Questions tagged [analytic-number-theory]
A beautiful 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.
2,895
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Distribution of square roots (mod m) on small intervals (with respect to m)
Fix a large positive integer $m$. Let $A$ be small positive number typically $\sim m^{1/3}$. Suppose $S(A, m)$ be set of solutions (normalized by dividing by $m$) to the quadratic congruences $x^2 = a ...
2
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1
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2D lattice sum with numerator
I've been struggling a bit with a double sum that arose as the trace of an operator:
$$\sum_{(j,k)\in Z^2 \setminus (0,0)} \frac{(j+k)^n}{(j^2+k^2)^n},$$
where $n$ is an even natural number. Is there ...
1
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0
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Convergence of Farey series integral of a "density" function as the order tends to infinity
Let $F_n$ denote the $n$-th Farey sequence, and let $q$ be a rational number such that $0 \leq q \leq 1$. I am studying the convergence of a specific integral related to Farey series, defined as ...
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Hardness of solving $0=\sum_{i=1}^k \operatorname{linear}_i(x_1,\ldots,x_n)^D$ over the rationals
This is related to cryptography and this question
and another question.
In short, we are asking about decomposing multivariate polynomial
as sum of perfect powers of linear polynomials.
Working over $\...
2
votes
1
answer
135
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Estimating the minimum number of distinct least prime factors found in range of $c$ consecutive integers
When I look at the count of distinct least prime factors for a range of consecutive integers, I am seeing the same minimum number appear again and again. I am wondering if this number represents the ...
2
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1
answer
148
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The asymptotic behavior of $F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$, $b>a>0$
Let $b>a>0$. Given $ \lambda>0$, what is the asymptotic behavior of
$$F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$$
as $\lambda\to 0^{+}$ and as $\lambda \...
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267
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Explanation for the sum of primitive roots modulo $p$ (taken from $[-(p-1)/2,(p-1)/2]$) being positive way more often than being negative?
An earlier version of this question received a few upvotes but no answers on math.stackexchange.
For $p$ an odd prime, let $S(p)$ denote the sum of the primitive roots modulo $p$, taken from $\left[-\...
2
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0
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What lower bounds are known on growth of the distribution of the abundancy index?
Let $a(n)=\sigma(n)/n$ be the abundancy index of $n$ and let $F(x)$ be the distribution function of this index: i.e., the proportion of integers $n$ with $a(n)\leq x$. (This function is well-defined ...
2
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0
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On a subset of the $abc$ triples
The $abc$ conjecture states that, for every positive real $\varepsilon$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers such that $a + b = c$ and
$$c > \...
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2
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Positivity of the coefficients of Taylor series associated to the Riemann hypothesis
The question below relates to the paper "Jensen Polynomials for the Riemann Zeta Function and Other Sequences" of Griffin, Ono, Rolen and Zagier. I'm asking it here because I am sure the ...
10
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Asymptotic behavior of a "strange" arithmetic function
Write $f(n)$ for the quotient of $n$ by its largest squarefree divisor. In other words, $f$ is a multiplicative function with $f(p^k) = p^{k-1}$ for all $k \geq 1$.
What, if anything, is known about ...
2
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0
answers
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Asymptotics for $\sum_{n \leq x} \frac{(-1)^{n-1} \overline{\chi}(n)M(n, \chi)}{n}$
Let $\mu$ denote the Möbius function, and let $\chi$ be a primitive Dirichlet character of modulus $q$. Define $M(n, \chi)=\sum_{j \leq n} \mu(j)\chi(j)$ and
$$f(x, \chi):=\sum_{n \leq x} \frac{(-1)^{...
2
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Whittaker function oscillation on diagonal near 0
In Proposition 1 of Blomer - Applications of the Kuznetsov formula on GL(3), bounds are given for the Whittaker function
$$W_{\nu_1, \nu_2}(y_1, y_2) = \mathcal W_{\nu_1, \nu_2}\begin{pmatrix} y_1y_2 &...
