Questions on the use of the methods of real/complex analysis in the study of number theory.
9
votes
0answers
225 views
Finding the integer $\le n$ with largest number of divisors
As mentioned in an answer to this question an integer less than $n$ with largest number of divisors can be found exploring the numbers $x$ of the form
$$ x = 2^{a_1} 3^{a_2} \dots p_k^{a_k} \dots $$
...
8
votes
0answers
199 views
Quadratic characters and Liouville's function
I'm working through the problems in Montgomery & Vaughan's Multiplicative Number Theory. In Section 11.2 'Exceptional Zeros', Exercise 9a says that for a quadratic character $\chi$, show that for ...
7
votes
0answers
147 views
Equidistribution of roots of prime cyclotomic polynomials to prime moduli
Here is a relevant - and longstanding, I'm told - conjecture. Let $f \in \mathbb{Z}[x]$ be irreducible and of degree > 1. Set
$E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { ...
7
votes
0answers
147 views
Weak version of Fortune's conjecture
Let $p\#=2\cdot3\cdot5\cdots p$ denote the primorial and $N(x)$ the smallest prime greater than or equal to $x$. Then Fortune's conjecture is that $N(p\#+2)-p\#$ is prime for all $p$. (Heuristic: to ...
7
votes
0answers
247 views
How can we prove a simple case of the High Indices Theorem?
Let $(a_n)$ be a sequence of real numbers such that
$$f(x) = \sum_{n=1}^{\infty} a_n x^{2^n}$$
converges for $|x| < 1$ and $f(x)$ converges to $a$ as $x \to 1^{-}$. Then I have to prove that $\sum ...
6
votes
0answers
225 views
Question about a proof in Iwaniec-Kowalski's Analytic Number Theory book
My question is about the end of the proof of theorem 1.1, in page 27.
Namely, it is stated that whenever we have a multiplicative function $f:\mathbb{N} \to \mathbb{C},$ let the sequence ...
6
votes
0answers
184 views
When does the “Zetor function” converge?
Let $p_n$ be the n'th non-trivial zero of the Riemann zeta function. We define the Zetor function (acronym of 'zeta' and 'zero') as follows: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}. ...
5
votes
0answers
175 views
Proof of Hardy-Ramanujan inequality in number theory.
On page 3 of http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf the author write that the following inequalities follow from "the Hardy-Ramanujan inequality", but he doesn't point to a proof. The ...
5
votes
0answers
539 views
one to one mapping between the floor function and the Riemann prime counting function
We have the following 'transform' of a real valued, piecewise continuous function $f(x)$ :
$$T[f(x)]=1+\sum_{n=1}^{\infty}\int_{\mathbb{R}^{n}_{+}}f\left(\frac{x}{\Lambda _{n}} \right ...
5
votes
0answers
302 views
Residue of Rankin-Selberg L-function for non-trivial nebentypus
Let $f\in S_k(\Gamma_0(N),\chi)$ be a normalized holomorphic newform (i.e. weight $k$, level $N$, nebentypus $\chi$) and write its Fourier expansion as
$$ f(z)=\sum_{n\ge 1} ...
5
votes
0answers
104 views
Dedekind sums + strange integral
I need to write $\displaystyle \int_0^1 f(x)\,dx$, where $f(x) = \# \Sigma_{pq} \cap (x, x+1)$ for $x \in [0, 1]$, where $$\Sigma_{pq} = \left \{ \frac{k}{p} + \frac{l}{q}:\ 1 \leq k \leq p-1, 1 \leq ...
5
votes
0answers
106 views
Limit inferior of the quotient of two consecutive primes
I have recently read an article about the prime number theorem, in which Mathematicians Erdos and Selberg had claimed that proving $\lim \frac{p_n}{p_{n+1}}=1$, where $p_k$ is the $k$th prime, is a ...
5
votes
0answers
106 views
Density of products of a certain set of primes
I have an infinite set S of prime numbers with relative density 0 (that is, $\lim_ns_n/p_n=\infty$ with $S=\{s_1,s_2,\ldots\}$ and $s_1 < s_2< \cdots$). I would like to find the size (in some ...
5
votes
0answers
150 views
$2, 5, 13, 17, 29, 421, 401, 53, 281,…,\rightarrow \infty$? $a_{n+1}=\operatorname{ GPF}(qa_n+p)$
I denote by $\operatorname{ GPF}(n)$ the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.
Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...
5
votes
0answers
169 views
How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$
I'd like to simplify
$$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
4
votes
0answers
68 views
Sum of rational numbers given some properties
Let $R(n)$ denote the sum of all positive rational numbers whose numerators and denominators are less than or equal to $n$ and have no common factors. I have estimated this sum to be
$$
\begin{align*}
...
4
votes
0answers
148 views
Sum of reciprocal of primes in arithmetic progression
In http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf on page 6 (top) the author states that:
$$
\sum_{p \le x, \ p \equiv 1 \bmod l} \frac{1}{p} =
\frac{\log \log x}{\phi(l)} + O \left ( \frac{\log ...
