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

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0
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1answer
22 views

Using Moebius Inversion to solve a functional equation.

I am reading about Moebius inversions, and I am given the following claim: $\sum_{d=1}^{\infty} f(z^d) = g(z)$ with $g(z) = O(z)$ at $z=0$, implies that $f(z) = \sum_{d=1}^{\infty} \mu(d)g(z^d)$ by ...
1
vote
0answers
46 views

Any heuristic explanation on why sieve methods can not prove Goldbach conjecture?

Any heuristic explanation on why sieve methods can not prove strong Goldbach conjecture ? I remember that Terence Tao published a blog and gave an heuristic explanation on why circle methods very ...
0
votes
1answer
50 views

Proving Euler Summation by Parts Without Using Integration by Parts

Assume $f$ has continuous derivative $f'$ on [a,b]. Prove the following summation formula, without using partial integration: \begin{equation} \sum_{a< x \le ...
0
votes
0answers
20 views

Any results to generalize weighted sieve to three parameters?

In Chen's theorem on Goldbach conjecture , he used two parameter weighted sieve method, and he proved every even number can be represented as a prime number and an almost prime ( 1 + 2 ). Are there ...
3
votes
1answer
29 views

Proving that $(2 \pi i)^{-1} \int e(\pi v^2/y^2) x^v y^{-1} dv = e(-\pi (\log x)^2 y^2 /4)$

I've seen a particular integral transform (an inverse Mellin Transform) used a few times, but I don't know how it's proved. In particular, I'm trying to prove $$\frac{1}{2\pi i} \int_{(2)} e^{\pi ...
0
votes
0answers
35 views

Definition of Hecke Operator on modular form of half-integral weight

We define, for $f\in M_{k/2}(\tilde\Gamma_1(N))$, $T_{p^2}f :=p^{k/2-2} f|[\tilde\Gamma_1(N)\zeta_{p^2}\tilde\Gamma_1(N)]_{k/2}$, where $\zeta_{p^2}$ is the lift of $\alpha=\begin{pmatrix} 1 & ...
3
votes
0answers
43 views

What is the relationship between GRH and Goldbach Conjecture?

We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, ...
0
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0answers
15 views

Please recommend the most easy to read analytic number theory book for self study [duplicate]

All: Can anyone recommend the most easy to read analytic number theory book for self study ? Prefer with hint to exercise. I have Apostol's, Introduction to Analytic Number Theory, just want to see ...
0
votes
0answers
15 views

Any 2D or 3D extension of Hardy Littlewood Circle Method?

All: Any 2D or 3D extension of Hardy Littlewood Circle Method ? Also any 2D or 3D extension of Vinogradov Trignometic Sum methods ?
0
votes
1answer
21 views

Finite measure on positive integers

Disclaimer: I am sure that this idea is not at all new, but I have had trouble locating content directly related. I humbly accept that this question may be the result of a brain fart. Suppose that ...
1
vote
2answers
37 views

Question about Mobius function.

Let $N \in \mathbb{N}.$ I would know if is it true that $$-\underset{k\mid N}{\sum}\mu\left(k\right)\log\left(k\right)>0.$$I know that $$-\mu\left(k\right)\log\left(k\right)=\underset{r\mid ...
10
votes
3answers
183 views

likely open number theory problem: finite sum of $\zeta(2)$ equal to a square of rationals

Which $n$ can let $S=1+\frac14+\frac19+\cdots+\frac1{n^2}$ be a square of a rational number? Obviously, $1$ and $3$ work, but how to prove they are the only ones? I think this problem is really hard. ...
2
votes
1answer
55 views

Periodicity over the prime indices of a multiplicative sequence implies periodicity?

I have a real sequence $(a_p)$ indexed by the prime numbers which takes values -1, 0, or 1, having the property that $a_p=a_q$ whenever $p\equiv q \pmod m$, where $m$ is a fixed integer $>2$. I'm ...
-1
votes
1answer
55 views

any computational analytic number theory book?

All: Can anyone recommend an introduction computational analytic number theory book ? I am mainly interested in using computer software to verify and confirm analytic number theorem, things related: ...
0
votes
1answer
85 views

who, by doing what, can make major contributions (breakthrough/discoveries) in math research?

