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
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Character sums over prime arguments
Let $f$ be a monotone decreasing, continuously differentiable function with $\lim_{x\rightarrow \infty}f(x)=0$. Let $\chi$ be a non-principal Dirichlet character. It is standard to show that $\sum_{...
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Who first proved that there are at least n^(1-ε) primes up to n?
It's well-known that Hadamard and de la Vallée-Poussin independently proved the Prime Number Theorem in 1896: that $\pi(n)=n/\log n+o(n/\log n)$. I'm curious as to a weaker result: that for any $\...
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Shortest/Most elegant proof for $L(1,\chi)\neq 0$
Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$?
Background: The non-vanishing ...
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Precise relation between prime number theorem and zero-free region
I was wondering about the following, and I was hoping that some expert here could answer, rather than me indulging in a search for a needle in the haystack of formulas in books like Titchmarsch.
...
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Is the Green-Tao theorem true for primes within a given arithmetic progression?
Ben Green and Terrence Tao proved that there are arbitrary length arithmetic progressions among the primes.
Now, consider an arithmetic progression with starting term $a$ and common difference $d$. ...
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Most squares in the first half-interval
It is well known that if $p$ is an odd prime, exactly one half of the numbers $1, \dots, p-1$ are squares in $\mathbb{F}_p$. What is less obvious is that among these $(p-1)/2$ squares, at least one ...
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BSD conjecture and L functions with zeroes of order g
If the group of rational points of $E$, which is finitely generated by the Mordell-Weil Theorem, has $g$ generators of infinite order, then the Birch-Swinnerton-Dyer conjecture gives
$L_E(s)$ has a ...
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A number encoding all primes
This may be a soft question, but it's just something I thought of one night before sleeping. It's not my field at all, so I am just asking out of curiosity. Has anyone studied the number which is the ...
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Lower bounds on zeta(s+it) for fixed s
This is most probably widely known and discussed here many times, so I am preliminay sorry.
Does Riemann conjecture imply some lower estimates on values, say $|\zeta(3/4+it)|$ for real $t$, when $|t|$...
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What is the value of the regularized incomplete beta function at x=0.5?
What is $I_{0.5}(a,b)$ where I is the regularized incomplete beta function?
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Lower bounds for split primes in Real quadratic fields
Snippet portion:
From Iwaniec and Kowalski's Analytic Number Theory:
If the class number $h=h(D)$ is small, then there are only few
prime ideals $\bf{p}$ of degree one with small norm. Indeed, if
$p=...
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distribution of coprime integers
Let $0 < a < 1$ be fixed, and integer $n$ tends to infinity. It is not hard to show that the number of integers $k$ coprime to $n$ such that $1\leq k\leq an$ asymtotically equals $(a+o(1))\...
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What information do the roots of the generating function of the nontrivial zeroes of the Riemann zeta function encode.
Let $a_{m}$ be the imaginary part of the nontrivial roots of the Riemann zeta function $\zeta(s)$. Suppose we have their generating function $u(x)=\sum_{m=1}^{\infty} a_{m}x^{m}=14.134725\ldots{}x^{1}+...
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Examples of asymptotic formulas with optimal error term
Many problems in analytic number theory regard the error term in asymptotic formulas. These problems usually take the form: prove that the number theoretic quantity $f(n)$ satisfies $f(n) = G(n) + O(n^...
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Mertens-like sum in arithmetic progressions
I find myself needing a good estimate for $\sum_{p\le x,\, p\equiv a\bmod q} 1/p$, perhaps something like
$$
\sum_{p\le x,\, p\equiv a\bmod q} \frac1p = \frac{\log\log x}{\phi(q)} + b(q,a) + O\big(\...
- 12.8k
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How to put "x^a => 1/(a+1) x^(a+1)" and "x^-1 => log(x)" together
This innocent-looking problem came to me some years ago.
These two most basic integration formulas are, of course, disturbingly different, in the eyes of any good mathematicians [ just joking ;-) ].
...
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What is the status of the Gauss Circle Problem?
For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) ...
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What are Mean Values of Ideal Densities in Galois Extensions?
In an unfinished (and as of now unpublished) article intended for the encyclopedia of mathematics, Arnold Scholz wrote:
"Classifying extensions according to the Galois group
of their normal closure ...
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Injectivity of Transfer (Verlagerung) map
Let $ K $ be a finite extension of a $p$-adic field or a number field, L a finite extension of $K$. The following fact holds: $
\text{Gal}(K^{\text{ab}} / K) \rightarrow \text{Gal}(L^{\text{ab}} / L) ...
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What is the relationship between the Bell numbers, the Bell polynomials, and the partition numbers?
A friend of mine and I were wondering what relationship exists between the partition numbers $p_{n}$ and the Bell numbers $B_{n}$ (and also possibly the Bell polynomials $B_{n,k}(x_1,x_2,\dots,x_{n-k+...
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Consequences of the Riemann hypothesis
I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice ...
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Reference for the expected number of prime factors of n larger than n^alpha is -log alpha
Let $0 < \alpha < 1$ be a constant. The expected number of prime factors of a "random" integer near $n$ which are greater than $n^\alpha$ is $-\log \alpha$.
It's my understanding that (...
- 14k
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Infinite sets of primes of density 0
Sorry if the question is too vague or if the examples I look for are too boringly well-known: my knowledge of analytic number theory is rather primitive......
So, here it goes: suppose that you want ...
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Is a "non-analytic" proof of Dirichlet's theorem on primes known or possible?
It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) | n \in \mathbb{Z} \}...
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Heuristic argument for the prime number theorem?
Here is a bad heuristic argument for the prime number theorem. Let $n$ be a positive integer and assume that PNT holds up to $n$. Then $n$ itself is prime if and only if for each prime $p<n$ the ...
