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
42 views

Are there any known asymptotics for $\sum_{p\leq x} p$? [duplicate]

As a prospective undergraduate who has really benefited from his time on MSE thus far, i recently learnt that there exists asymptotic approximations for $\sum_{p\leq x} 1, \sum_{p\leq x} p, \sum_{p\...
2
votes
0answers
38 views

Question on a proof of the Euler product of zeta function

Let $\zeta(s)$ be the Riemann zeta function. Then we know it satisfies the Euler product for $\text{Re}(s) > 1$, $$ \zeta(s) = \prod_{p} (1 - p^{-s})^{-1}. $$ The proof I read, if I recall ...
2
votes
1answer
49 views

How are the nontrivial zeros of the Riemann zeta function calculated?

The Riemann zeta function, is the function of the complex variable $s$, defined in the half plane $\Re(s)>1$ by the absolutely convergent series $\zeta(s) = \sum_{n} n^{-s}$ and extends to the ...
0
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1answer
33 views

An upper bound for the Chebyshev function?

The Chebyshev functions are defined as $\psi(x) = \sum_{p^m \leq x} \log n$ and $\theta(x) = \sum_{p\leq x} \log p$, where $p$ is a prime, $m\geq 1$ is an integer and $n=p^m$ in $\psi(x)$. It is known ...
0
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1answer
37 views

Is the prime counting function differentiable?

Let $\pi(x)$ denote the number of primes not exceeding $x$. Is $\pi(x)$ differentiable ? My attempt: It is well known that $\log \zeta(s) = \int_{2}^{\infty} \dfrac{s\pi(x)}{x(x^s - x)} \mathrm d{x}$ ...
1
vote
1answer
45 views

estimation for n-th prime

The famous theorem of Hadamard and Vallee-Poussin https://en.wikipedia.org/wiki/Prime_number_theorem implies that $p_n\sim n\ln n$, so $C_1 n\ln n \le p_n \le C_2 n\ln n$ holds for all $n\ge 2$ with ...
0
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0answers
65 views

An asymptotic involving fractional parts

I guess this is quite well known, but I was not able to find the related result. I want to find an asymptotic estimate for the expression $\sum_{k=1} ^{C\lfloor L \rfloor} \sum_{n=1} ^{\infty} \...
1
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1answer
56 views

On the sum of the reciprocals of the zeros of $\zeta(s)$

It is well known that whenever $\rho$ is a nontrivial zero of the Riemann zeta function $\zeta(s)$, then $1-\rho$ is also a zero. But does the equality $\Re \sum_{\rho} \dfrac{1}{\rho} = \Re \sum_{\...
1
vote
1answer
33 views

Dense on the unit circle

I am reading: "It is sufficient to show that the points $z_n = e^{2\pi in \xi}$ $\:\:n = (1, 2, 3...)$ are dense on the unit circle. ( $\xi$ is an irrational number)" How is this possible? Can ...
0
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0answers
66 views

An analytic formula for the sum of the logs of primes.

I just read in Martin Klazar's Intoduction to Number Theory (page 53), that $\sum_{p\leq x} \log p - \log (p-1) = \log\log x + \gamma + O(1/\log x)$. Where $\gamma$ is the Euler-Mascheroni constant, $...
0
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1answer
37 views

On the sum of the logarithms of primes.

Let $p$ be a prime and $x$ be an integer. It is known that $\sum_{p\leq x} \log p = O(x)$, and i think this is equivalent to the Prime Number Theorem. ...
0
votes
1answer
41 views

What is an upper bound for number of prime powers and semi primes in the interval $[n^2+1,n^2+n]?$

What is an upper bound for number of prime powers in the interval $[n^2+1,n^2+n]?$ What is an upper bound for number of square free semi primes in this interval$?$
2
votes
1answer
81 views

Goldbach Conjecture, what are new research methods after Chen's work?

For Goldbach Conjecture, my understanding is that there are three major methods to attempt it: Schnirelmann density circle method sieve method (Chen used two parameter sieve method to get his ...
0
votes
1answer
155 views

What would the Riemann Hypothesis mean for the Prime Number Theorem?

The Prime Number Theorem states $\pi(n)\sim \dfrac{n}{\ln n}$. Would there be an equally simple expression if Riemann's Hypothesis were proved true? From Chebyshev Function, would $\pi(n)\sim \dfrac{...
4
votes
1answer
81 views

Probability that a number has $m$ indistinct factors

I just discovered Matlab's factor()-function, and I randomly typed in 20081294819, and to my surprise it only had two factors (5099 and 3938281)! I had expected many more factors for such a big number ...
0
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0answers
23 views

$p$-adic digits via character sums

Let $p$ be a prime and let $n = \sum_{k=0}^\infty n_k p^k$ be a $p$-adic integer with each $0 \leq n_k \leq p-1$. Fix $0 \leq c \leq p-1$. Is there a way to check whether the $i$-th digit $n_i$ equals ...
1
vote
1answer
78 views

Elementary proof of the prime number theorem?