I am a Math Ph.D student, had already published two small articles. I want to ask more experienced mathematician a question. What kind of person, by doing what, can make major contributions ...
5
votes
1answer
108 views

How many numbers are products of $p^p$?

Consider the set $\mathcal{S}=\{1,4,16,27,\ldots\}$ of numbers which are products of numbers of the form $p^p$ for $p$ prime. ($\mathcal{S}$ is A072873 in the OEIS.) Note that multiple primes are ...
2
votes
1answer
86 views

Using Dirichlet's hyperbola method and Dirichlet's formula

Dirichlet Hyperbola Method. For $x \geq 2$: $$ \sum_{n \leq x} \frac{d(n)}{n} = \frac{1}{2} \log^2 x + 2\gamma \log x + \gamma^2 + O(\frac{\log x}{\sqrt{x}})$$ I know already that the summation of ...
1
vote
0answers
47 views

Any results for small number Goldbach conjecture research?

It seems to me that most research results on Goldbach conjecture research are for large number. (Example: results of Vinogradov, Terence Tao, Harald Helfgott, etc). My understanding is that those ...
3
votes
0answers
36 views

Trigonometric sum evaluation

Let $q$ a prime number and $1 \leq a<q$ a positive integer. We know from Ramanujan identity that $$\underset{h=1,\left(h,q\right)=1}{\overset{q}{\sum}}e^{2\pi ...
6
votes
0answers
93 views

Books to read to understand Terence Tao's Analytic Number Theory Papers

I tried to understand Terence Tao's Analytic Number Theory Papers. For example, this paper, Every Odd Number Greater Than 1 is The Sum of at Most Five Primes. Which books shall I read to prepare ...
0
votes
0answers
21 views

Questions on Heath-Brown's paper “Kummer’s Conjecture for Cubic Gauss Sums”

On page 21 in Heath-Brown's paper "Kummer’s Conjecture for Cubic Gauss Sums" (http://eprints.maths.ox.ac.uk/158/1/kummer.pdf), a formula says $$\sum_{j\in \mathbb{Z}[\omega]}f(j)=\sum_{k\in ...
1
vote
0answers
15 views

a question on upper bound for Bessel function $K_{2it}(x)$

Can we have $$K_{2it}(x)\sinh(t)\ll_{x} 1$$ for $1<x< (1+|t|)^3,$ where $K_{2it}(x)$ deotes the ordinary K-bessel function and $t>1$. This is true when $x\ge (1+|t|)^3$ from some references. ...
1
vote
1answer
46 views

Solution of Pell equation over field of p-adic numbers

Right now I am studying Pell equation. Using continued fractions, we can find the solution of Pell equation. Now my question, is it possible for me to find a solution in the field of p-adic numbers ...
1
vote
0answers
28 views

Looking for proof of formula in WolframMathWorld article [duplicate]

I came across the formula (24) in the WolframMathWorld article on Web page http://mathworld.wolfram.com/TrigonometryAngles.html where no source of the proof could be identified by me. The formula is ...
0
votes
0answers
20 views

Estimation with logarithm

I have to prove: $q$ an integer $\geq 2$, $\tau$ a real number $\geq 2$ and $\sigma \leq 1- \frac{1}{\log q\tau}$, $M\geq 0$. Then $M^{1-\sigma }\leq (q\tau )^{-\sigma }$ and $(M+1)^{-\sigma }\leq ...
0
votes
0answers
38 views

Dirichlet L series estimation

let $\chi$ be a non-principal character modulo $q$, $M\geq 1$. I have to prove that, if $\vert \sigma - 1 \vert \leq \frac{1}{\log M}$, then $\vert \sum_{n=1}^M \chi (n)n^{-s}\vert\leq 1+e\log M$ and ...
2
votes
1answer
26 views

Dirichlet character modulo p

How can I prove that if $\chi$ is a non-principal character modulo $p$ prime, then $\chi (-1) = \overline{\chi} (-1)= \pm 1$ and $\sum_{x=1}^p \chi (x) e^{2\pi i x}=0$? For the first question, I just ...
0
votes
1answer
19 views

Any books on Trigonometrical Sums (for the Theory of Numbers )?