4
votes
0answers
134 views
Questions about the L-function for Eisenstein Series
$E_a(z,s)$ denotes the Eisenstein series expanded at the cusp $a$. For each cusp $a=\frac{u}{w}$ of $\Gamma_0(N)$, we define the Eisenstein series
$$
...
4
votes
0answers
217 views
Evaluating a series with the Möbius function and greatest common divisor.
Problem: Let $\gcd(a,b,c,d)$ refer to the largest integer $r$ such that $r$ divides each of $a,b,c,d$. Evaluate the series ...
4
votes
0answers
71 views
Easiest proofs for why $L(1,\chi)$ won't vanish?
The usual proofs in number theory books seem rather annoying if the goal is to actually be able to remember the proof in the future. At the same time these books typically assume that the reader's ...
4
votes
0answers
123 views
Euler summation and its transformation
The following results:
For any function $f \in C^1[a,b]$ and any $q \in \mathbb{N}$,
$$\sum_{a<k \leq b, (k,q)=1} f(k)=\frac{\varphi(q)}{q} \int_a^b f(x) dx + O(\tau(q) (\sup_{x \in [a,b]} ...
3
votes
0answers
56 views
A proof concering $\Re(\log\zeta(\sigma+it))$
I have been trying to prove that
$$\Re(\log\zeta(\sigma+it))=\sum_{n=2}^\infty\frac{\Lambda(n)}{n^\sigma\log n}\cos(t\log n),$$
but now I've given up, so I looked up the answer in the back of the ...
3
votes
0answers
87 views
existence of closed forms of certain Dirichlet series
In the following post we find an interesting calculation of
$$ \sum_{n\ge 1} \frac{\mu(n)^2}{n\varphi(n)}.$$
I had been trying to do this calculation myself, setting the slightly more ambitious goal ...
3
votes
0answers
34 views
Schneider's theorem about the transcendence of values of the $j$-function
It is known that the $j$-function takes algebraic values when evaluated at imaginary quadratic integers. This is a result that was proved by Schneider in 1937 apparently. To be precise, Schneider ...
3
votes
0answers
111 views
Show that $n \sum\limits_{p \leq n} \frac{\log(p)}{p} = n \log(n) + \mathcal{O}(n)$
Using the fact that $\log(n!) = n \log(n) - n + \mathcal{O}(\log(n))$ I am asked to show that:
$$ n \sum_{p \leq n} \frac{\log(p)}{p} = n \log(n) + \mathcal{O}(n) $$
Prior to this result it was ...
3
votes
0answers
78 views
Products of primes of the form $an + b$
What is the asymptotic order of numbers divisible by no primes except those of the form $an+b$ ($a$, $b$ fixed)?
Surely (except for the trivial cases) they are of order strictly between that of he ...
3
votes
0answers
117 views
Dirichlet series represents an analytic function
Let $$T(x)=\sum_{n \leq x} t_n$$ and $T(X)=O(x^a)$ for $a \geq 0$. Now let $$F(s)=\sum_{n=1}^{\infty} \frac{t_n}{n^s}$$ What needs to be checked to prove that this Dirichlet series represents an ...
2
votes
0answers
47 views
Rationality of Polynomial Coefficients. Integral Question.
We are entertaining polynomials with roots, all unique, on the curve $\Upsilon_s = \{{( 1-\cos[\theta])^{-s} \exp(i \theta)} \ | \ \theta \in \mathbb{R} \}$, where $s>0.$ $\Upsilon_s$ looks like a ...
2
votes
0answers
108 views
Partial summation of a harmonic prime square series (Prime zeta functions)
I am trying to find the following series:
$S=\displaystyle\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\dfrac{1}{p_ip_j},A\leq p_1 < p_2 < \dots < p_n \leq B, \lbrace A,B\rbrace \in \mathbb{N}$
...
2
votes
0answers
78 views
references for an arithmetic function
I was wondering if anyone is aware of any existing literature on the arithmetic function defined as $$f(n):=2^{\omega(n)}\tau(n).$$ Here $\omega(n)$ is the number of distinct prime divisors of $n$ and ...
2
votes
0answers
85 views
Dirichlet series 'shifted' by a polynomial
Let $F(x) \in \mathbb{Z}[x]$ and
$$
\xi(s) = \sum^\infty_{n=1}g(n)n^{-s}
$$
be the Dirichlet series associated an arithmetic function $g(n)$. Define a new Dirichlet series
$$
\xi_F(s) = ...
2
votes
0answers
117 views
Subadditive in analytic number theory
I have just encountered the following question:
Let $C_n$ be a sequence of real numbers with the following three properties:
1) $C_n$ is subadditive, such that $$C_{m+n} \leq C_m +C_n$$
2) ...
