Integer points of $f(x)=\frac{1}{c-\frac{1}{x}}$, where c is fixed

So I am looking for the integer solutions of $f(x)=\frac{1}{c-\frac{1}{x}}$ for fixed $c\in \mathbb{Q}$ i.e. points $(x,f(x))\in \mathbb{N}\times \mathbb{N}$. (The c equals $\frac{4}{n}-\frac{1}{k}$ ...
10
votes
2answers
181 views

Numbers divisible by the square of their largest prime factor

Let $p(n)$ be greatest prime factor of $n$, denote $A=\{n\mid p^2(n)\mid n,n\in \mathbb N\}.$ $A=\{4,8,9,16,18,25,27,32,36,49,50,\cdots\},$ see also A070003. Define $f(x)=\sum_{\substack{n\leq ...
1
vote
2answers
107 views

A set with zero density

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that $$n \mid a^{f(n)}-1.$$ Prove that $S$ has density $0$; that ...
-1
votes
1answer
60 views

Fermat Last theorem on Poly-Euler numbers

The poly-Euler numbers, denoted as $E_{n}^{(k)}$, are defined by the following generating functions :$${2\operatorname{Li}_k(1-e^{-x}) \over 1+e^{-x}}=\sum_{n=0}^\infty E_n^{(k)}{x^n\over n!}$$ The ...
1
vote
1answer
124 views

how to find an upper bound for this series

I don't understand why $$\left|\sum_{b<y}\frac{\mu(b)\rho(bc)}{b}\right|\ll\tau(c)log(y)^{-A}$$ with $A>1$, where $\mu(b)$ is the mobius function and $\rho(n)$ is the number of solution of ...
0
votes
0answers
25 views

Riemann's hypotesis and some equivalences [duplicate]

Use that $\psi(x)=x+O(\sqrt{x}\:log^{2}x)$ to show that $\pi(x)=li(x)+O(\sqrt{x}\:logx)$ where $li(x)=\int_{2}^{x}\frac{dt}{logt}$ I tried but I get confused. Many pdf's say it's very easy to show. I ...
1
vote
1answer
50 views

On Newman/Zagier's proof of PNT

I have just got this paper: http://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf and I have a serious doubt: When proving that soft Tauberian theorem he explicitly uses ...
0
votes
0answers
25 views

Zeta Riemann Function

Use that $\zeta(s)=1+\frac{1}{s-1}-s\int_{1}^{\infty}\frac{\left\{u\right\}}{u^{s+1}}du$ if $Re(s)>0$ to show that. 1) $\zeta(s)=s\int_{1}^{\infty}\frac{\left[ u\right]}{u^{s+1}}du$ for ...
1
vote
1answer
37 views

Based on prime number theorem

I have a problem... If $A(x)=\sum_{n\leq x}a(n)$ where $a(n)=\frac{1}{k}$ if $n=p^{k}$ and $a(n)=0$ in other case, show that $A(x)=\pi(x)+O(\sqrt{x}\:log\:logx)$ I think I should use the theorem of ...
3
votes
0answers
54 views

Application of Dirichlet Theorem in AP to elementary number theory problems.

I have learnt this theorem in my class, however, "elementary" examples are very limited. (focusing more on analytic machinery) Are there any interesting applications to elementary number theory that ...
0
votes
1answer
34 views

Summation formula in dimension 2

One of the most common tools in analytic number theory is the summation by parts, my question is what is the similar formula when we are, for example, in dimension two and we have the sum $$ ...
0
votes
0answers
54 views

Finding an asymptotic formula for $f(m,n)=\sum_{\substack{d\mid m \\ d\leq n}}1$?

$$f(m,n)=\sum_{\substack{d\mid m \\ d\leq n}}1$$ Here $n<m$ and $m$, $n$ are positive integers.
0
votes
3answers
115 views

Calculating the class group of $\mathcal{O}_K$, for $K=\mathbb{Q}(\sqrt{7})$?

How to calculate the class group of $\mathcal{O}_K$, for $K=\mathbb{Q}(\sqrt{7})$ without using the Minkowski bound?
0
votes
1answer
20 views

bounding gaps between points in an interval

I've been reading Davenport's Multiplicative Number Theory and came across something that I didn't understand. On p. 108, there is an argument for finding a lower bound on the imaginary parts $\gamma$ ...
4
votes
1answer
34 views

Identities of Hecke operators

While studying, I recently came across the following interesting problem. Let's say that the (level one) weight $k$ modular forms $M_k(\Gamma(1))$ have dimension $d$. We know by the ring structure ...
1
vote
3answers
160 views

Basic Questions about the 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. ...
5
votes
1answer
252 views

A general explicit formula for the generalized divisor summatory function?

Mertens function has, by residues, an explicit formula of $M(x)=\displaystyle\sum_{\rho}\frac{x^\rho}{\rho\zeta'(\rho)}-2+\sum_{n=1}^\infty\frac{(-1)^{2 n}(2\pi)^{2n}}{(2n)! n \zeta(2n+1)x^{2n}}$ ...