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

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0answers
8 views

Low bound of Dirichlet eta function

every one. Suppose that $\eta(s)$ is Dirichlet eta function, I may find a low bound of this function, namely $\eta(2n)>\frac{2^{2n-1}-2}{2^{2n-1}-1}$ with $n>1$ and $n$ is a integer. But is ...
4
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2answers
53 views

Squares modulo 2^n

How many squares are there modulo $2^n$? If we would deal with $p^n$, where p an odd prime, then we could use Hensel's Lemma, which clearly doesn't work with $2^n$.
4
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2answers
73 views

What does this $\asymp$ symbol mean? (subject: analytic number theory)

I'm reading a survey article by Andrew Granville on analytic number theory. On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot ...
2
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1answer
21 views

Error term of the prime number theorem in arithmetic progressions

It is known that if $(a, q)$ and $q\le (\ln x)^N$, then the following is true $$\sum_{k\le x, k\equiv a\pmod{q}}\Lambda(k) = \frac{x}{\phi(q)} + O(x\exp(-C\sqrt{\ln x}))$$ where $C$ depends only on ...
2
votes
1answer
47 views

Rewriting $\tau(p)\Delta(\tau)$ when $p$ is prime

$p$ is a prime, and $\tau$ is Ramanujan's tau function: $$p^{11}\Delta(p\tau)+\frac{1}{p}\sum_{k=0}^{p-1}\Delta\bigg(\frac{\tau + ...
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2answers
52 views

Apostol - Analytic Number Theory, Chapter 3 problem 4a

The problem comes from "Introduction to Analytic Number Theory" by Tom M. Apostol, Chapter 3, Problem 4a: Question: Prove $\sum_{n \le x} \mu(n)[ \frac xn]^2 = \frac{x^2}{\zeta(2)} + O(x \log(x))$ ...
1
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1answer
30 views

Asymptotic result about analytic number theory

I don't know if there is any done work done about ehis matter, and I don't have access to research news. I'm interested in this question (I haen't tried to answer it myself, but it seems very ...
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1answer
37 views

Rational vs irrational

If two points on a number line is shown, are rational numbers between the two points is more or irrational number is more ? I have tried using probability , my collegue who was like my teacher also ...
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0answers
35 views

Generalizations of results on the sum of divisors function over $\mathbb{Q}$ to number fields

Consider the sum of divisor function $$ \sigma_0(n) = \sum_{d\mid n} 1. $$ This is known to satisfy $\sum_{n\leq x} \sigma_0(n) = (x\log x)+2\gamma x+\mathcal{O}(\sqrt{x})$. If, instead, we shift the ...
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2answers
66 views

Riemann zeta, why are the residues either zero or one?

One more question, probably equally simple to answer but I don't know how this is true either: Why is the residue of Riemann zeta zero - trivial or non-trivial: $$\text{residue}\left(\zeta ...
1
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1answer
22 views

Prime Counting: Relationship between Chebyshev's function and the Prime counting function

How do I show that if $\psi(x)=x+O(x^{1/2}\log^2(x))$ then $\pi(x)=\int_2^x \frac{dt}{logt} + O(x^{1/2}\log x)$ Where $\psi(x)$ is Chebyshev's second function and $\pi(x)$ is the prime counting ...
2
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1answer
48 views

Pole of Riemann zeta and Riemann zeta zeros, prove this relation.

Prove this relation: $$\displaystyle \lim_{s\to 1} \, \left(\zeta (s)-\frac{\zeta '(s-1+\rho _n)}{\zeta \left(s-1+\rho _n\right)}\right)=\gamma -\frac{\zeta ''(\rho _n)}{2 \zeta '(\rho ...
0
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1answer
21 views

Equation with a sum for the prime-counting function involving the Mobius function

I have come across the statement that $$ \sum_{n\leq x}\sum_{d\mid(n,P_z)}\mu(d) = \sum_{d\mid P_z}\mu(d) \left[\frac{x}{d}\right], $$ where $P_z=\prod_{p\leq z}p$ where $p$ is prime, $\mu(d)$ is the ...
1
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0answers
15 views

Relation between Dirichlet convolution and Bell series and convolution of functions and the Fourier transform?

We define the Dirichlet convolution of two arithmetical functions $a,b:\mathbb{N}\to\mathbb{C}$ to be $$ (a*b)(n)=\sum_{d\mid n}a(d)b\left(\frac{n}{d}\right). $$ Given a prime $p$, we define the Bell ...
2
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1answer
28 views

Sums Involving the Mobius Function

Are there any good approximations for the following sums in terms of $n$? $$\sum_{k=1}^{n}\mu(k)$$ $$\sum_{k=1}^{n}\mu(k)\log^m(k)$$ $$\sum_{k=1}^{n}\frac{\mu(k)}{k}.$$ I realize that the third sum ...
3
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1answer
61 views

Can the exact value of the product over the Riemann zeta function at even arguments be evaluated?

