Questions tagged [analytic-number-theory]

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

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143 views

How to use an inverse Mellin transform to get to the $\mathrm{core}(n)$?

From this question here: Moreover, if multiplicative function $\mathrm{core}(n)$ is defined to map positive integers "$n$" to square-free numbers by reducing the exponents in the prime power ...
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175 views

Proof of Prime Number Theorem

I am looking for a detailed proof of the Prime Number Theorem using analytic methods (that is, using $\zeta(s)$). What is a good reference to read?
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3answers
178 views

is there a Globally convergent series for Riemann Xi function?

According to Wikipedia, there is a global convergent series for Riemann Zeta function: https://en.wikipedia.org/wiki/Riemann_zeta_function#Globally_convergent_series Is there a similar global ...
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1answer
100 views

What are some recursive properties of Merten function or Summatory Liouville function?

Both Merten function and Summatory Liouville function show some kinds of "scale invariance" properties. (Those functions also display some kind of "periodic" behavior.( Just wonder if those "scale ...
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36 views

Convergence of a sum involving the divisor function and characters

Could someone please show me that $\sum_{n = 1}^{\infty} \sigma_{x}(n)n^{-(x + 1)/2} \chi(n)$ converges where $\chi $ is a Dirichlet character of modulus $m$?
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50 views

Limit of the ratio of the square root of a Mersenne number to the product of its prime factors

Mersenne numbers with prime exponents are numbers of the form $M_p = 2^p-1$, where $p$ is prime. Suppose that $p$ is such that $M_p$ has exactly two prime factors, $\rho, P$. Given $\epsilon > 0$, ...
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64 views

Kloosterman Sums and Lattice Hyperbolas

Part of this blog discussing the twin prime conjecture mentions a connection between three objects: $ \sum_{x \leq n \leq 2x} \tau\Big(n(n+2)\Big) $ average over twin primes where $\tau(n) = (1 \ast ...
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1answer
186 views

Out of all the proofs of the PNT, which one is the most accessible?

I have been studying the continuation of the Riemann zeta function $\zeta(s)$ for the past while. I can prove that all the zeroes must lie in the critical strip.I am currently in the process of using ...
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1answer
26 views

Definition of “Contractive Invariant Plane”

Can someone please explain the definition of a contractive invarient Plane found in: the paper It is nearly at the very beginning of the Introduction. By contractive do they mean a contractive map? ...
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45 views

What is general Riemann's Hypothesis? [duplicate]

What makes it so important in analytic number theory?
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17 views

Prove that $L(s)$ converges for $s > 0,$ or more generally for all complex $s \in \mathbb{C}$ with $\Re(s) > 0.$ [duplicate]

Let $L(s) = \sum_{n=1}^{\infty} a_n/n^s$ be a Dirichlet series. Suppose that the partial sums of the coefficients $A_n = \sum_{i=1}^n a_i$ are bounded, i.e. there exists a constant $C$ such that $|A_n|...
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300 views

“Necessary” condition for Power Diophantine Equation.

Motivation: Brocard’s problem $n!+1$ being a perfect square Observations: Given a power Diophantine equation of $k$ variables with a “general solution” (provides infinite integer solutions) to ...
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2answers
115 views

On the summatory function of $\Lambda(n)/n$

In this paper is written that the prime number theorem in the form $\psi(x) = ( 1 + o(1) ) x$ is elementary equivalent to $$\sum_{n \le x } \frac{\Lambda(n)}{n} = \log x - \gamma + o(1) $$ I started ...
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1answer
30 views

want a better bound the expression

Consider the expression $$ f_n(x)=\sum_{d|n,1<d\leq x} \Lambda(d)\left(\frac{1}{\log d}-\frac{1}{\log x}\right) $$ I've got $f_n(x)=\operatorname{O}\left(\frac{ x}{(\log x)^2}\right)$, but I've ...
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82 views

definition of the L-function $L(f, \chi, s): \mathbb{A}_K \rightarrow \mathbb{C}$, what is smoothness and what is $f$?

To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for ...
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1answer
65 views

Proof Janusz Algebraic number fields, convergence of Dirichlet Series.

The book Algebraic number fields, Janusz Please, Could you explain the proof of the part b) a little more? Thank you all.
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1answer
92 views

Elliptic curves $\mathbb C/\Gamma , \mathbb C/\Gamma'$ are isomorphic iff $\Gamma=\lambda\Gamma'.$

Let, $\Gamma, \Gamma'$ be $lattices$ of $\mathbb C$, define $elliptic$ $curves$ by $\mathbb C/\Gamma , \mathbb C/\Gamma'$, then $\mathbb C/\Gamma , \mathbb C/\Gamma'$ are isomorphic $\...
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1answer
206 views

Elementary proof that $\omega(n)$ is bounded $\frac{\log n}{\log( \log n)}$ in the limit?

