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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Accelerating convergence for some double sums
I am interested in the following general double sums, for integers $a\geq 1$ and $b\geq 2$,
$$Z(a,b) = \sum_{k,\ell \geq 0} \frac{2k+3}{\binom{k+2}{2}^a} \frac{2\ell+3}{(\binom{k+2}{2}+\binom{\ell+2}{...
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Explanation of a step in a preprinted work
I have been studying this preprinted paper and the references therein. I believe that there are some typos; However, since I am not an expert yet, I would like to make sure I am correct.
I do not ...
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A condition on $(a_{j})_{j\in \mathbb{N}}$ so that for all $x \in \mathbb{R}$ we have $\min_{1 \leq j \leq N}\|a_{j}x\|=o(1)$
Suppose that the sequence $(a_{j})_{j \in \mathbb{N}}$ is an increasing sequence of positive integers that satisfies $$(1)\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } d | a_{d}$$ and $$ (2)\text{ }...
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Has this number-theoretic constant been studied?
Unless I made a mistake, the expected value of the largest exponent in the prime factorization of random positive integer (defined in the appropriate way) is $$\eta := \sum_{n=1}^\infty \Big(1-\zeta(n)...
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On probability of coprimality of a list of numbers
We know $r$ randomly chosen integers are coprime with probability $\frac1{\zeta(r)}$.
Pick a bound $N$ and pick $k\lceil N^{\alpha}\rceil$ uniformly random integers in $[0,N^{\alpha+\beta}]$ where $\...
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What can be said about the primality of Zsigmondy numbers?
I am cross-posting this from math.stackexchange, as it has received upvotes but no comments/answers after a couple months.
Let $\mathcal{Z}(n,a,b)=\frac{\Phi_n(a,b)}{\gcd (\Phi_n(a,b),n)}$ be the $n$-...
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Specific application of Cauchy-Schwarz and Large Sieve
Im reading a paper by Matomaki here, and the following is stated (I'm paraphrasing):
"By the Cauchy-Schwarz inequality and the large sieve, we have
$$\sum_{q \leq Q}\frac{q}{\phi(q)}\sum_{\...
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A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis
Let $p_n$ be the $n$th prime. Assuming the Riemann hypothesis, Harald Cramér proves that $p_n-p_{n-1}\le C(\sqrt p_n \log p_n)$ for sufficiently large $n$. Is there a value known for the constant $C$ ...
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Advanced texts on analytic number theory?
So a friend of mine is very interested in analytic number theory, and is looking for resources past the basic level.
He has studied analytic number theory from several books, among them are Hardy’s ...
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Non-negativity of an infinite absolutely convergent sum
The infinite sums involving mobius function and a multiplicative function has got quite interest in past. In particular, sums of the form $$\sum_{d=1}^{\infty}\frac{\mu(d)}{f(d)}$$ for mobius function ...
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Is there a way to specify a special kind of reciprocals of natural numbers?
Any number with of a form $\frac{1}{n}$ has a decimal with a repetend of finite length that is never longer than $n$ (provable by Dirichlet principle). (Example: $\frac{92}{99}=0.929292\ldots$ in ...
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Asymptotic behavior of the sum $\sum_{k\le x}\frac{1}{\varphi(k)}$
Suppose $x>0$ and let $f(x)=\sum_{k\le x}\frac{1}{\varphi(k)}$, where $\varphi(k)$ is the Euler totient function. It is well known that $\sum_{k\le x}\frac{1}{k}\sim\log x$. What is the asymptotic ...
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One series converges iff the other converges
In Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges it is said that this sequence of partial sums converges
$$
\begin{split}
\sum_{1<n\leq N}\frac{a_{n}}{\...
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$\lim_{x\to \infty} \left(\sum_{n\leq x} (\log n)^k/n - \int_1^x (\log t)^k/t\right) = \text{?}$
It is easy to see (by Euler-Maclaurin, say, or just by thinking of a graph) that
$$\lim_{x\to \infty} \sum_{n\leq x} \frac{(\log n)^k}{n} = \int_1^x \frac{(\log t)^k}{t} + C + O\left(\frac{(\log x)^k}{...
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Resources where I can find open problems in number theory along with their level of difficulty
NOTE: I will not accept an answer because a lot of answers are really good and if anyone want to post under this question later then they are most welcome to post as comment or answer because it will ...
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Convergence of zeta Euler product with additional term
Let's consider the following Euler product ($s=\sigma+it)$:
$$ P(s)=\prod_{p \; \text{prime}} \frac{1}{1-p^{-s}} \; e^{-p^{-s}}$$
So for $\sigma>1$, it is clear the product converges and we have:
$$...
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Reference request Re Vinogradov's ternary Goldbach proof
I believe that I.M. Vinogradov's proof of the ternary Goldbach conjecture used the observation that the number of ways $n$ can be written as a sum of three primes equals
$$
\int_0^1 \sum_{p , q , r \...
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Question about the definition of divisor sum functions at 0
I was working on convolution sums of divisor sum functions, and found it very curious that some authors would define the function at 0 as:
$$\sigma_k(0):= {1\over 2}\zeta(-k)$$
But I cannot understand ...
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Small near-reciprocals
Let $p$ be large prime in $[T,2T]$ where $T$ is a parameter.
Can we have an integer pair $a,b$ satisfying $ab\equiv c\bmod p$ such that $|c|$ is of size $O(\operatorname{polylog}(T))$ and $a,b$ are ...
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Constructing an integer with small residues for two distinct primes in polynomial time
Given two primes $p,q\in[T,2T]$, how many integers $m$ of size $O(T^{3/2+\epsilon})$ are there such that the residues $m\bmod p$ and $m\bmod q$ are both $O(polylog(T))$? Looking for an answer
Is it ...