6
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1
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277
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The error term for the second moment of Fourier coefficients of cusp forms with the level explicitly determined
There is a basis question which puzzles me for a while. The question is the following:
Let $p$ be a prime and $X\ge 2$. Let $f$ be a $GL_2$-newform of level $p$ and non-trivial
nebentypus, with the $n$...
10
votes
1
answer
325
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Vinogradov-Korobov prime number theorem for number fields
Without assuming the Riemann hypothesis, the traditional error bound of the prime-counting function $\pi(x)$ is $O(x\exp(-c(\log(x))^{1/2}))$. As shown by the Wikipedia page for the Landau prime ...
5
votes
1
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267
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Asymptotics of the Liouville sum at the primes
Let $\lambda$ denote the Liouville function, and let $L(x):=\sum_{n \leq x} \lambda(n)$ be the Liouville sum. Define $c$ to be the supremum of the real parts of the zeros of the Riemann zeta function. ...
0
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A question on the Hilbert-Kamke problem
The Hilbert-Kamke problem consists in studying the integral solutions of the Diophantine system
$$
x_1^i + \dots + x_s^i = n_i \text{ for } 1\leq i\leq k
$$
with $x_i\geq 0$ for $i = 1,\dots,k$.
I am ...
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2
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Mertens-like theorem
Mertens' first theorem states that
$$
\sum_{p \leq n} \frac{\log p}{p} = \log n + O(1).
$$
I read in this paper that the following variant is "classical":
$$
\sum_{p \leq n} \frac{\log p}{p -...
2
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0
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147
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Tools to prove lower bounds in analytic number theory
Perron's formula and related methods are used to relate statements such as the Riemann hypothesis to upper bounds of functions occurring in analytic number theory. For example, Perron's formula is ...
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795
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Convolution sum of divisor functions
Let $\sigma_0(n)$ be the divisor counting function
$$\sigma_0(n) = \sum_{d \vert n} 1.$$
I'm interested in the convolution sum
$$ S(n) := \sum_{k=1}^{n-1} \sigma_0(k) \sigma_0(n-k)$$
I ran some quick ...
3
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2
answers
714
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Do all rigorous proofs of Euler's product for $\zeta (s)$ use infinitude of primes?
There are two proofs of
$$\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{p \text{ prime}}\frac{1}{1-p^{-s}},\quad \Re (s)\gt 1$$
which I'm aware of. I'll call the first one the Sieve proof and the second one ...
9
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1
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502
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Is the divisor counting function equidistributed mod $p$?
Let $\sigma_0(n)$ be the divisor counting function:
$$\sigma_0(n) = \sum_{d \vert n} 1.$$
I ran some numerical experiments that showed when $p$ is prime, the function $\sigma_0(n)$ is equidistributed ...
4
votes
2
answers
518
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Explicit formula for Artin L-functions
The classical explicit formula for the Riemann Zeta function states that
$$
\psi(x)=x-\sum_{\rho} \frac{x^{\rho}}{\rho}+O(1),
$$
where $\psi(x)=\sum_{n \leq x} \Lambda(n)$ and the sum is over all non-...
4
votes
1
answer
270
views
The function $f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}$ for $g\in \text{SL}(2,\mathbb R)$
For $g\in \text{SL}(2,\mathbb R)$ and the Hilbert-Schmidt norm $\|\cdot\|$ (square root of sum of squares), define
$$f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}.$$
It is ...
1
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1
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115
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Bound for some trigonometric polynomials
Let $e(x)=e^{2\pi i x}$ and consider the following functions defined for $x\in [0,1]$:
$$
f_1(x)=\frac{e(10x)-e(x)}{e(x)-1}, \quad f_2(x)=\frac{e(110x)-e(11x)}{e(11x)-1},
$$
and
$$
f_3(x)=\frac{e(...
