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

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1answer
36 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
51 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
111 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
33 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
46 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 ...
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1answer
53 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
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
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1answer
44 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
39 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
36 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
55 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 ...
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1answer
37 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
81 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
27 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
57 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
51 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
13 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
11 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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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 ...
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1answer
117 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
66 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
120 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
31 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 ...
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1answer
67 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
75 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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53 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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75 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
43 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
55 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
34 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
72 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 ...
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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 ...
5
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1answer
127 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
134 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
88 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 $\frac{\pi^4}{90}$ ...
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2answers
69 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 ...
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97 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 ...
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1answer
93 views

Volume of first cohomology of arithmetic complex

Let $K$ be a number field and consider the Arithmentic complex $\Gamma_{Ar}(1)^\bullet$ be defined by $$\begin{array} A\Bbb R^{r_1+r_2} & \stackrel{\Sigma}{\longrightarrow} & \Bbb R \\ ...
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1answer
59 views

Product of the logarithms of primes

I would like to know if there is a result for the product $$f(x)=\prod_{p\leq x}\log p,\quad \text{where $p$ is prime}.$$ A simple upper bound is $f(x)<(\log x)^{\pi(x)}$, where $\pi(\cdot)$ is ...
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1answer
119 views

How to prove the convergence of a series of prime numbers

I have a bit of a problem proving that the series: $$ \sum_{p\leq x} \frac{p\ln\left(p\right)}{x^2} $$ where the sum is extended over all prime numbers, converges to 0.5. Any ideas? Thanks in ...
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0answers
56 views

Specific form of integral representation of the Riemann zeta function

Is there an integral represenation of the Riemann zeta function of the form: $$\zeta(s) = f(s)+c\int_a^b\frac{g(x)}{x^{p(s)}}dx,$$ where $a,b,c\in\mathbb{R}$ with $a\neq b$, $p(s)$ is some ...
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2answers
112 views

$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$ for integer k

Can anyone compute $$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$$ for integer k? Can it be done in closed form, using only elementary functions, without the use of the Gamma function? For k=1, the closed ...
4
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0answers
70 views

Prime number theorem for Dedekind domains

Let $\mathscr P\subseteq \mathbb N$ be the set of prime numbers. The prime number theorem tells us that if $\pi(x)=|\{p\in\mathscr P\colon p\leq x\}|$ then $\pi(x)\sim \frac{x}{\log x}$. Now one could ...
2
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1answer
142 views

Integral representation of the Riemann zeta function

I've come across the following integral representation for the Riemann zeta function, $$\zeta(s) = \frac{s+1}{2(s-1)} + \frac{s}{8} - \frac{s(s+1)}{2\pi^2}\int_1^\infty \frac{(\tan^{-1}\cot(\pi ...
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0answers
35 views

Convergence of a series concerning the multiplicative order of 2

I was trying to bound the value of $v_p(2^n-1)$ and some of the series I obtain made me wonder about the following problem. Problem : When does the series $$\sum_{prime \: p} \frac{1}{(ord_p ...
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0answers
106 views

Are $ut + 1$ and $ut + t + 1$ both prime for some t for any $u$?

Conjecture : For any natural number $u$, there is a natural number $t$ such that $ut + 1$ and $ut + t + 1$ are both prime. So we get a solution of the equation $$au - b(u+1) = -1$$ with prime ...
3
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0answers
116 views

Ratio of maximal to minimal jump in the set of angle multiples (corrected)

(This is the corrected version of the question I asked here: Ratio of maximal to minimal jump in the set of angle multiples.) Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times ...
4
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1answer
123 views

Estimating integrals involving $\pi(x)$

While solving an exercise in analytic number theory, I ran into difficulty of estimating an integral of the form $\displaystyle\int_{1}^{x} \frac{\pi(t)}{t} dt$ where $\pi(x)$ is the prime counting ...
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0answers
68 views

Equivalence of three asymptotic statements

Question 22 of chapter 4 in Apostol's "Introduction to Analytic Number Theory" asks to show that the following three statements are equivalent: $$\psi(x) \log(x) + \sum_{n \le x} \Lambda(n) ...