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.

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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 ...
Melanka's user avatar
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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 ...
R Grady's user avatar
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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 ...
swami's user avatar
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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 $\...
joro's user avatar
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2 votes
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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 ...
Larry Freeman's user avatar
2 votes
1 answer
148 views

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 \...
Medo's user avatar
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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[-\...
Mastrem's user avatar
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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 ...
Steven Stadnicki's user avatar
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104 views

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 > \...
Olivier Rozier's user avatar
7 votes
2 answers
822 views

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 ...
Jon Bannon's user avatar
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10 votes
1 answer
449 views

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 ...
JSE's user avatar
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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)^{...
user501735's user avatar
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101 views

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 &...
Mayank Pandey's user avatar
6 votes
1 answer
277 views

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 $n$...
hofnumber's user avatar
  • 553
10 votes
1 answer
325 views

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 ...
George Bentley's user avatar
5 votes
1 answer
267 views

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. ...
user501735's user avatar
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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 ...
Puzzled's user avatar
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11 votes
2 answers
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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 -...
Charles Bouillaguet's user avatar
2 votes
0 answers
147 views

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 ...
EGME's user avatar
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7 votes
2 answers
795 views

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 ...
Adithya Chakravarthy's user avatar
3 votes
2 answers
714 views

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 ...
Vestoo's user avatar
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9 votes
1 answer
502 views

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 ...
Adithya Chakravarthy's user avatar
4 votes
2 answers
518 views

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-...
Dekimshita's user avatar
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 ...
taylor's user avatar
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1 answer
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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(...
Itachi's user avatar
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0 votes
0 answers
62 views

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}^{...
fractal's user avatar
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2 votes
1 answer
298 views

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 ...
user501735's user avatar
1 vote
0 answers
87 views

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 $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$ ?
Li Xnu's user avatar
  • 11
3 votes
2 answers
238 views

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 $n$ ...
Nick Belane's user avatar
16 votes
1 answer
1k views

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|...
CHUAKS's user avatar
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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 ...
Dekimshita's user avatar
9 votes
1 answer
310 views

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 $T \leq t_1 < t_2 < \cdots < t_R \leq 2T$, $t_{r+1} - t_r \geq T_0$. ...
Joshua Stucky's user avatar
10 votes
1 answer
969 views

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 $$ r(B) \sim C \frac{B}{\sqrt{\log B}} $$ ...
Johnny T.'s user avatar
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0 votes
1 answer
266 views

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. ...
Beta's user avatar
  • 365
1 vote
0 answers
61 views

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 ...
Joshua Stucky's user avatar
9 votes
1 answer
1k views

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 ...
Mark Lewko's user avatar
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3 votes
0 answers
157 views

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 ...
Nilotpal Kanti Sinha's user avatar
1 vote
1 answer
428 views

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 $p$...
user500926's user avatar
0 votes
1 answer
147 views

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&...
Harmonic4352's user avatar
2 votes
0 answers
75 views

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 ...
Marty's user avatar
  • 13.1k
10 votes
2 answers
899 views

$\psi(x)-x$ 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)...
Dr. Pi's user avatar
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1 vote
2 answers
234 views

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 ...
Puzzled's user avatar
  • 8,842
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)...
H A Helfgott's user avatar
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3 votes
0 answers
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'...
Jon Aycock's user avatar
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 ...
tomos's user avatar
  • 1,096
2 votes
1 answer
163 views

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}\\...
user avatar
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 \...
Ruy's user avatar
  • 2,233
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 ...
SKNEE's user avatar
  • 81
0 votes
2 answers
158 views

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 \\...
Anish Ray's user avatar
  • 311
0 votes
0 answers
83 views

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\} $...
khattab's user avatar
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