1
vote
0answers
42 views
Prime-Like sets
I need some examples of "prime-like" sets of numbers. May be this term is already known by some other standard name. Let me define it. A set $S=\{s_1,s_2,\ldots ,s_n\}\subset \mathbb{R}$, is called ...
1
vote
0answers
43 views
Consequencesof the Hadamard product expression of $L(s, \chi)$
I'm going through my lecture notes and I'm stuck on the proof of
For any $t>0$ and primitive $\chi$ modulo $q$
$$\sum_{\rho=\beta+i \gamma: \Lambda(\rho, ...
1
vote
0answers
17 views
Using Gamma function to show the limiting case of Gordon's continued fraction as q approaches i.
A question similar to: How to derive the Golden mean by using properties of Gamma function?
The limiting case of Gordon's continued fraction when $q$ approaches $i$ yields:
$$\sqrt2 + 1 = ...
1
vote
0answers
21 views
Existence of zeros of Mellin transform and properties of function to be transformed
Mellin transform of function $f(x)$ defined for $x\geqslant 0$ is given by
$$
f^\ast(z) =\int\limits_0^\infty x^{z} f(x) \frac{dx}{x}.
$$
I consider only exponentially decreasing (there exist such ...
1
vote
0answers
68 views
By establishing a recurrence relation and using induction, or other-wise, show that this sequence is 3-adically Cauchy?
this is a question from a book I'm struggling with, please could you provide a clear proof
Consider the sequence of rational numbers
$a_1 = 1+3,a_2 = 1+\frac{3}{1+3},a_3= 1 + \cfrac{3}{1
...
1
vote
0answers
65 views
Applications of prime-number theorem in algebraic number theory?
Dirichlet arithmetic progression theorem, or more generally, Chabotarev density theorem, has applications to algebraic number theory, especially in class-field theory.
Since we might think of the ...
1
vote
0answers
51 views
modulus of a series
My question is about the possibility of calculating the modulus of the Dirichlet Eta Function for complex numbers with positive real part. This series is uniformly convergent but it is not absolutely ...
1
vote
0answers
71 views
relationship between Connes trace formula and Weil's trace formula
Connes trace formula $$ Tr{U(h)}=2h(1)log\Lambda + \sum_{v} \int d^{*}x \frac{h(u^{-1})}{|1-u|} $$
Weil's trace $$ \int_{C}h(u)|u|d^{*}u- ...
1
vote
0answers
45 views
Asymptotic bounds: $\ll$ vs. $\ll_{\epsilon}$?
I am feeling a bit slow today. In Analytic Number Theory it is usual to express asymptotic bounds by specifying the relation of the constant to a specific variable, i.e.
$\log n \ll_\epsilon ...
1
vote
0answers
94 views
Evaluate a sum with Chebyshev's and Mertens' estimation
So if using only Chebyshev's and Mertens's estimation, how to evaluate the following series:
$$\sum_{p\geq 3}\frac{1}{p \log{\log{p}}} ?$$
1
vote
0answers
117 views
Binary sequences in primes
Is anything known about these problems?
If we make a string S of 0's and 1's with 1 in n'th position if the the nth prime $p_n$ is of the form $1+m 2^{9^{9^{9^{9}}}}$, else 0, does every finite string ...
1
vote
0answers
184 views
Dirichlet's Class Number and its connections with the $GL(2)$
i posted the same question on MO,but cant get an answer so i am trying here
note:all those who answer my question just mention the question number in their reply so that i can tally them,thanks a ...
1
vote
0answers
117 views
Bounds for Fourier coefficients of cusp forms
I've asked the background question here, which still left unanswered.
Now I have a more precise question. In my homework I've been asked to prove that
$$\left| \sum_{1\leq n \leq N} a_f (n)e^{2\pi i ...
0
votes
0answers
43 views
Vanishing of Dirichlet Series
Suppose the function
$\sum_{n=1}^{\infty}{a_{n}n^{-s}}$ is $0$ on some open set $U\subset\mathbb{C}$. (Can assume the sum converges absolutely on $U$.)
Is it true that $a_{n}=0$ for all $n$?
(This ...
0
votes
0answers
96 views
Show that the field of p-adic numbers is complete
this is a question from a book I'm struggling with, please could you provide a clear proof
Show that the field of p-adic numbers is complete
i.e. that a sequence of p-adic numbers converges if and ...
0
votes
0answers
89 views
Divergence of the Derivative of the Prime Counting Function
On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written
$$
\pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1}
$$
with $ \operatorname{R}(z) = ...
0
votes
0answers
87 views
$L(1+it,\chi)\neq 0 $ whenever $t \neq 0 \in \mathbb{R}$
I understand that the proof of the assertion in the title uses the same method which proves that zeta function satisfies $\zeta(1+it)\neq 0$, where the above $L$ is Dirichlet L-function.
I.e, you ...
0
votes
0answers
143 views
Has it been ruled out that the Riemann hypothesis fails for only finite number of zeros?
Has it been ruled out that the Riemann hypothesis fails, but fails only for finite number of zeros?