The prime number theorem is equivalent to $\lim_{x \to \infty} \dfrac{1}{x} \left| \sum_{n\leq x} \mu(n) \right| = 0$, where $\mu(n)$ is the Mobius function. We know that $\left| \sum_{n\leq x} \mu(...
0
votes
1answer
57 views

On the log of the Riemann zeta function.

Let $\pi(x)$ denote the prime counting function. It is well known that $\log \zeta(s) = \int_{2}^{\infty} \dfrac{s\pi(x)}{x(x^s - x)} \mathrm d{x}$ where $\Re(s)\geq 2$. Inserting $s=4$, we have $\...
0
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1answer
35 views

Comparing Euler products

I have this $a(n)$ is unknown multiplicative function and $b(n)=n$. Let $\zeta(x)$ be Riemman zeta function. And $$B(x)=\zeta^2(x)A(x).$$ where $B(x)=\sum_{n\in \mathbb{N}}\frac{b(n)}{n^x}$ (same ...
3
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0answers
114 views

Combining Firoozbakht's conjecture and abc conjecture

Firoozbakht's conjecture states that for all $n\geq 1$ $$p_n^{\frac{1}{n}}>p_{n+1}^{\frac{1}{n+1}},$$ where $p_k$ the kth prime number. By asumption of this conjecture, for a fixed $n$, there is a ...
0
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0answers
28 views

What is Dirichlet class number formula for d when d is NOT a fundamental discriminant?

According to Wikipedia, Dirichlet published a proof of the class number formula for quadratic fields in 1839, but it was stated in the language of quadratic forms. Let d be a fundamental discriminant,...
2
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1answer
40 views

What are the equivalent statements of GRH using the Möbius or Liouville functions?

We all know that Riemann Hypothesis can be stated as properties of $\mu$ or $\lambda$, particularly in terms of the random behaviour of those functions with "square root" bounds. Are there similar ...
2
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1answer
57 views

What is an upper bound for number of semiprimes less than n?

A semi prime is a number which is product of two distinct prime number. What is an upper bound for number of numbers in the form pq less than n? $p,q$ are prime numbers smaller than $n$.
2
votes
1answer
72 views

Counting squarefree numbers which have $k$ prime factors?

How to find an asymptotic formula for this function below? $$f(n)=\sum_{pq\leq n}1$$ where $p$ and $q$ are different prime numbers. I guess we can write $$f(n)=\sum_{p\leq \sqrt{n}}\pi (\frac{n}{p})...
3
votes
2answers
100 views

What is an upper bound for number of prime powers less than $n$?

What is an upper bound for number of prime powers less than $n$? I mean the numbers in the form $a^b$ in which $b \ge 2$ and $a$ is a prime number. I have found that $\frac {\log n} {\log 2} + \frac ...
1
vote
1answer
40 views

Can class number $h(d)$ equal to zero for some $d$?

We know that $L(1, \chi)$ is related to the class number $h(d)$ with a constant. And this is one way that we can prove $L(1, \chi)$ not vanish on $s = 1$. What confused me is: we know that class ...
1
vote
2answers
62 views

an upper bound for number of primes in the interval $[n^2+n,n^2+2n]$

What is an upper bound for the number of primes in an interval of $n$ consecutive numbers? What is an upper bound for the number of primes in the interval $[n^2+n,n^2+2n]$?
3
votes
2answers
68 views

Convergence of prime zeta function for $\mathfrak R(s)=1$?

By doing some estimates for the partial sums of the Prime zeta function $P(s)=\sum_p p^{-s}$ for $\mathfrak R(s)=1$ I got that $P(1+i\alpha)$ converges for every $\alpha\neq0$... Since I did not ...
0
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0answers
23 views

dirichilet class number and non-vanish of L function at s = 1

All: I have been confused by dirichilet class number formula. We know that L ( 1 , χ ) is related to the class number h(d) with a constant. And this is one way that we can prove ...
0
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0answers
27 views

A question about a sequence of sets of prime numbers deduced from Euclid strategy

Let the sequence of sets of prime numbers defined by $$S_1=\{2\},$$ and for $n>1$ $$S_n=S_{n-1}\bigcup\{\text{p prime such that p divides } 1+\prod_{s_i\in S_{n-1}}s_i\}.$$ Examples. We have $...
1
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2answers
48 views

Dirichlet Convolution of Mobius function and distinct prime factor counter function.

Let us define an Arithmetical function $\nu(1)=0$. For $n > 1$, let $\nu(n)$ be the number of distinct prime factors of $n$. I need to prove $\mu * \nu (n)$ is always 0 or 1. According to my ...
2
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0answers
69 views

Proof that the spectrum of prime distribution will give zeros of Riemann Zeta function

All: Many of us have read that the spectrum of prime distribution will give zeros of Riemann Zeta function. For example, Mazur and Stein's book: (http://wstein.org/rh/rh.pdf ) have many nice pictures ...
1
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0answers
33 views

about sums in analytic number theory

I have attended my first course in analytic number theory (undergrad). I have encountered sums like Gauss' sum, Ramanujan sum and Kloosterman sum. My professor said that they are all over the place in ...
1
vote
1answer
121 views