All: Can anyone recommend good books on Trigonometrical Sums ? The only book I found is Vinogradov's book: Method of Trigonometrical Sums in the Theory of Numbers. but it is really old. I am ...
0
votes
1answer
37 views

Partial sums of powers of the divisor function

It is easy to establish that $$\sum_{n\le x}\tau(n) \sim n\log n$$ How would one find good bounds on $$\sum_{n\le x} \tau(n)^k $$ for some $k > 0$
2
votes
0answers
60 views

Mellin transform on $\mathbb{Z}[\omega]$

Let $\omega=\frac{-1+i\sqrt{3}}{2}$ be a complex cube root of unity. The Eisenstein integers $\mathbb{Z}[\omega]$ (a unique factorization domain) are of the forms $a+b\omega$ where $a$ and $b$ are ...
0
votes
1answer
18 views

Why does $f(x) \asymp g(x) \implies log(f(x)) = log(g(x)) + O(1)$?

Why does $f(x) \asymp g(x) \implies log(f(x)) = log(g(x)) + O(1)$? Has it got something to do with the fact that \begin{align} f(x) \asymp g(x) \implies \exists c_1,c_2, \text{ such that}\\ ...
0
votes
1answer
15 views

Show that $\frac{1}{e^\gamma \text{log }x + O(1)} = \frac{1}{e^\gamma\text{log }x} + O\left(\frac{1}{(\text{log }x)^2}\right)$

Show that $\frac{1}{e^\gamma \text{log }x + O(1)} = \frac{1}{e^\gamma\text{log }x} + O\left(\frac{1}{(\text{log }x)^2}\right)$ I'm using one of Merten's estimates in a proof, the one that states ...
1
vote
1answer
27 views

Why is it impossible that $\frac{\phi(n^*)}{n^*} < \frac{\phi(n)}{n}$ when $n^* < n$

Why is it impossible that $\frac{\phi(n^*)}{n^*} < \frac{\phi(n)}{n}$ when $n^* < n$ and $n$ has $k$ prime factors, and $n^*$ is the product of the first $k$ prime factors? I understand that ...
1
vote
1answer
74 views

a problem about averages of fractional parts

I know a little analytic number theory , for example i know that: $\sum_{d\le x}(\frac{x}{d}-[\frac {x}{d}])=\frac{x}{a}(1-\gamma)+O(\sqrt x)$ when the $\sum$ is over $d$'s the integers less or equal ...
0
votes
1answer
48 views

Trying to show that $\phi(n) > c_1 \frac{n}{\text{log log }n}$ for some constant $c_1 > 0$

Trying to show that $\phi(n) > c_1 \frac{n}{\text{log log }n}$ for some constant $c_1 > 0$ where $\phi(n)$ is the euler phi function. I was wondering if I could use something like ...
0
votes
2answers
25 views

Is it true that $\sum\limits_{n=1}^N e^{in} = \frac{e^i(1-e^{iN})}{1-e^i}$ even if $| e^i | > 1$?

Is it true that $\sum\limits_{n=1}^N e^{in} = \frac{e^i(1-e^{iN})}{1-e^i}$ even if $| e^i | > 1$? I know this question is quite trivial and I will understand if it gets removed. I am trying to ...
0
votes
1answer
35 views

Is $\sum\limits_{\text{p prime}, p \geq 2}\frac{(-1)^{\frac{p^2-1}{8}}}{p}$ convergent or divergent??