I'm trying to show that $\omega(n)$ is less than $\frac{\log n}{\log(\log n)}$ as it's stated without proof in an analytic number theory text. It's a corollary of the PNT, but I want to not use that ...
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73 views

Do you know any answer for equation y^2 = x^3 + k? [duplicate]

As you know, the equation y^2 = x^3 + k for k like (4n-1)^3 - 4m^2 that m , n are integers & no prime number that p is congruent to 1 modulo 4 counts m, doesn't have any answer & its proof is ...
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1answer
38 views

How to answer the following question regarding a certain number of primes in a certain interval?

For an analytic number theory homework assignment, we are asked to prove the following (using the Prime number theorem $\pi(x) \sim x/\log(x)$ as $x \to \infty$ ): For every $\epsilon > 0 $ and ...
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1answer
59 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 \...
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462 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 ...
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70 views

Continuation of the Zeta Function

I already showed that für $\sigma >1$, $$\zeta (s) = \frac{1}{s-1} + \frac{1}{2} + \sum_{j=1; 2\mid j+1}^{k-1}\left( \prod_{i=0}^{j-1}(s+i) \right) b_{j+1}(0) - \left( \prod_{j=0}^{k-1}(s+j)\right) ...
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120 views

Question about Riemann zeta function + my proof

First let me say that I am 16 years old so I am not very professional in math. English is also a second language so I apologize for any mistakes. Now i have been reading about the Riemann zeta ...
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140 views

Upper bound for number of primes in an interval

Let $S(x,y)$ be the number of primes $p$ in $(x, x + y]$ such that also $p + 6$ and $p + 12$ are primes. I know that $$ T(x, y) \leq 48 c \frac{y}{\log^3 y} \left( 1 + O \left ( \frac{\log \log y}...
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110 views

A short question on the estimation of $\sum_{1\leq n\leq x} \mu(n)n^{-1}$.

$ \ \ $ I want to ask an estimation of $\sum_{1 \leq n\leq x} \mu(n)n^{-1}$. According to a paper: http://arxiv.org/pdf/0908.4323v5.pdf of Terry Tao (See the theorem 1.3 on page 4 if you want), for an ...
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30 views

How many solution are possible for this multivariable equation? [duplicate]

$$2(a+b+c+d+e+f)+g=N$$ where $$a,b,c, \cdots ,N \in \mathbb{N}$$ Any lead will be appreciated.
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1answer
98 views

Size of N in primes in arithemtic progression algorithm

I've been implementing the search for Primes in Arithmetic Progression (PAP) as explained by Weintraub (1976), and in his paper he refers to a number N which he sets to what seems to be an arbitrary ...
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38 views

Gamma function whose argument is a reciprocal power with integer base and exponent

Consider the analytic continuation of the factorial function $n!$ given by $\Gamma(z)$ (note $n!=\Gamma(n+1)$), and suppose $z=a^{-n}$, where $a,n\in\mathbb{N}$ are positive integers. Are there any ...
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88 views

Wolstenholme Number

Does Wolstenholme Numbers have perfect squares other than 1 and 49? The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749 seems to be a complicated problem
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63 views

Zeta Riemann Function

Use that $\zeta(s)=1+\frac{1}{s-1}-s\int_{1}^{\infty}\frac{\left\{u\right\}}{u^{s+1}}du$ if $Re(s)>0$ to show that. 1) $\zeta(s)=s\int_{1}^{\infty}\frac{\left[ u\right]}{u^{s+1}}du$ for $Re(...
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1answer
63 views

Summation formula in dimension 2

One of the most common tools in analytic number theory is the summation by parts, my question is what is the similar formula when we are, for example, in dimension two and we have the sum $$ \sum_{|z|&...
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1answer
52 views

How do you generate results for various n in the following formula:

Let f be the arithmetic function defined by $f(n)$ = $3^{w(n)}$, where $w(n)$ is the number of distinct prime factors of n. Let $f^{-1}$ be the inverse of f with respect to the convolution product. ...
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1answer
127 views

Evaluation of Riemann-Stieltjes integral in Laurent expansion of zeta function

I'm probably being really stupid but in a proof of the Laurent expansion of the Riemann zeta function the quantity \begin{equation} S_r(t) = \sum_{n \leq t} \frac{(\log (x/n))^r}{n} \end{equation} is ...
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1answer
97 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
50 views

Sieve dimension of union of two sets.