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Analytic continuation of Dirichlet series with completely multiplicative coefficients of modulus 1
A sequence $a_n \in \mathbb{T}$ ($n \geq 1$) satisfying $a_{mn} = a_m a_n$ for all $m, n \geq 1$ defines a Dirichlet series $f(s) = \sum_n a_n n^{-s}$, absolutely convergent for $\Re s > 1$. If in ...
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Dirichlet L series and integrals
If $f : t \to e^{-xt}$ with $x \geqslant 1$, and $d_n$ is the number of positive integers that divide $n$, I can show that
$$ \lim_{\epsilon \to 0^+} \sum_{n\geqslant 1}\frac{\epsilon^2 d_n f(\...
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at which rational points does the Hypergeometric function take rational values
A generic example is ${}_2 F_1(\frac{1}{3},\frac{2}{3},\frac{5}{6};\frac{27}{32})=\frac{8}{5}$. So my question: Is there any description of the set of rational points at which the hypergeometric ...
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Riemann hypothesis generalization names: extended versus generalized?
This is a "names" question. There are two standard directions of generalization of the Riemann hypothesis: one to L-functions (which is used quite a bit in analytic number theory, and for ...
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Primes $p$ for which $p-1$ has a large prime factor
What are the best known density results and conjectures for primes $p$ where $p-1$ has a large prime factor $q$, where by "large" I mean something greater than $\sqrt{p}$.
The most extreme case is ...
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Primes of the form a^2+1
The fact that the Riemann zeta function $\zeta(s)$ and its brethren have a pole at $s=1$ is responsible for the infinitude of large classes of primes (all primes, primes in arithmetic progression; ...
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PNT for general zeta functions, Applications of.
When I read it for the first time, I found the whole slog towards proving the Prime Number Theorem and the final success to be magnificent. So I am curious about more general results.
We talk of ...
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constants in Gamma factors in functional equation for zeta functions.
Usually the Riemann zeta function $\zeta(s)$ gets multiplied by a "gamma factor" to give a function $\xi(s)$ satisfying a functional equation $\xi(s)=\xi(1-s)$. If I changed this gamma factor by a non-...
- 41.4k
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Modular forms and the Riemann Hypothesis
Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions?
What I'm thinking of is this: under the Mellin transform, the Riemann zeta function $...
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Why does the Riemann zeta function have non-trivial zeros?
This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ...
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Evaluating Shintani cone zeta functions
Hi everyone
I am trying the evaluate sums of the form
$$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)}$$
...
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Question concerning the arithmetic average of the Euler phi function:
Let $\varphi(n)$ denote Euler's phi-function. If we let
$$ \sum_{n\leq x} \varphi(n) = \frac{3}{\pi^2}x^2+R(x),$$
then it is not hard to show that $R(x)=O(x\log x)$. What is the best known bound for $...
- 5,037
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orthogonality relation for quadratic Dirichlet characters
Hello,
I've been working deriving the orthogonality relation for quadratic Dirichlet characters $\chi_d(n)$ (or real primitive characters). The statement I'm trying to prove is
$$\lim_{X \...
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Historical question in analytic number theory
The analytic continuation and functional equation for the Riemann zeta function were proved in Riemann's 1859 memoir "On the number of primes less than a given magnitude." What is the earliest ...
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Eisenstein series and the Kronecker limit theorem
It is well known that the first Kronecker limit theorem gives the Laurent expansion of the Eisenstein series $E(z,s)$ over $SL(2,Z)$ at $s=1$; see, for example, Serge Lang's book Elliptic Curves, ...
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Applications of Artin's holomorphy conjecture
I wonder why the Artin conjecture is so important. The only reason I could figure out is that one could use the holomorphy of Artin L-series and Weil's converse theorem to show modularity of two-...
user19475
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Continuation up to zero of a Dirichlet series with bounded coefficients
Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a ...
- 7,442
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Partial sums of multiplicative functions
It is well known that some statements about partial sums of multiplicative functions are extremely hard. For example, the Riemann hypothesis is equivalent to the assertion that $|\mu(1)+\mu(2)+\dots+\...
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Analytic density of the set of primes starting with 1
In 'Cours d'arithmetique', Serre mentions in passing the following fact (communicated to him by Bombieri): Let P be the set of primes whose first (most significant) digit in decimal notation is 1. ...
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Smallest k-term AP of primes
Let $S(k)$ denote the smallest integer such that there exists a k-term arithmetic progression of primes among the integers $[1,S(k)]$. Green and Tao have an unpublished note that gives a very large ...
- 13k
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Reference for learning global class field theory using the original analytic proofs?
I'm wondering if anyone knows of a reference for learning global class field theory using the original analytic proofs developed in the 1920s and 1930s. Almost every book I can find either does local ...
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Why does the Gamma-function complete the Riemann Zeta function?
Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or ...
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When does the zeta function take on integer values?
Here $\zeta(s)$ is the usual Riemann zeta function, defined as $\sum_{n=1}^\infty n^{-s}$ for $\Re(s)>1$.
Let $A_n=${$s\;:\;\zeta(s)=n$}. The behaviour of $A_0$ is basically just the Riemann ...
- 7,013
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Typical value of totient function
Can anyone tell me what the expected value of Euler's totient function φ$(n)$ is (roughly) if you choose a random integer $n$ in the range $[N,N+M]$, where $M$ is large and $N$ is larger than $M$? ...
- 29k
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Integral of the error estimate in the prime number theorem
This seems like something that should be in discussed in the literature, but I can't find anything. Here $\pi(x)$ is the prime counting function and $\psi(x)$ is the usual sum of the Von Mangoldt ...
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