Questions tagged [analytic-number-theory]
On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.
3,066 questions
-1
votes
1
answer
304
views
Consequences of the degree conjecture
the title is quite explicit: I would like to know the consequences of the degree conjecture for the Selberg class.
Thank you in advance.
- 6,984
7
votes
2
answers
275
views
Estimating the size of reduction of rational points on $\mathbb{G}_m^2$
Hi,
Let $\Gamma$ be a free subgroup of rank 2 in $\mathbb{G}_m^2(\mathbb{Q})$. For all but finitely many primes p we can reduce $\Gamma$ modulo p. Let $S$ be the of primes for which $\Gamma$ does not ...
- 527
8
votes
5
answers
3k
views
Exponential sums for beginner.
What are the good books, online lecture notes or starting material on exponentials sums with applications in number theory for a beginner, apart from N. M. Korobov's book? The book or notes should ...
7
votes
2
answers
1k
views
Salie-type sum bound
I am interested in bounding the following Salie-type ("twisted Kloosterman") sum
$$
S(a,b,\beta) = \sum_{x \in \mathbb{Z}/{p^{\beta}}\mathbb{Z}} \left( \frac{x}{p^{\beta}} \right) \chi(ax + bx^{-1}).
...
- 197
24
votes
2
answers
2k
views
Does the quadratic form $x^2 - 7y^2$ represent infinitely many primes, with the restriction that $0 < y < x/10$?
Surely yes, and in more generality, but can it be proved?
It seems that most, if not all, statements about quadratic forms representing primes fall back on algebraic number theory (i.e. splitting of ...
- 7,347
9
votes
1
answer
3k
views
Values of the Riemann zeta function and the Ramanujan summation - How strong is the connection?
(This Question was taken from MSE. As Eric Naslund pointed out there, this question is relevant. The summation method mentioned in this question is actually a good answer to it.)
The Ramanujan ...
- 4,829
7
votes
3
answers
1k
views
Asymptotic Formula for a Mertens Style Sum
Hello,
I am wondering if there is a simple asymptotic formula for
$$\sum_{p\leq x}\frac{\left(\log p\right)^{k}}{p},$$
where $k\geq0$ is some integer. If $k$ is $0,$ by using the Prime Number ...
- 11.4k
18
votes
2
answers
1k
views
Lower bounds on the easier Waring problem
The easier Waring problem asks for the least number $v=v(k)$ such that every every integer is a sum of $v$ $k$'th powers with signs, i.e. every $n\in \mathbb{N}$ is of the form $$n=x_1^k\pm x_2^k\pm\...
- 7,836
15
votes
2
answers
1k
views
Evaluating the integral $\int_{1}^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$
I am trying to find a formula for the following integral for non-negative integer $k$:
$$\int_1^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$$
My first thought was to use the formula $$\...
- 11.4k
8
votes
3
answers
2k
views
Evaluating the integral $\int_0^\infty \frac{\psi(x)-x}{x^2}dx.$
Let $\psi(x)=\sum_{n\leq x} \Lambda(n)$ be the weighted prime counting function. I am trying to evaluate the integral $$\kappa:=\int_{1}^{\infty}\frac{\psi(x)-x}{x^{2}}dx$$ in several different ways. ...
- 11.4k
15
votes
1
answer
1k
views
Is there an elegant algebraic proof of this formula for quadratic field discriminants?
Consider the Dirichlet series counting discriminants of real quadratic fields. Quadratic field discriminants are "basically" squarefree integers, so the associated Dirichlet series $\sum D^{-s}$ is "...
- 7,347
21
votes
4
answers
3k
views
Is the Euler product formula always divergent for 0<Re(s)<1?
It is known that the Euler product formula converges for $\Re(s)>1$
(and there it represents the Riemann zeta function).
My question: Is the Euler product always divergent for
$0 < \Re(s) < ...
- 255
20
votes
3
answers
2k
views
Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field.
Let $M$ be the splitting field of
x^8 + 3*x^7 + 13*x^6 + 17*x^5 + 45*x^4 + 37*x^3 + 11*x^2 + 112*x + 108
over the rationals. If I've understood some tables ...
- 41.4k
15
votes
5
answers
9k
views
Upper bounds for the sum of primes up to $n$
Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the sum of the first $n$ ...
- 3,463
18
votes
2
answers
1k
views
Why isn't meromorphic continuation enough for converse theorems?
This is a very naive question which really does little more than highlight my ignorance of how converse theorems really work.
Take an algebraic gadget which should be conjecturally associated to an ...
- 41.4k
4
votes
1
answer
1k
views
Exact formula for the number of integers in an interval which are the sum of two squares.
Denote by $\lambda(n)$, the number of numbers between $0$ and $n$ which are the sum of two squares. Landau, and Ramanujan have proven independently, that $$\lambda(n) \sim \frac{n}{\sqrt{\ln(n)}}$$
...
- 829
16
votes
1
answer
2k
views
Why is the functional equation of the Riemann zeta function equivalent to the Poisson summation formula?
