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

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10 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
24 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
47 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
23 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
28 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
35 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
88 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 ...
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0answers
23 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
51 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 ...
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1answer
29 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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72 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 ...
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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.
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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?
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1answer
40 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
107 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?
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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
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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
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2answers
141 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
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1answer
34 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 ...
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0answers
36 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 : ...
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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
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1answer
46 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
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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
72 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 $\lfloor x \rfloor=\sum_{\substack{ k\leq x }}1.$) ...
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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
votes
1answer
54 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
10 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
8 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
98 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 ...
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1answer
42 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
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1answer
101 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 ...
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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
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1answer
43 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 ...
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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 $P_n$ ...
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1answer
55 views

Rational and trascendental numbers: $\pi$, $e$ and $\pi+e$ [duplicate]

The numbers $\pi,e$ are trascendentals, but if consider: $\pi+e$ then is rational, trascendental? Thanks
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47 views

Convergence of $\sum_{n\leq x} \frac{\chi(n) \Lambda(n)}{n}$

I am trying to prove convergence of certain series related to non-principal Dirichlet series. In the proof, I want to use the following fact: $$ \sum_{n\leq x} \frac{\chi(n)\Lambda(n)}{n} \tag{1} $$ ...
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70 views

Question about the proof of Goldbach's weak conjecture

H.A. Helfgott recently proved Goldbach's weak conjecture here: http://arxiv.org/pdf/1305.2897v2.pdf In (1.1), he explains that he is trying to show that $$\sum_{n_1 + n_2 + n_3 = ...
2
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1answer
42 views

Second part of Eloi's Conjecture

We know that "There exist some real k such that ∀ integer n>1 the integer part of k∗nln(n) is always prime?" is false (prove here Is there a $k$ for which $k\cdot n\ln n$ takes only prime values? ) ...
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0answers
46 views

Integration by parts of the Logarithmic Integral

I am doing work on analytic number theory, and I am currently looking at the Prime Number Theorem, that is $$\pi(x) \sim Li(x)$$ Some of my sources say that I can do integration by parts on the ...
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0answers
27 views

Upper bound on $\zeta(s)$

I'd like to know an upper bound for $\zeta(s)$ in the critical strip, and hopefully one that is not too difficult to prove. For instance, ...
2
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0answers
61 views

Bounding $\sum_{p\leq x} \chi(p )$ for non-principal character $\chi$

Suppose $\chi$ is a non-principal Dirichlet character mod $k$. Let $A(x)=\sum_{n\leq x} \chi(n)$. Since $\sum_{n\leq k} \chi(n)=0$, we easily get the bound $|A(x)|\leq \varphi(k)$ where $\varphi$ is ...
2
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1answer
78 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
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1answer
113 views

Bounding this arithmetic sum

I am interesting in bounding the arithmetic sum $$ \sum_{n \leq x} \frac{\mu(n)^2}{\varphi(n)}$$ (The motivation is that this is a sum that comes up a lot in sieving primes, in particular in the ...
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1answer
69 views

How to make Dirichlet character table modulo $5$

There are four reduced residue classes $\mod 5$, namely $1, 2, 3, 4$ and thus four Dirichlet characters $\mod 5$ since $\phi(5)=4$. I understand how to deduce that the character can be $1$ or ...
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2answers
65 views

Existing Algorithm / Code to calculate exact values of the Riemann Zeta function at even natural numbers?

I wanted to know if there's any existing algorithm to compute exact values of the Riemann Zeta function at even natural numbers? For example, it should compute $\zeta(4)$ as exactly $\frac{\pi^4}{90}$ ...
2
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2answers
50 views

How to determine growth rate of coefficients of generating function

For a given ordinary generating function $f(x)=a_0+a_1x+...$, are there any methods to determine the growth rate of its coefficients based on that of $f$ ? In particular if we are given the extra ...
0
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0answers
92 views

n-th harmonic number when n is prime

the $n$-th harmonic number $H\left(n\right)$ is usually defined by the sum $\sum_{k=1}^{n}\frac{1}{k}$. Now, we know there is no closed form for this number, however, in the Apostol "introduction to ...