According to wolframalpha, the product over the Riemann zeta function at even arguments converges : $$\prod_{n=1}^{\infty} \zeta(2n) \approx 1.82 $$ Q1: Can it be proved that this product actually ...
2
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1answer
25 views

Numerical verification of the ternary Goldbach conjecture

In his proof of the ternary Goldbach conjecture, H.A. Helfgott says that it has been verified that every odd number less than $N_0 = 10^{30}$ is the sum of at most 3 primes. How would one verify this ...
2
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0answers
29 views

Zero Free Regions of Zeta'

I'm interested in calculating all of the zeroes of the first derivative of the Riemann $\zeta$ function. I know that (on the RH), all of these zeroes will have real part $\geq \frac{1}{2}$. I am ...
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1answer
40 views

Derivation of a generalization of Mertens' Third Theorem.

One of Mertens' Theorems states $$\prod_{p\le x}(1-\frac{1}{p})\sim \frac{e^{-\gamma}}{\ln(x)}.$$ I have seen a generalized version that states $$\prod_{m<p\le x}(1-\frac{m}{p})\sim ...
4
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0answers
91 views

Multiplicative subgroup of a finite field with prescribed trace.

Any suggestions/methods/estimates for the following problem would be very appreciated. $l,p$ are primes with $p \equiv 1 \!\! \pmod l$. $r$ is a positive integer with $r \equiv 1 \!\! \pmod p$ and $q ...
0
votes
1answer
33 views

Sum of Euler Phi equalities

Show: $\sum_{n\le x} \phi(n) [\frac{x}{n}] = \sum_{n \le x} \sum_{m\le \frac{x}{n}} \phi(m)$ I know the left most sum boils down to $\sum_{n\le x} n$. If we know that $m|\frac{x}{n}$ then we know ...
2
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0answers
54 views

Good book on analytic continuation?

For my Bachelor's thesis, I am investigating divergent series summation methods. One of those is analytic continuation. There are quite a few books on complex analysis that include a chapter or two on ...
1
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1answer
30 views

Euler totient function sum of divisors. Theorem 2.2 Apostol

Prove that : $If $ $ n\ge{1} $ $ \sum_{d|n}\phi(d)=N $ $ N \in{\mathbb Z} $ Let S denote the set {1,2,...,n}. We distribute the integers of S into disjoint sets as follows. For each divisor d ...
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0answers
74 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
1answer
33 views

Proof of convergence of $L'\left(1,\chi\right)$

can someone give me a good reference for a clear proof of the convergence of $L'\left(1,\chi\right)$, $\chi$ real-valued, non-principal Dirichlet character? Thanks in advance.
1
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1answer
25 views

Question about Abel summation

Now in this sum. $A(y) =\sum_{n=1}^y a_n$ or $A(y) =\sum_{n=0}^y a_n$ Which one is true where does n starts?
1
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1answer
42 views

Analytic proof of quadratic reciprocity

Is there any proof of quadratic reciprocity that is more analytic than those described on Wikipedia (http://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity)?
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0answers
109 views

What are going to change of our view if $\pi+e$ is a rational? [closed]

It is well known that there's no conclusion now whether $\pi+e$ is a rational or not. Just for curiosity, what will happen if we know the answer?
0
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0answers
28 views

Evaluation of certain trigonometric sums

In trying to approximate the number of solutions to the equation $3^n - 2 = k^2$ for positive integers $n, k$, I tried to use the circle method. In doing so, I had to bound the trigonometric sum for ...
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1answer
41 views

Questions about the proof that every odd integer is the sum of 5 primes

In http://arxiv.org/pdf/1201.6656.pdf, Tao proved that all odd numbers greater than 1 are the sum of 1, 3, or 5 primes. In page 36-37, he uses the fact that for all $x > 1.1\times10^{10}$, there ...
2
votes
1answer
44 views

Explanation for a theorem pertaining on Dirichlet character sums

A very well known theorem pertaining on Dirichlet characters sums states that if $\chi$ is a Dirichlet character modulo $k$, defining $$ A\left(n\right)=\sum_{d\mid n}\chi\left(d\right) $$ Then ...
3
votes
2answers
144 views

Find the number of series

Find the number of series $(a_1,..., a_{2n})$ that have terms from ${\{0,...9\}}$ so that: $$ 11|\sum_{i=1}^{n}a_i-\sum_{i=n+1}^{2n}a_i $$ (this is not a homework) There is a similar problem ...
3
votes
1answer
35 views

Lower bound on certain exponential sums and expressions related to them

Let $$G(\alpha, x) = \sum_{n\le x}e(\alpha n^2)$$ Clearly, $r_k(n)$, the number of representations of a number as the sum of $k$ squares is given by the following expression: $$r_k(n) = \int_0^1 ...
1
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0answers
37 views

Factorization Patterns for Ideals

Let $K/\mathbb{Q}$ be a Galois Number field. Let $p$ be an unramified rational prime. In this extension, for any $P,Q | p\mathcal{O}_K$ then the relative degrees $f(P) := [\mathcal{O}_K/P : ...
1
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1answer
35 views

Tightest constant factor for error term of the prime number theorem

What are the best known (unconditional) bounds on the following: $$\mid\psi(x) - x\mid$$ (With a known constant factor)
5
votes
1answer
47 views

Intuitive basis of Mobius inversion?

