# Tagged Questions

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

214 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 ...
163 views

520 views

### Some identities with the Riemann zeta function

Can someone either help derive or give a reference to the identities in Appendix B, page 27 of this, http://arxiv.org/pdf/1111.6290v2.pdf Here is a reproduction of Appendix B from Klebanov, Pufu, ...
96 views

### Between Mertens' theorems

It is well-known that $$\sum_{p\le x}\frac{\log p}{p}=\log x+O(1)$$ and $$\sum_{p\le x}\frac1p=\log\log x+M+o(1).$$ What is the order of $$\sum_{p\le x}\frac{\sqrt{\log p}}{p}$$ ?
789 views

### Proofs of trivial zeros of zeta function?

I know that the trivial zeros of zeta function are negative even integers . I have seen the wiki-proof using the functional equation of zeta function, I might have seen a proof using Bernoulli ...
504 views

### Newman's “Natural proof”(Analytic) of Prime Number Theorem (1980)

I am trying to understand this short proof by newmann. I faced some problems while grasping this very proof. Please help me out. 1 . I am not clear, why in step (1)'s proof he says that from unique ...
321 views

98 views

### Prove that $\sum_{n=1}^\infty \frac{\sigma_a(n)}{n^s}=\zeta(s)\zeta(s-a)$

I would appreciate a hint concerning how to surpass the roadblock I've encountered in my attempt at a proof below. A nicer proof than mine would also help (Edit: The latter part is now done by Gerry ...
157 views

### Elementary Proof of Landau's count on number representable as sum of two squares

In Analytic Number Theory by Iwaniec and Kowalski, there is an elementary proof of Landau's result of $\#\{n \le x: \exists a, b,\ s.t.\ n = a^2 + b^2\} \sim Cx/\sqrt{\log x}$ with an explicit ...
80 views

### Goldbach weak conjecture verification

I found from http://en.wikipedia.org/wiki/Talk:Goldbach%27s_weak_conjecture that Goldbach's weak conjecture might have been proven but the proof has not been peer reviewed yet. What results of the ...
104 views

### Prove that the number of primes satisfying this is $\log(n)$

Let $\textrm{ord}(n)$ denote the number of primes $p$ such that order of $10$ modulo $p$ is $n$. Prove that $\textrm{ord}(n) \sim \log(n)$.
45 views

### Apostol ANT chapter 13 Question 9

Given $L(s,\chi)$ has a zero of order $m\ge1$ at $s=1+it$, prove that for this t we have: (a) $\frac{L'}{L}(\sigma+it,\chi)=\frac{m}{\sigma-1}+O(1)$ as $\sigma \to 1^{+}$ and (b) there exists an ...
102 views

### Dirichlet series for prime sequence

With $p_n = n^{th}$ prime and $f(s):=\sum_{n=1}^\infty 1/p_n^s$ when the series converges. What is the status of the following questions: What is the abscissa of convergence of $f$? What are the ...
67 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 ...
177 views

54 views

### Closed forms for $\lim_{x\rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$

Im looking for closed forms for $\lim_{x \rightarrow \infty} \ln(x) \prod_{x>(p-a)>0}(1-(p-a)^{-1})$ where $x$ is a positive real, $a$ is a given real, $p$ is the set of primes such that the ...
84 views

### To which extent distribution of Riemann non-trivial zeros follow a gauss process?

I am trying to clearer and preciser understand to which extent the distribution of the non-trivial zeros of the Riemann $\zeta$-function follow a Gauss process? Yet, what I figured out from readnigs,...
43 views

### What is known about meromorphic functions agreeing with $\pi(n)$?

Let $f$ be a meromorphic function in some region containing the positive real axis such that $f(n) = \pi(n)$ for all but finitely many positive integers $n$, where $\pi(n)$ is the number of primes ...
206 views

35 views

### Proving that $\sum_{i=2}^{M}\frac{\pi(x^{1/i})}{i}=O(x^{1/2})+O(Mx^{1/3})$

How do I prove that $$\sum_{i=2}^{M}\frac{\pi(x^{1/i})}{i}=O(x^{1/2})+O(Mx^{1/3}).$$ I tried to use Prime Number theorem for $\pi(x)$ and then approximating the summation by integral, but when I used ...
93 views

### Primes in binary

Let $$S_n(k)=\{1\leq m\leq n: m\ \mbox{has k ones in its binary representation and m is prime}\}\ \\ \forall \ n\geq 2^k-1,\ k\geq1.$$ Let $\pi(x)$ be the prime number function. Then what can be ...