Is $\sum\limits_{\text{p prime}, p \geq 2}\frac{(-1)^{\frac{p^2-1}{8}}}{p}$ convergent or divergent? So far I have that \begin{align} \sum\limits_{\text{p prime}, p \geq 2} ...
4
votes
2answers
49 views

$(n+1)^{\textrm{st}}$ prime less than $2^{2^n}$

Using elementary means, show that the $(n+1)^{\textrm{st}}$ prime is less than $2^{2^n}$ please do not use fancier stuff like the prime number theorem or beyond. using this how can you show that ...
0
votes
0answers
37 views

Singularities of zeta function

I have to prove (if $\gamma \ne 0$) that there is a analytic continuation for $\Re s >0$ of the function $$f(s)=\frac{\zeta (s)^2 \zeta(s-i\gamma )\zeta(s+i\gamma ) }{\zeta(2s)} $$ and that this ...
0
votes
0answers
22 views

Conductor of Dirichlet character divides every quasiperiod

Let $\chi$ be a Dirichlet character which has quasiperiods $d_1, d_2$. I.e., if $(n(n + kd_i), q) = 1$ then $\chi(n + d_i) = \chi(n)$ for any $k \in \mathbb{Z}$. Supposedly we can then show that ...
1
vote
2answers
80 views

Prove the Inequality on $\pi$-function

Prove that for each $y \geq 2$ , we have $\pi(x)+\pi(y)>\pi(x+y)$ for all sufficiently large $x$. I tried searching in the Internet for quite a while. The best result that I have found is L. ...
1
vote
1answer
53 views

Find all positive integer pairs $(x,y)$ and $(u,v)$ with certain relations.

Is there exists any positive integer pairs $(x,y)$ and $(u,v)$ for which, the relations, $x^2+y^2=u^2+v^2$ and $x^3+y^3=u^3+v^3$ are satisfied simultaneously?
3
votes
1answer
33 views

Can we have $\sum_{n\leq [x]}e^{-\sqrt{\frac{\log x}{r}}}\ll \frac{x}{e^{c \sqrt{\log x}}}$ for some constant $c>0$, where $x>1.$

Let positive interger $n$ is square-free, that is $n=p_1p_2\cdots p_r$ some $r$. Can we have $$\sum_{n\leq [x]}e^{-\sqrt{\frac{\log x}{r}}}\ll \frac{x}{e^{c \sqrt{\log x}}}$$ for some constant ...
2
votes
0answers
34 views

Sum of reciprocals of natural numbers with numerator being Legendre symbol mod 7 (L-series)

How do I show that $$\sum_{k=1}^\infty \left({k \over 7}\right)\Big/k = \sum_{k=0}^\infty \left(\frac{1}{7k+1} + \frac{1}{7k+2} - \frac{1}{7k+3} + \frac{1}{7k+4} - \frac{1}{7k+5} - ...
2
votes
1answer
82 views

Sum of inverse prime numbers

How can the following equation be proven? $\sum\limits_{p \le n; p \in P} \frac{\ln p}{p} \sim \ln(n) + O(1).$ I just wanna understand this sum Sum of reciprocal prime numbers
1
vote
1answer
40 views

Dirichlet characters - proof in a book

I found the following in a book and don't understand. Let $\chi$ denote a non-principal character modulo $q$ and $S(x)=\sum_{n\leq x}\chi (n)$. Then $\sum_{m>y} \frac{\chi(m)}{m} = \int_y^{\infty ...
1
vote
0answers
78 views

What are the “hidden” symmetries in Goldbach Conjecture?

What are the "hidden" symmetries in Goldbach Conjecture ? If Goldback conjecture is true, the basic instinct is that there must exist some "symmetries" which ensure (and lead) such properties. As we ...
2
votes
0answers
62 views

What are the missing gaps to prove Goldbach Conjecture?

When Andrew Wiles proved FLT, all he needed to do was to prove "semi-stable elliptic curve case" of Shimura-Taniyama conjecture. He did not need to start from scratch, he just needed to fill this ...
0
votes
2answers
54 views

Twin Prime Constant

How would one prove that the twin prime constant $$C_2 = \prod_{p > 2}1-\frac{1}{(p-1)^2} > 0$$ Simply computing the product for a large number of terms isn't rigorous, and simply establishes ...
1
vote
1answer
37 views

Prove the complex conjugate of an analytic function is analytic in the set of conjugates.

Given a function $f(z) \in C$ that is analytic, prove that $g(z) = \overline{f(\bar z)}$ is analytic in the set $\{\bar z : z \in C \}$. This is for homework: tips would be appreciated.