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votes
0
answers
62
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Calculate the number theory function $\sum_{n \equiv a(\bmod q)} \frac{\phi(n)-\mu(n)}{n^{s+1}}$
I am trying to simplify the function:
$$\sum_{n \equiv a(\bmod q)} \frac{\phi(n)-\mu(n)}{n^{s+1}}=\frac{1}{\phi\left(q^{\prime}\right)} \sum_\chi \bar{\chi}\left(a^{\prime}\right) \sum_{n^{\prime}=1}^{...
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What are the best known upper bounds for $\frac{1}{L(s, \chi)}$?
Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding Dirichlet L-function. What are the best known bounds for $\frac{1}{L(s, \chi)}$ in the half-plane of convergence?
I'm aware of ...
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0
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what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$?
what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$ ?
3
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Proof of an asymptotic formula by Tricomi
Firs of all I ask my question, then I explain how this question arises in my mind and lastly what I tried to solve it.
QUESTION:
Let $P_{n,N}(k)$ be the number of composition of an integer $k$ in $n$ ...
16
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1
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Does the sum of reciprocal of integers with average power at least two converge?
$\DeclareMathOperator{\ap}{ap}$ $\DeclareMathOperator{\rad}{rad}$ The average power of an integer $m$ is given by
$$
\ap(m):=\log_{\rad(m)}(m)=\frac{\log(m)}{\log(\rad(m))},
$$ where $\rad(m)=\prod_{p|...
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0
answers
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Reference for explicit formula used by Ramanujan
In his work on highly composite numbers http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf , Ramanujan used a version of an explicit formula (equation (329) on page 133) relating primes and zeros of ...
9
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1
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Large values of $\zeta(1/2+it)$ from sums of short moments
In a now classical paper, Iwaniec proved the following theorem.
Theorem. Let $T \geq 2$, $T^{1/2} < T_0 \leq T$ and $T \leq t_1 < t_2 < \cdots < t_R \leq 2T$, $t_{r+1} - t_r \geq T_0$. ...
10
votes
1
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969
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A generalisation of theorem of Landau on sum of two squares?
Let
$r(B)$ be the number of integers $1 \leq n \leq B$ such that $n = x^2 + y^2$ for some $x, y \in \mathbb{Z}.$
Then it is a known theorem of Landau that
$$
r(B) \sim C \frac{B}{\sqrt{\log B}}
$$
...
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1
answer
266
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Estimating a sum involving the von Mangoldt function
I'd like to know the estimate of the following sum
$$\sum_{n\leq x}\sum_{d|n}\Lambda(d)\frac{\phi(d)}{d} $$
where $\Lambda(d)$ is the von mangoldt function and $\phi(d)$ is the Euler totient function. ...
1
vote
0
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Optimal exponents in upper bound for 4-dimensional exponential sum
A classical result of Fouvry and Iwaniec states that if $\alpha_1,\ldots, \alpha_4$ are nonzero, $M_1,\ldots, M_4 \geq 1$, $X > 0$, and $|\varphi_{m_1,m_2}|,|\psi_{m_3,m_4}|\leq 1$ are complex ...
9
votes
1
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1k
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Why are Deligne-type exponential sum estimates so hard to use?
Let $F$ by a finite field, and $R(x_1,x_2,\ldots,x_n) := r_1(x_1,x_2,\ldots,x_n)/r_2(x_1,x_2,\ldots,x_n)$ a rational function in $n$ variables, frequently in analytic number theory or harmonic ...
3
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0
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What is the density of numbers which have at least two divisors whose sum is a perfect square?
Note: This question was posted in MSE about two years ago but it not receive an answer. Hence posting in MO.
A positive integer is said to have square-sum divisors if it has at least two divisors ...
1
vote
1
answer
428
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Artin's conjecture for polynomials and rational functions over finite fields
Artin's conjecture on primitive roots over the integers states that a given integer
$0\ne h\in \mathbb{Z}$ that is neither a square number nor $-1$ is a primitive root modulo infinitely many primes $p$...
