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

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53 views
+50

Is it possible to use Möbius inversion on the last equation to get $\mathrm{core}(n)$?

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 ...
0
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1answer
25 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 ...
2
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0answers
20 views

Question on a proof of the euler product of the zeta function

Let $\zeta(s)$ be the Riemann zeta function, then we know it satisfies the Euler product for Re$(s) > 1$, $$ \zeta(s) = \prod_{p} (1 - p^{-s})^{-1}. $$ The proof I read, if I recall correctly, was ...
2
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1answer
35 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 ...
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3answers
65 views

At what rate are composites removed in a set after each prime multiple is cancelled out?

I was looking at sieves today, mainly sieving for primes and I noticed a pattern type thing. As I crossed out primes in a small set, the number of composites that were crossed out decreased. I haven't ...
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0answers
14 views

On the upper bound for the Chebyshev function: What am i missing here?

The Chebyshev second function is defined as $\psi(x) = \sum_{p^m \leq x} \log p$, where $p$ is a prime, $m\geq 1$ is an integer and $n=p^m$. It is known that there exist positive constants $c_1$ and ...
7
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1answer
101 views

Arithmetic/category theoretic information encoded in $q$-series reciprocals

According to the pentagonal number theorem: $$\prod_{n=1}^{\infty} (1-q^{n}) = \sum_{k=-\infty}^{\infty} (-1)^{k}q^{k(3k-1)/2}$$ Now the reciprocal of this has the partition numbers $p(k)$ in its ...
9
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1answer
348 views

Interpolating the primorial $p_{n}\#$

The primorial $p_{n}\#$ is given by the product $p_n\# = \prod_{k=1}^n p_k$ (where $p_{k}$ is the $k$th prime) -- is there a natural (a la the gamma function $\Gamma(z)$) way of interpolating it for ...
0
votes
1answer
29 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 $\log \zeta(s) = \int_{2}^{\infty} \dfrac{s\pi(x)}{x(x^s - x)} \mathrm d{x}$ ...
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0answers
22 views

How to prove $M(x)\ll \frac{x}{\log(x)^A}$ implies that $\psi(x)\ll \frac{x}{\log(x)^A}(\log\log x)^2$?

I've been working on a variant of a proof of the prime number theorem using Granville's pretentious methods. I want to prove that $$M(x)\ll \frac{x}{\log(x)^A}\implies \psi(x)\ll ...
5
votes
0answers
154 views

Relationship between Dixonian elliptic functions and Borwein cubic theta functions

In this paper, it says that the three Borwein cubic theta functions obey the identity $a(q)^{3}=b(q)^{3}+c(q)^{3}$, which is strongly reminiscent of the identity that Dixonian elliptic functions obey ...
1
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1answer
38 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 ...
8
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0answers
198 views

Modular transformation of $\eta(\tau)$

I know that the Dedekind $\eta$ function can be represented in the form$$\eta(\tau) = q^{1\over{24}} \prod_{n = 1}^\infty (1 - q^n) = \sum_{n = -\infty}^\infty (-1)^n q^{{3\over2}\left(n - ...
1
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0answers
47 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 ...
0
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0answers
59 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} ...
2
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4answers
84 views

Probability that a Mersenne number is prime

Let $p$ be a prime and let $M_p = 2^p-1$ be a (Mersenne) number. Is there any known result on the probability that $M_p$ is prime? In particular is it known whether the probability tends to $1$ as $p ...
1
vote
1answer
25 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
62 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
32 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. ...
6
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1answer
448 views

Evaluating a limit of the truncated exponential series motivated by the prime number theorem for $k$ distinct prime factors.

If $\pi_k(n)$ is the cardinality of numbers with k factors (repetitions included) less than or equal n, the generalized Prime Number Theorem is: $$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln \ln ...
0
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1answer
26 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
28 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 ...
4
votes
1answer
154 views

Brun's sieve bounds

Working from Halberstam-Richert they state the following bounds \begin{align} S(\mathcal{A}; \mathfrak{P}, z) \leq XW(z)\left(1 + 2 \frac{\lambda^{2b + 1}e^{2\lambda}}{1 - \lambda^2 e^{2 + ...
0
votes
1answer
106 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 ...
0
votes
1answer
28 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 ...
2
votes
1answer
63 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 ...
4
votes
1answer
58 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 ...
4
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4answers
377 views

What is the best way to supplement a complex variables class to make it more complete for a math major?

For the upcoming semester I plan on a taking a “complex variables” course that many people, including myself, would not consider a true complex analysis class. I know that the course will likely use a ...
3
votes
0answers
96 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 ...
3
votes
2answers
59 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
19 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
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1answer
58 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} ...
0
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1answer
52 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 ...
-1
votes
0answers
44 views

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

A semi prime is a number which is product of two distinct primes. What is an upper bound for number of semi primes in the interval $[n^2,n^2+2n]$?
0
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0answers
18 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 ...
2
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1answer
40 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
44 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 ...
3
votes
2answers
76 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
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1answer
35 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 ...
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vote
2answers
55 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 $[n^2+n,n^2+2n]$?
2
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1answer
334 views

How do we prove $p_n\sim n\log(n\log(n))$ from the Prime Number Theorem?

Let $p_n$ be the $n$th prime. Could someone please help me with the steps between $\pi(n)\sim\dfrac{n}{\log(n)}$ and $n=\pi(p_n)$, to the statement $p_n\sim n\log(n\log(n))$?
0
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0answers
20 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
24 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
vote
2answers
27 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
votes
0answers
52 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
26 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
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1answer
46 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
1answer
74 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 ...
1
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1answer
39 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
43 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 ...