We can derive from the Poisson summation formula the modularity of the Theta function, which results in the functional equation. In his book on the Riemann Zeta function, Patterson mentions also that ...
- 11.2k
1
vote
3
answers
492
views
Cos[Im[zetazero(n)]Log(prime)] spans a countable dense set in [-1,1]?
It is known that cos(N) spans a countable dense set in [-1,1].
(N: any natural number)
As far as I know generally, for any continuous function f defined in [a,b],
f is Riemann integrable where its ...
- 255
4
votes
1
answer
708
views
Calculating the constant in the Bateman-Horn-Stemmler conjecture
Bateman & Horn [1], building on Bateman & Stemmler [2], give a conjectured formula for the density of numbers that produce simultaneous primes in a number of fixed polynomials.
The constant ...
- 9,114
15
votes
3
answers
1k
views
There is no lattice in PSL(2,R) which contains PSL(2,Z) properly?
How can I see that there is no lattice in $G=\mathrm{PSL}_2( \mathbb{R})$ which contains $\Gamma_1=\mathrm{PSL}_2( \mathbb{Z})$ properly, or equivalently, that $X_1 =\mathrm{PSL}_2(\mathbb{Z}) \...
- 11.2k
40
votes
5
answers
8k
views
Is $\zeta(3)/\pi^3$ rational?
Apery proved in his paper from 1979 that $\zeta(3)$ is irrational, and we know that for all integers $n$,
$\zeta(2n)=\alpha \pi^{2n}$
for some $\alpha\in \mathbb{Q}$. Given these facts, it seems ...
- 7,013
19
votes
3
answers
6k
views
Are the nontrivial zeros of the Riemann zeta simple?
A few years ago, I found on arXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann zeta function were all ...
- 6,984
18
votes
4
answers
4k
views
are there infinitely many triples of consecutive square-free integers?
The title says it all ... Obviously, any such triple must be of the form
$(4a+1,4a+2,4a+3)$ where $a$ is an integer. Has this problem
already been studied before ? The result would follow from Dickson'...
- 3,595
1
vote
1
answer
365
views
Selberg's orthonormality conjecture and density
Let $F$ and $G$ be two primitive functions of the Selberg class, and let $\mathbb{A}$ be the set of values taken by the function which maps a prime number $p$ to $a_{p}(F)\overline{a_{p}(G)}$. $\...
- 6,984
4
votes
2
answers
1k
views
Fourier coefficients of newforms
I apologize in advance for what is probably a very naive question:
I'd like to understand the Fourier coefficients of newforms, and so I was wondering what exactly was known about them (I do know that ...
- 661
4
votes
1
answer
391
views
A question concerning products of finite cyclic groups
Let $m_1,\ldots, m_n$ be pairwise coprime natural numbers $\geq 1$. We consider the product $$G(m_1,\ldots,m_n) := \prod_{i=1}^{n} \mathbb{Z} / m_i \mathbb{Z}.$$ We define $M(n)$ as the set $n$-tuple ...
- 397
4
votes
2
answers
323
views
Multiplicity one prime in the factorisation of p-N
I'm wondering if analytic number theorists can prove results which have the following flavor:
So let $N$ be a large positive integer.
Q: Can you always find a prime number $p$ in the interval $(N, ...
- 7,609
0
votes
1
answer
485
views
greatest common divisor of p-1 and q-1 [closed]
Hi there,
Can we say that if $p$ and $q$ are distinct prime number diving $n$
$\Omega(gcd(p-1,q-1)) \leq \Omega(n)$
Where $\Omega(n)$ denotes the number of prime powers dividing $n$
Best
rahmi
- 167
7
votes
2
answers
1k
views
Contour integration of $\zeta(s)\zeta(2s)$
I have been looking at this for days and I am going insane.
I need to show that for a Dirichlet series equal to $\zeta(s)\zeta(2s)$, the sum of the coefficients less than $x$ is $x\zeta(2)+O(x^{(3/4)...
7
votes
1
answer
1k
views
Is it possible to improve on Siegel's theorem for exceptional zeroes?
Let $\chi$ be a real nonprincipal character modulo $q$. Siegel's theorem on exceptional zeroes states that for any $\epsilon >0$ there exists a positive number $C(\epsilon)$ such that, for any real ...
- 489
2
votes
1
answer
1k
views
What is the Stirling formula for x(x+1)(x+2)...(x+n-1)?
Let x be a complex number.
What is the Stirling formula for x(x+1)(x+2)...(x+n-1) when n goes to infinity?
- 3,804
40
votes
8
answers
12k
views
How does one motivate the analytic continuation of the Riemann zeta function?
I saw the functional equation and its proof for the Riemann zeta function many times, but usually the books start with, e.g. tricky change of variable of Gamma function or other seemingly unmotivated ...
- 3,806
54
votes
2
answers
8k
views
Walsh Fourier transform of the Möbius function
This question is related to this previous question where I asked about ordinary Fourier coefficients.
Special case: is Möbius nearly orthogonal to Morse
August Ferdinand Möbius (November 17, 1790 – ...