0
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1
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147
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Equidistribution of $(2^nx)$, $x$ irrational
I've run into a scenario where it would be extremely useful to know if, for $x$ irrational, the sequence $\{(2^nx)\}$ is equidistributed on in $[0,1]$, where $(\cdot)$ denotes the ``fractional part&...
2
votes
0
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Which sets of natural numbers are "lambda-analytic"?
Begin with a bit of notation. Let $t = t_0, \ldots, t_d$ be a finite sequence of real numbers. Define
$$\lambda^t(x) = x^{t_0} \log(x)^{t_1} \log(\log(x))^{t_2} \cdots.$$
for all real numbers $x \in ...
10
votes
2
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899
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$\psi(x)-x$ on average
This is a reference question:
Let $\psi(x)$ be the psi-Chebyshev function. Is there any unconditional result in the literature that proves that there exists $0<a<2$ such that
$$
\int_2^x (\psi(y)...
1
vote
2
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234
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Solutions of a linear diophantine equation
Let $N(h)$ be the number of solutions of the following linear diophantine equation:
\begin{equation}
x_1 + 2x_2 + 3x_3 + \dots + (h-1)x_{h-1} = 6h-6;
\end{equation}
where $h\geq 2$ and solution means ...
5
votes
0
answers
216
views
Function on $\mathbb{Z}/p^k \mathbb{Z}$ with small Fourier transform?
For $f:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$, define the Fourier transform $\widehat{f}:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$ in the usual way, viz., $\widehat{f}(\xi) = \sum_x f(x) e(-\xi x/p^k)...
3
votes
0
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146
views
Are there infinitely many twin primes p and p+2 with p+1 divisible by 4?
Every pair of twin primes $p$, $p+2$ will have either $p \equiv 1 \pmod{4}$ or $p \equiv 3 \pmod{4}$. Is there any work towards whether one of these is "more common", the way that Dirichlet'...
4
votes
1
answer
179
views
Replacing a sharp cut-off by a smooth one
This question is more of a check/validation of a concept.
Suppose I want to study $$\sum _{n\leq X}a_n$$ (e.g. $a_n=d(n)$, the divisor function). As is well-known it's standard practice to replace ...
2
votes
1
answer
163
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What does it mean to have a number of size $B$?
I have a really stupid question that I don't seem to know the answer to and have been too embarassed to ask. In some number theory papers, I encounter sums of the form $$\sum_{\substack{{x \asymp B}\\...
2
votes
2
answers
327
views
When does $\lim_{s\to 1_-} (1-s)\sum_{n=0}^\infty a_ns^n$ exist?
Let $a=\{a_n\}_{n\geq 0}$ be a sequence of positive real numbers with $a_n\leq 1$, for all $n$, and observe that, for
any real number $s\in [0,1)$, one has that
$$
\sum_{n=0}^\infty a_ns^n \leq \...
8
votes
1
answer
241
views
Spectral decomposition of $\Gamma\backslash X$
Let $X$ be a reasonable manifold of non-positive curvature (could be $\mathbb{H}^n$, symmetric or locally symmetric space, homogeneous Hadamard manifold etc.), and let $\Gamma$ be a reasonable group ...
0
votes
2
answers
158
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Asymptotic equivalence of two infinite products of prime numbers in residue classes
I am trying to figure out if the infinite product $$\omega=\frac{5\sqrt{3}}{12}\prod\limits_{\substack{p\equiv 1\pmod3 \\
p\ge 13}}\left(\frac{p-2}{p-1}\right)\prod\limits_{\substack{p\equiv 2\pmod3 \\...
0
votes
0
answers
83
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Cardinality of a subset of smooth numbers
Recall: An $n$-smooth number is an integer whose prime factors are all less than or equal to $n$.
Let $k$ and $N$ two integers with $0\leq k\leq N$. Let's put $H= \{n\in[0,N]\ s.t.\ n\ is\ k-smooth\} $...