Let $P$ be a set of primes $\leq p$. Let $A$ be a set of all integers $\leq x$ in which the elements in $A$ would avoid two classes mod $p_i$ for all $p_i \leq p$ (except $2$,$3$). My understanding ...
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128 views

Is the explicit formula for the second chebyshev function unique?

Is the explicit formula for the second chebyshev function unique ? Or is it possible there are multiple explicit formula ? Are there explicit formula's given as an infinite product over the zero's ...
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92 views

Count of numbers with the given prime factors in a range [duplicate]

Given two primes: $p$ and $q$, $p \neq q$ and $n \in N$ find count of numbers $u$, so that $u \leq n$ and $u = p^k q^l$; $k, l \in N$. If we'd given with just one prime $p$ this count would be ...
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108 views

estimate $\sum_{x<p\le x+y} \log{p}/p$

In his paper the prime number theorem via the large sieve, A. Hildebrand made use of the following inequality $$\sum_{x<p\le x+y} \frac{\log{p}}{p} \le (2+o(1))\log{\frac{x+y}{x}}$$ where $x\ge y$ ...
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98 views

By establishing a recurrence relation and using induction, or other-wise, show that this sequence is 3-adically Cauchy?

this is a question from a book I'm struggling with, please could you provide a clear proof Consider the sequence of rational numbers $a_1 = 1+3,a_2 = 1+\frac{3}{1+3},a_3= 1 + \cfrac{3}{1 +...
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1answer
893 views

Equivalent to Riemann Hypothesis

Through last number theory, I did learn that Riemann hypothesis is equivalent to the following inequality : $|\pi(x)-Li(x)| \leq \sqrt{x} log(x)$ where $Li(x)$ is the Logarithmic integral function and ...
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1answer
67 views

Sequence of numbers with a special property [closed]

Prove that the sequence a(n) = 2013 + 317n, where n is any nonnegative integer, generates infinitely many palindromic numbers.
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1answer
60 views

Semiprime of the form $x^2 + 5 y^2$

Let $a$ be a prime of the form $20 n + 3$ or $20 n + 7$ Let $b$ be a prime of the form $20 m + 3$ or $20 m + 7$. Then the semiprime $ S = ab $ is always of the form $ S = x^2 + 5 y^2 $. For some ...
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1answer
15 views

A Converse For A Particular Case of the Hardy-Littlewood Tauberian Theorem

Let $V$ be a set of positive integers, and let: $$\varsigma_{V}\left(x\right)\overset{\textrm{def}}{=}\sum_{v\in V}x^{v}$$ Defining the natural density of $V$ by the limit: $$d\left(V\right)\overset{\...
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2answers
40 views

Generalised sum of divisors.

Let $$F(x,k)=\sum_{a_{1}a_{2}...a_{k}\le x}1$$ Find asymptotic formula for $F(x,k)$ For example it is known. $$F(x,2)=\sum_{n\le x}\tau(x)=x\log x+(2\gamma-1)x+O(x^{\frac{1}{2}})$$ The proof of ...
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1answer
83 views

Density zero subsets of primes

Let $P$ denotes the set of all primes. Find a subset $Q$ of $P$ such that the following holds: (i) Relative density of $Q$ is zero in $P$ and (ii) $ \sum_{p \in Q} \frac{1}{p} = \infty $.
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1answer
77 views

Given $N$ find the number of natural numbers less than $N$ that may be written in the form $\frac{(k)(k+1)}{2}$

Given $N$, find the number of natural numbers less than $N$ that may be written in the form $$\frac{k(k+1)}{2},$$ where $k\in \Bbb N$. I know that the answer to this problem is approximately $\sqrt {...
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1answer
136 views

Problem with tau function

Show that $\tau(n)^a$ is $o(n)$ (small $o$) for all $a$ real, with $\tau(n)$ the tau function (number of divisors of n). I have not completed the demonstration of this.
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1answer
89 views

Show that the natural density is $1/2$.

Let $$A_b= \left\lbrace{p \in \mathbb{P}| \left(\frac{b}{p}\right)=1 } \right\rbrace $$ and $$ \nu(A_b)=\lim\limits_{x \to \infty} \frac{\#\lbrace {p \in A_b|p\le x}\rbrace}{\pi(x)}$$ the natural ...
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2answers
83 views

On the asymptotic behaviour as $X\to\infty$ of the sum $\sum\frac{1}{x}$ over every $x\leq X$ such that $\pi_2(x)<2C_2\int_2^x\frac{dt}{\log^2t} $

Let $\pi_2(x)$ the twin prime-counting function that counts the number of twin primes $p,p+2$ with $p\leq x$, and $C_2$ is the the twin prime constant. We assume the Twin Prime conjecture, see it as ...