If we're given $f(n)= \sum_{d|n}g\left(\frac{n}{d}\right),n \in \mathbb{N},$ then Mobius inversion gives $$g(n)=\sum_{d|n}\mu \left( d\right) f \left( \frac{n}{d}\right).$$ Also, the generalised ...
4
votes
1answer
32 views

Precise Error Term in Chebotarev's Theorem

Let $K/\mathbb{Q}$ be a Galois Number Field with Galois group $G$ and discriminant $\Delta_K$. Chebotarev's theorem states that the number of (unramified) rational primes with Frobenius conjugacy ...
2
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1answer
74 views

Möbius function [duplicate]

For any natural number $x$, determine the sum; $$ \sum_{\substack{ n\leq x }} \mu(n)\left\lfloor \frac{x}{n} \right\rfloor.$$ (Hint: Use $\lfloor x \rfloor=\sum_{\substack{ k\leq x }}1.$) ...
0
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0answers
25 views

$g(x) = \sum_{m=1}^{\infty}f(mx)$ if and only if $f(x) = \sum_{m=1}^{\infty}\mu(m)g(mx)$

This is Problem 1.1.10 from book Problems in analytic number theory by Ram Murthy. It says, given the condition $$ \sum_{k=1}^{\infty} d_3(k)|f(kx)| < \infty $$ where $d_3(k)$ denotes the number of ...
5
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1answer
55 views

Sum of a certain series related to the primes

It is well known that $$\sum_{n > 0}\frac{1}{n}$$ diverges, but $$\sum_{n > 0}\frac{1}{n^2} = \frac{\pi^2}{6}$$ converges. Similarly, $$\sum_{p}\frac{1}{p}$$ diverges, but $$\sum_{p} ...
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1answer
43 views

estimation of sum over primes

The problem is this: I have read that if i have a function $f: \mathbb{N}\to\mathbb{C}$ and we are interested in estimation of the sum $V(x)=\sum_{p<x} f(p)$ where $p$ runs on primes numbers then ...
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0answers
11 views

Fourier transform of a quasi character on a local field

Why is the twice iterated fourier transform of a quasi-character $c$ on local field $k$, $c$ itself? That is, why is $\hat{\hat c}(\alpha) = c(\alpha)$? In general, when we apply the fourier transform ...
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0answers
10 views

When is the fourier transform of a quasi-character $\hat c(\alpha)=|\alpha|c^{-1}(\alpha)$?

This is from lemma $2.4.2$ of Tate's thesis. Let $c$ be a quasi-character on $k^{*}$, the multiplicative group of a number field completed at a non-archimedian place. Lemma 2.4.2 For $c$ in the ...
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0answers
19 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 ...
4
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1answer
101 views

Sum of squares function usually larger than $(\log x)^{1/2 - \epsilon}$?

Let $r_2(n)$ be the sum of squares function, i.e. the number of different pairs $a,b\in \mathbb{Z}$ such that $a^2 + b^2 = n$. Let $R$ be the set of representable integers, i.e. the subset of ...
0
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1answer
43 views

Embedding into $p$-adic complex numbers

As I'm reading notes about the Leopoldt conjecture, the following question came to my mind: Let $\mathbb{C}_p$ be the $p$-adic complex numbers, i.e. the completion of the algebraic closure of the ...
5
votes
1answer
104 views

Riemann Zeta Function Non-Vanishing on the Line $\mathrm{Re} \; z = 1$

The result quoted in the title is usually a stepping stone in the proof of the prime number theorem and I am familiar with the usual argument for this result. The other day my professor was telling ...
1
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0answers
26 views

Trouble Identifying a “Psi” Function in Number Theory

In these lecture notes on number theory I am reading I came across the notation $\Psi(e^t;a,q)$ in connection with the Dirichlet theorem on arithmetic progression. I was hoping someone could help me ...
2
votes
1answer
44 views

Continuation of the Riemann Zeta Function

I am actually aware of the argument showing $\zeta$ has a meromorphic extension to $\mathbb{C}$ with a single pole at $z = 1$. On a recent number theory exam, however, one of the questions asked to ...
1
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2answers
67 views

Question on product of primes

How to prove the following result: $$\prod_{i=1}^{n}P_i=\frac{2^{(P_n+3)/2}}{\sqrt{\pi}} \gamma (1+P_n/2) \cdot \frac1R$$ where $R$ is the product the odd composite natural numbers less than $P_n$ ...