0
votes
2
answers
690
views
A simple question regarding the sum-of-divisors function
A good day to everyone.
Consider the following "Conjecture":
If $a, b \in M \subset \mathbb{N}$, then $1 < a < b$ and ... [plus some more conditions on $a, b$ and $M$...] if and only if $\...
20
votes
2
answers
1k
views
Median largest-prime-factor
Let $P(n)$ denote the largest prime factor of $n$. For any integer $x\ge2$, define the median
$$
M(x) = \text{the median of the set }\{P(2), P(3), \dots, P(x) \}.
$$
Classical results of Dickman and ...
- 12.8k
25
votes
3
answers
3k
views
Discrete Fourier Transform of the Möbius Function
Consider the Möbius function $\mu (m)$. (Thus $\mu(m)=0$ unless all prime factors of $m$ appear once and $\mu (m)=(-1)^r$ if $m$ has $r$ distinct prime factors.) Next consider for some natural number $...
- 24.7k
4
votes
0
answers
424
views
generalization of the Brauer-Siegel bound?
For imaginary quadratic number fields $K$ of fundamental discriminant $-D$, the Brauer-Siegel theorem implies that the class number $h(D)$ of $K$ is "close" to $\sqrt{D}$, more precisely for any $c\in(...
- 1,305
10
votes
1
answer
813
views
Importance of large gaps between zeros of zeta function?
I have noticed that there are quite a few publications, many of them recent, on trying to determine the supremum of the gaps (normalized) between zeros of $\zeta \left(\frac{1}{2} + i t \right)$. ...
- 1,077
12
votes
2
answers
616
views
Are there any notion of 'almost primes' known to have small gaps?
A notorious question with prime numbers is estimating the gaps between consecutive primes. That is, if $(p_n)_{n \geq 1}$ is the canonical enumeration of the primes, then set $g_n = p_{n+1} - p_n$. It ...
- 26.9k
20
votes
1
answer
2k
views
On a Conjecture of Schinzel and Sierpinski
Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) states the following:
A conjecture of Schinzel and Sierpinski asserts that every positive rational number $...
- 4,795
9
votes
2
answers
1k
views
On rational functions with rational power series
Let $f(z)=\sum_{n\geq 0}a_n z^n$ be a Taylor series with rational coefficients with infinitely non-zero $a_n$ which converges
in a small neighboorhood around $0$. Furthermore, assume that
\begin{...
- 7,609
9
votes
3
answers
989
views
sharper minkowski bound
If we want to bound the norm of the smallest ideal which generates a nontrivial ideal class, is there a better bound than Minkowski's bound?
(Note that Minkowski's bound is to guarantee something ...
- 1,832
14
votes
2
answers
3k
views
Zeta Function: Zero Density Theorems.
I was reading about some of the zero density Theorems for my Analytic Number Theory Topics course. While looking over some more complicated results and proofs a few simple questions came up:
First, ...
- 11.4k
9
votes
1
answer
521
views
Zeroes of complete L-functions
Hello,
Let $F$ and $G$ be two functions belonging in the Selberg class, $\Lambda_{F}$ and $\Lambda_{G}$ the complete L-functions associated to $F$ and $G$. I would like to know whether this assertion ...
- 195
22
votes
4
answers
2k
views
Given a prime $p$ how many primes $\ell<p$ of a given quadratic character mod $p$?
There was this question for which my response was unusally popular, so I dare to ask the following:
(1) Given a prime $p>2$, how many primes $\ell < p$ there exist which are quadratic residues ...
- 105k
4
votes
1
answer
2k
views
When is the Siegel-Walfisz theorem non-trivial?
The following paragraph appears in Analytic Number Theory (Iwaniec, Kowalski):
The Siegel-Walfisz theorem asserts that:
$\displaystyle \hspace{5cm} \psi(x;q,a) = \frac{x}{\phi(q)} + O(x(\log x)^{-A})...
- 489
24
votes
4
answers
8k
views
What does log convexity mean?
The Bohr–Mollerup theorem characterizes the Gamma function $\Gamma(x)$ as the unique function $f(x)$ on the positive reals such that $f(1)=1$, $f(x+1)=xf(x)$, and $f$ is logarithmically convex, i.e. $\...
- 11.1k
4
votes
2
answers
1k
views
Some Dirichlet series questions.
I asked this question on m.SE in an attempt to find out the right words to say for these questions I am about to ask.
In his great answer, Matthew Emerton explained that (cuspidal) automorphic L-...
49
votes
3
answers
6k
views
The Hardy Z-function and failure of the Riemann hypothesis
David Feldman asked whether it would be reasonable for the Riemann hypothesis to be false, but for the Riemann zeta function to only have finitely many zeros off the critical line. I very rashly ...
- 13.1k
49
votes
4
answers
6k
views
If the Riemann Hypothesis fails, must it fail infinitely often?
That is must there either be no non-trivial zeros off the critical line or
infinitely many?
I'm sure that no one believes otherwise, but I've never seen a theorem in the
literature addressing this. ...
- 17.6k