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

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1
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1answer
32 views

a possible period of 124 for the sign of Ramanujan $\tau(3^n)$ function

The Ramanujan $\tau(n)$ seemed to have random positive/negative signs: ...
0
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0answers
10 views

Proving a certain function involving the Riemann-Zeta function is non-increasing

Show that $ f(x) = \frac{\zeta(x -2)}{\zeta(x-1)} \qquad x > 3, $ where $\zeta$ is the Riemann-Zeta function, is non-increasing. My attempt was to use $\zeta(s) = \frac{1}{\Gamma(s)} ...
4
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1answer
63 views

Sign of Ramanujan $\tau$ function

The Ramanujan $\tau(n)$ seemed to have random positive/negative signs: ...
2
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1answer
29 views

Can we tell if a number is prime by the number of its partition ?

Can we tell if a number is prime by the number of its partition ? Or in general, how much can we know about a number itself from its partition function ? I understand that Ramanujan has some ...
-1
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1answer
31 views

Basic definition of exponent and $a^0$? [on hold]

How can we define $a^0$ as being equal to $1$? Why?
5
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0answers
85 views
+50

Number of ways to express a binary number in a certain way

So I'm working on a problem where I get to a point where I have to count the number of solutions to an equation or at least find a decent upper bound to be used in an estimate I need later. The ...
3
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1answer
46 views

Determining the asymptotics of the Summatory function of an Arithmetic Function

We define the arithmetic function: $\displaystyle f(n) = \max\limits_{p^{\alpha} || n} \alpha$, that is if $\displaystyle n = p_1^{\alpha_1}\cdots p_k^{\alpha_k}$ (prime factorization of $n$) then ...
0
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0answers
29 views

Koch's version of the Riemann hypothesis for $x=p^2$

According to von Koch 1991, if the Riemann hypothesis is true, then the for the prime counting function $$\pi(x)=Li(x)+O(\sqrt x \log x)$$ For this equation, does there exist any reference or does ...
0
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0answers
24 views

Modular Equation and Modular Forms?

I'm reading Ramanujan's Notebook now and I see some kinds of "Modular Equation". At first I think Modular Equation is just a set of interesting fomulaes, but wikipedia says that Modular Equation is ...
3
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0answers
92 views

Divisor summatory function for squares plus one

As an exercise for my Analytic Number Theory course, I need to prove, using Dirichlet hyperbola method, that: $\sum_{n\leq x}\tau(n² + 1)= {3\over\pi}x\log(x) + O( x)$, where $\tau(n)=\sum_{d|n}1$ ...
0
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0answers
21 views

How to prove that $\zeta(s)<0$ for $s \in (0,1)$ using a particular expression for the Riemann zeta function?

Like in this question, I would like to prove that $\zeta(s)<0$ for $s \in (0,1)$. However, I have to use the expression $$\zeta(s) = \frac{1}{s-1} + 1 -s \int_{1}^{\infty} ...
1
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1answer
33 views

Upbounds for Ramanujan $\tau(n)$ function

For prime $p$, $|\tau(p)|\le 2p^{11/2}$. I am looking for the upbound for $|\tau(n)|$,$n\in \mathbb{N}$. Thanks- mike
1
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1answer
29 views

express the dirichlet series for the sequence d(n)^2 in terms of riemann zeta.

Prove that $$\sum_{n=1}^\infty d(n)^2n^{-s}=\zeta(s)^4/\zeta(2s)$$ for $\sigma>1$ what i did: I already proved this formally, that is, without considering convergence. I use euler products, ...
3
votes
3answers
61 views

How to prove that $ \lim_{u \downarrow 1} (u-1) \zeta(u) =1 $?

I would like to prove that $$ \lim_{u \downarrow 1} (u-1) \zeta(u) =1 \quad .$$ However, I am not sure which form of the Riemann-zeta function I ought to pick in order to compute this limit. I ...
4
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1answer
37 views

Existence of primes $q$ such that $p\mid q-1$ and $q\mid p^n-1$

Let $p$ be a prime. By Dirichlet's theorem on arithmetic progressions, there are infinitely many primes $q$ such that $p\mid q-1$. Must there be also primes $q$ such that $p\mid q-1$, and also, $q ...
2
votes
2answers
36 views

Can sieve method prove ternary (three) prime Goldbach conjecture?

Can sieve method prove ternary (three) prime Goldbach conjecture (Vinogradov Theorem) ? I had done some research, I could not find any articles on this. Can anyone provide some help on this ? I ...
0
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0answers
15 views

Do Sieve Methods all depend on some properties of prime number distribution?

Do Sieve Methods all depend on some properties of prime number distribution to reach any meaningful results ? Particularly, those sieve methods that are used in Goldbach Conjecture. Such as: Brun, ...
0
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0answers
15 views

what is the best Schnirelmann Constants?

what is the best Schnirelmann Constant for Goldbach Conjecture ? On http://mathworld.wolfram.com/SchnirelmannConstant.html the best Schnirelmann Constant is 7 ( from Ramaré ) My understanding is ...
2
votes
1answer
80 views

Corollaries of Green-Tao Theorem?

there is already a good thread which discusses some corollaries of the Green-Tao Theorem, here: Constructing arithmetic progressions The question I was wondering about is of a similar flavor but ...
0
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1answer
20 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
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0answers
42 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
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1answer
47 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
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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
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1answer
27 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
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0answers
29 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
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0answers
39 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
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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
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1answer
20 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
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2answers
33 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
176 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
53 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
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1answer
50 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
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1answer
82 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
79 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
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0answers
44 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
33 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 ...
-2
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0answers
33 views

Any complete analytic number theory course online?

All: I am studying analytic number theory by myself. Just wonder if there is any complete analytic number theory course online ? (either for undergraduate or graduate level). I did a search on ...
-1
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0answers
23 views

Why does Trigonometrical Sums often appear in prime number research?

All: Why does Trigonometrical Sums often appear in prime number research ? Particularly in Goldbach conjecture related research. Is there fundamental reason for this ? Can anyone provide intuitive ...
6
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0answers
84 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
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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
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0answers
14 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
41 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
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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
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0answers
19 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
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0answers
36 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
25 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
18 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
35 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 $k > 0$