4
votes
1answer
123 views

Estimating integrals involving $\pi(x)$

While solving an exercise in analytic number theory, I ran into difficulty of estimating an integral of the form $\displaystyle\int_{1}^{x} \frac{\pi(t)}{t} dt$ where $\pi(x)$ is the prime counting ...
4
votes
1answer
36 views

Integer solutions to an ellipsoid surface

Given the equation $$x^2+2y^2+5z^2+xz =n$$ where $n$ is any positive integer, what is the smallest odd integer for which no integer solution $(x,y,z)$ exists (i.e. $x,y,z$ are integers)? I know that ...
1
vote
0answers
59 views

Liouville function and PNT

The Big Omega function is defined as the number on non-distinct prime factors of an integer. I.e. $\Omega (2^a3^b...p^z)=a+b+...+z$, and the Liouville function is defined as ...
3
votes
3answers
120 views

What does the integer span of one irrational, and one (possibly irrational) real number look like in $\mathbb{R}$?

My title was rejected a few times, here is what it was initially: If you take two real numbers- one irrational and one possibly irrational - how close does their $\mathbb{Z}$ span come to any ...
30
votes
3answers
623 views

proving that $\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$

Prove that $$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$ ($H_n=\sum_{k=1}^{n}\frac{1}{k}$)
14
votes
3answers
710 views

A closed form for the sum $\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$

How can I find a closed form for the following sum? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
1
vote
1answer
77 views

An upper bound for $-\frac{\zeta'}{\zeta}(s)-\frac{1}{s-1}$

Let $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$. We have $\frac{\zeta'}{\zeta}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}$ for $s>1$, where $\Lambda$ stands for the von Mangoldt function ...
2
votes
0answers
103 views

Integer values of the Riemann function - II

For what value of $n \ge 2$ can we have an real $x > 0$ such that both the numbers $$ \zeta\Big(1+\frac{1}{x}\Big) \text{ and } \zeta\Big(1+\frac{1}{nx}\Big) $$ are positive integers.
5
votes
1answer
155 views

Prime harmonic series

We have following identity: ($p$ is a prime number) $$\left(1+\frac{1}{p}\right)\sum_{k=0}^n\frac{1}{p^{2k}}=\sum_{k=0}^{2n+1}\frac{1}{p^k}$$ Now, How to derive the following inequality from the above ...
3
votes
1answer
131 views

Convergence of the Zeta and Phi functions

I want to show that the following functions (in the picture) are absolutely and locally uniformly convergent if real part of complex number $s$ is bigger than 1. Absolute part for zeta function is ...
1
vote
0answers
77 views

Entropy Rate of a sequence of Random Variables with Distributions related to Primes

Let us consider a stochastic process $\mathcal{X}=\{X_i\}_{i \in \mathbb{N} }$ where $X_i$'s are independent and $X_i$ is distributed as $$X_i=p_k \ \mbox{w. p.}\frac{p_k}{\sum_{l=1}^{i}p_l},\ 1\leq ...
5
votes
3answers
180 views

What is the set $\{x\in\Bbb R\mid \forall q\in\Bbb Q: q^x\in\Bbb Q\}$?

What is the set $\{x\in\Bbb R\mid \forall q\in\Bbb Q: q^x\in\Bbb Q\}$? Of course $\Bbb Z$ is a subset of this set. Are there any other? if not what is the proof? is there a good reference for it?
6
votes
5answers
212 views

$(1+\frac{1}{n\log n})^n-1=O(\frac{1}{n})$.

When I solved a problem, I could solve it if I assumed that $(1+\frac{1}{n\log n})^n-1=O(\frac{1}{n})$ I tried to prove it, but I failed. Actually, I don't convince if it is true. Is it correct? If ...
5
votes
0answers
591 views

one to one mapping between the floor function and the Riemann prime counting function

We have the following 'transform' of a real valued, piecewise continuous function $f(x)$ : $$T[f(x)]=1+\sum_{n=1}^{\infty}\int_{\mathbb{R}^{n}_{+}}f\left(\frac{x}{\Lambda _{n}} \right ...
5
votes
4answers
477 views

Evaluating $\sum\limits_{n=2}^{\infty} \frac{1}{ GPF(n) GPF(n+1)}\,$, where $\operatorname{ GPF}(n)$ is the greatest prime factor

$\operatorname{ GPF}(n)=$Greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. $\operatorname{ LPF}(n)=$Least prime factor of $n$, eg. $\operatorname{ ...
21
votes
4answers
623 views

Evaluating $\sum\limits_{n=1}^{\infty} \frac{1}{n\operatorname{ GPF}(n)}$, where $\operatorname{ GPF}(n)$ is the greatest prime factor

$\operatorname{ GPF}(n)=$Greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. How to evaluate convergence/divergence/value of the sum $$\sum_{n=1}^{\infty} ...
26
votes
3answers
1k views

Ramanujan's First Letter to Hardy and the Number of $3$-Smooth Integers

A positive integer is $B$-smooth if and only if all of its prime divisors are less than or equal to a positive real $B$. For example, the $3$-smooth integers are of the form $2^{a} 3^{b}$ with ...