If a 10 digit number is formed using all the digits from 0 to 9 then find the following . [closed]

A) Find the largest such number divisible by 11111 . No matter what I try , I end up with atleast a digit repeating . Since the question says that the no. has all from 0 to 9 , therefore I cant ...
2
votes
2answers
102 views

Definition of Dirac Delta function on the surface of a unit sphere

I am looking for a definition of a Dirac Delta function which is defined on the 2D unit sphere surface in 3D. In other words, I am looking for a function which is zero everywhere on the 2D spherical ...
2
votes
1answer
78 views

Sums of digits of prime numbers: reference request

I wonder if someone could point out to me a paper on the following problem, if it has been considered at all. If not, it would still be nice to have some good references to good papers related to the ...
2
votes
1answer
49 views

What inequalities similar Lagarias' statement are easy to prove?

Let $$H_n=1+\frac{1}{2}+\cdots+\frac{1}{n},$$ the nth harmonic number and $$\sigma(n)=\sum_{d\mid n}d,$$ the sum of divisor function, for example $\sigma(6)=12$. I believe that this could be a nice ...
1
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0answers
32 views

Applying Green's formula to Petersson inner product.

I'm reading book by Motohashi: spectral theory of Riemann zeta function. And after defining the set of automorphic functions $L^2(\mathcal{F}, d{\mu})$ with Petersson inner product $$\langle f_1, f_2 ...
1
vote
1answer
61 views

Euler product for sum of multiplicative function times log

Let $g$ be a multiplicative function. Iwaniec and Fouvry claim the following identity on p. 273, identity (7.19). Why is this Euler product identity true? $$-\sum_n \mu(n)g(n)\log n = \prod_{p} (1-g(...
4
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1answer
75 views

Is there an upper bound for $\pi (n)-\pi (n/2)$?

Is there a nice upper bound for $\pi (n)-\pi (n/2)$ where $\pi$ is the prime counting function?
0
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0answers
29 views

Upper bounds for the number of roots of polynomials, over finite fields, lying in given extensions

Let $F$ be the finite field with $q$ elements, where $q$ is a power of a prime, and let $E$ be its degree $n \geq 2$ extension. Let $f(x) \in F[x]$ such that $f(E) = F$. Clearly the number of distinct ...
2
votes
0answers
57 views

Lower bound for the values of cyclotomic polynomials evualuated at integers

Let $b,n \geq 2$ be integers and let $\Phi_n(b)$ be the value of the $n$-th cyclotomic polynomial evaluated at $b$. I've recently noticed by computer experiments that whenever $n$ is odd, we seem to ...
1
vote
1answer
43 views

Approximation race : Chebyshev theta vs Mertens third theorem

If $O(R(x))$ is the error term in the PNT what is it for the two different problems $\theta(x)-x$ and Mertens third theorem? Is it $O(xR(x))$ vs $O(R(x))$? Or is there a sharper bound for the first ...
3
votes
3answers
118 views

On the asymptotic growth of the products of prime numbers

Something must be known about the asymptotic growth of the products of prime numbers. Let $p_n$ be the sequence of prime numbers and define $$P_k=\prod_{n=1}^k p_n$$ I'm looking for a sequence $n_k$ ...
0
votes
2answers
64 views

Number of proper divisors $d_1 < \cdots < d_j$ of $n$ such that $\gcd(d_1, \ldots, d_j) = d$

Let $n$ be a positive integer and let $D^*(n)$ be the set of proper divisors of $n$, i.e., positive divisors of $n$ excluding $n$. For every $j \geq 1$, define the function $f_j : D^*(n) \to \mathbb{N}...
-3
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1answer
154 views

Wouldn't the Riemann hypothesis rule out a formula to predict primes? [closed]

Prime formula: a deterministic way to predict primes. Riemann hypothesis: implies "primes are random". If RH is true will we never have a useful prime formula?
6
votes
1answer
178 views

Product of two series to get a series decomposition of zeta in the critical strip

$\def\sfrac#1#2{% \small#1% \kern-.05em\lower0.1ex/\kern-.025em% \lower0.4ex\small#2}$I've been working on gaining an intuitive understanding of the analytic continuation of the zeta ...
0
votes
1answer
51 views

Does this limit involving the Dirichlet eta function and the Riemann zeta function make sense?

Let $p_n$ the sequence of prime numbers (and you will consider below, too, the sequence $\frac{1}{n}$ with $n>1$). And if it isn't wrong for $0<\Re s<1$ the known equation between Dirichlet ...
4
votes
0answers
80 views

Equidistribution theorem of Weyl

Have you examples of applications of Equidistribution theorem of Weyl in proofs of irrationality of numbers? I don't know if "if and only if" is true for this theorem.
3
votes
0answers
68 views

What is the series expansion of reciprocal of theta function $\frac{1}{\theta(z;q)}$?

"The" theta function is an ambiguous concept, but one definition I have found is: $$ \theta(z;q) = (z;q)_\infty(q/z;q)_\infty = \frac{1}{(q;q)_\infty}\sum_{k \in \mathbb{Z}}z^k q^{\binom{k}{2}} \tag{$...