# Tagged Questions

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

52 views

29 views

65 views

### On the proof of “The infinite series $\sum_{n=1}^{\infty} p_n^{-1}$ diverges”.

The following text is from the book Introduction to Analytic Number Theory by T. M. Apostol : Theorem 1.13 $\$ The infinite series $\sum_{n=1}^\infty 1/p_n$ diverges. Proof. The following ...
44 views

### Estimates of $\Omega_{\text{av}}(n)$

Ramanujan proved that the average number of distinct divisors of $x$ for $x$ on $[1,n], ~\omega_{\text{av}}(n),$ and the average number of divisors including repetitions, $\Omega_{\text{av}}(n),$ are ...
58 views

### Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...
68 views

### Combinations of four consecutive primes in the form $10n+1,10n+3,10n+7,10n+9$

Here $n$ is some natural number. For example, among the primes $< 1000$ I found four such combinations: \begin{array}( 11 & 13 & 17 & 19 \\ 101 & 103 & 107 & 109 \\ 191 &...
49 views

### Which values of $n$ is this inequality related to prime numbers true for?

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the $i$-...
125 views

39 views

36 views

I am trying to learn Bertrand’s postulate. I can not understand two steps Why $\displaystyle\sum_{n \leq x}\log n=\sum_{e \leq x} \psi\left(\frac{x}{e}\right)$, where $\psi(x)=\displaystyle\sum_{p^\... 0answers 30 views ### Upper Bound on Li's criterion Background: Bombieri and Lagarias showed that a function$f$with roots$\rho=x+iy$satisfies has all its roots lying on$x=\frac12$if and only if $$\lambda_n :=\sum_\rho 1-\left(1-\frac{1}{\rho}\... 0answers 82 views ### Finite Messy Trigonometric Sum Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$The source of this ... 1answer 75 views ### Asymptotic expression for sum of first n prime numbers? Is one known? If not, what are the best known bounds? Is there reason to think that an asymptotic expression is beyond current methods if none exists? 0answers 31 views ### \eta(s)+\eta(1-s)=F(s)-G(s) and roots of F(s),G(s) are on the critical line Wusheng Zhu in 2012 uploaded to arxiv.org an interesting preprint titled "Riemann Zeta Function Expressed as the Di fference of Two Symmetrized Factorials Whose Zeros All Have Real Part of 1/2" (arxiv:... 0answers 32 views ### Inequality of the Logarithmic Derivatives of a Sequence of Hypergeometric Functions For brevity's sake, define the following sequence of (Gaussian) hypergeometric functions for each n\in\mathbb{N}:$$f_n(z)={}_2 F_1(-n,-(n-1);2;z).$$I wish to show that the logarithmic derivatives ... 0answers 10 views ### Relation between support of a function and that of its DFT Let f : \mathbb{Z}_N \to \mathbb{C}, let \zeta_N be a primitive N-th root of unity and let \hat{f} : \mathbb{Z}_N \to \mathbb{C} be the DFT of f given by \hat{f}(m) = \sum_{n \in \mathbb{Z}... 0answers 24 views ### Evaluating Certain L-functions I have found a systematic way to find the exact value of the L-series$$L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$for s a positive even integer if \chi(-1)=1 and s odd and positive if ... 1answer 82 views ### How do you refute these conjectures that seem imply contradictory statements? I've formulated two conjectures that seems to imply a strong result when are combined with well known equivalences of the Riemann hypothesis, and I would like to know how get a disproof of such ... 1answer 52 views ### Context of this problem: \sum_{n\,\text{odd}} (-1)^{\frac{n-1}2}\frac{\log n}{\sqrt{n}} /\sum_{n\,\text{odd}}(-1)^{\frac{n-1}{2}}\frac{1}{\sqrt{n}} [duplicate] I remember seeing this somewhere a while ago - I'd given it a go but it was - and still is - beyond my capabilities. The problem came with the tag: "requires knowledge of analytic number theory". I am ... 0answers 27 views ### Questions about totient function See this image (https://en.m.wikipedia.org/wiki/Euler%27s_totient_function#/media/File%3AEulerPhi.svg) from wikipedia. I can make out two lines that have a high density of values along them. The top ... 1answer 29 views ### Residue of Rankin-Selberg Dirichlet series Let f\in S_k(N,\chi) be a cusp forms, and let R_f(s)=\sum_{n=1}^{\infty}\frac{a(n)^2}{n^s} the Rankin-Selberg Dirichlet series then R_f(s) hase a pole at s=k. Can someone suggest to me a ... 0answers 41 views ### Natural numbers, divisors, primes and their generalized means Let div, nat and pri the finite sequences given in increasing order for an integer n\geq 1 of its divisors 1=d_1<d_2<\ldots d_{\sigma_0(n)}=n, the first n natural numbers, and the first n... 0answers 19 views ### A function related to divisior counting function Let d(n) be the divisor function. Let d_{2}(n)=d(d(n)), d_{3}(n)=d(d(d(n))), d_{4}(n)=d(d(d(d(n)))) and so on... We're gonna define f(n), the smallest number satisfies d_{f(n)}(n)=2. For ... 1answer 34 views ### \sigma _{0}(n)=\sigma _{0}(n+1) will occur infinitely often. [closed] In 1984, Roger Heath-Brown proved that will occur \sigma _{0}(n)=\sigma _{0}(n+1) infinitely often. How did he prove that? I couldn't find the paper on the internet. 2answers 54 views ### What is the asymptotic behaviour of \sum_{p_k\leq x}kp_k, where p_k is the kth prime number? I would like to study the asymptotic behaviour of this sequence A014285, see as OEIS, that seems has few references and a good behaviour (see the sequence as graph)$$\sum_{k=1}^nkp_k,$$where p_k ... 0answers 58 views ### On the asymptotic limit of the divisor function. It is known that$$ \limsup_{k \to \infty} \frac{\sigma(N_k)}{e^{\gamma}N_k \log \log N_k} = \frac{6}{\pi^2}$$Where$N_k$is the$k$-th primorial,$\sigma$is the divisor function and$\gamma$is ... 0answers 32 views ### Finite sequence created by reducing$n$with each prime under$n$ends in$0$? Given$n$a fixed integer we constuct the following sequence:$a_0=n$,$a_i=\lfloor \frac{a_{i-1}(p_i-1)}{p_i}\rfloor$. For what values of$n$do we have$a_{\pi(n)}=0$? Computer calculation shows ... 0answers 27 views ### A linearly uniform but quadratically non-uniform set I have been working on this problem for a while but have no clue at all. Fix a smooth cutoff function$\varphi: \mathbb R/\mathbb Z\rightarrow [0,1]$supported on$[−\varepsilon-\delta, \...
Suppose we have a function $f : \mathbb{N_0} \to \mathbb{R}$ for which we can give an estimate of its values, and say its values $f(n)$ are roughly uniformly distributed for $n$ in some range $[1,N]$. ...