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Simultaneous small near-reciprocals at two distinct primes
Let $p$ and $q$ be large primes in $[T,2T]$ where $T$ is a parameter.
Can we have same integer pair $a,b$ satisfying $ab\equiv c'\bmod p$ and $ab\equiv c''\bmod q$ such that both $|c'|$ and $|c''|$ ...
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The monodromy in the proof of Little Picard via Klein's $J$
First of all, my question is in the context of formalising proofs in the proof assistant Isabelle/HOL, so my questions are a bit guided by what material we have in the complex analysis library there.
...
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The smallest volume possible for a lattice with integer distances?
Let $\Lambda \subset\mathbb{R}^n$ be a lattice satisfying $\|x-y\|_2^2 \in \mathbb{Z}$ for all $x,y\in\Lambda$. How small can $\text{vol}(\Lambda)=\det(\Lambda)$ be?
For example, in dimension $2$, the ...
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Asymptotic estimate for $\sum_{\substack{ab\le x \\ a,b\in A}}f(a)f(b)$
Suppose that $A\subseteq \mathbb{N}$ and suppose that you have an estimate of the form
$$
\sum_{\substack{a\le x \\ a\in A}}f(a) \sim g(x).
$$
With this information is it possible to get an asymptotic ...
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Sum of squares squared in an arithmetic progression
Let $r(n)$ be the number of ways to write $n$ as a sum of two squares and $(a,q)=1$.
What is known about
$$
\sum_{n \le x,n \equiv a (\text{mod} \, q)} r(n)^2 \quad?
$$
I am looking for uniform ...
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Could you please confirm or deny two identities involving weighted Stirling numbers of the second kind?
In the paper [1] below, among other things, Carlitz introduced weighted Stirling numbers of the second kind $R(n,k,r)$. He also proved that the numbers $R(n,k,r)$ can be generated by
\begin{equation*}%...
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Generalizations of the Brun-Titchmarsh theorem
Let $\pi(x;q,a)$ count the number of primes $\leq x$ congruent to $a$ mod $q$. The Brun-Titchmarsh Theorem states that for all $q< x$, $(a,q)=1$, we have
$$
\tag{1}
\pi(x;q,a) \leq \frac{2x}{\...
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Analytic approximation of the step function in $L^p$ norm
Motivation: Euler-Maclaurin formula uses calculus to estimate discrete sums. I wonder what one can do by reverse engineering. A concrete problem I ran into is the following.
Question: Let $\chi: \...
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Around the equation $\sigma\left(\square\right)=\text{prime}$: counterexamples or a proof for some of these conjectures
For integers $A,B\geq 1$ we define the difference $\sigma(A)\sigma(B)-\sigma(AB)$, denoting it as $[A,B]$, where $\sigma(n)=\sum_{1\leq d\mid n}d$ denotes the sum of divisors function. It is possible ...
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Proof of an explicit formula for $\pi_0(x)$
Let $\pi(x)$ denote the prime counting function and $$\pi_0(x) = \lim_{\epsilon \to 0} \frac{\pi(x+\epsilon)+\pi(x-\epsilon)}{2}.$$
I've seen noted in a few references the explicit formula
$$\pi_0(x) =...
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Is $\frac{1}{L(1+it)}$ unbounded?
Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding L-function. Is $$\frac{1}{L(1+it, \chi)}$$ unbounded for $t \in \mathbb{R}$? I'm aware that this is true if $L=\zeta$, but I'm ...
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Estimate $ \sum_{\substack{n\leq x\\ n\in Q\\}}\log n$ and $\sum_{\substack{n\leq x\\ n\in Q\\}} n$ where $Q$ is the square-free numbers
Let $Q$ be the set of squarefree numbers. I'd like to know estimates of following sums:
$$ \sum_{\substack{n\leq x\\ n\in Q\\}}\log n \qquad\text{and}\qquad \sum_{\substack{n\leq x\\ n\in Q\\}} n. $$
...
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Ask for a proof of an identity involving the product of two Bernoulli numbers
It is well known that the Bernoulli numbers $B_{k}$ for $k\in\{0,1,2,\dotsc\}$ can be generated by
\begin{equation*}
\frac{z}{\textrm{e}^z-1}=\sum_{k=0}^\infty B_k\frac{z^k}{k!}=1-\frac{z}2+\sum_{k=1}^...
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Transcendence of sum of reciprocals of factorials
For $A \subseteq \mathbb{N}$, define $\displaystyle x_A = \sum_{n \in A} \frac{1}{n!}$. It is easy to see that for every infinite $A$, $x_A$ is irrational.
Question: Is there an infinite $A \subseteq \...
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$\zeta(s) = 1 + X^{1-s}/(s-1) + ...$?
Let $s = \sigma+ i t$ with $0\leq \sigma\leq 1$, $|t|\leq X$, where $1\leq X<2$.
It is easy to use the Euler-Maclaurin formula to prove a result of the form
$$\left|\zeta(s) - 1 - \frac{X^{1-s}}{s-...
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Ask for a reference or a proof of an identity involving a finite sum and the Bernoulli numbers
Let $B_{n}$ for $n\ge0$ denote the Bernoulli number generated by
\begin{equation*}
\frac{z}{\textrm{e}^z-1}=\sum_{n=0}^\infty B_n\frac{z^n}{n!}=1-\frac{z}2+\sum_{n=1}^\infty B_{2n}\frac{z^{2n}}{(2n)!},...
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Divisibility based on central binomial coefficients
For some prime $p$, it is a standard approach based on Kummer's criterion to bound the number of positive integers $n<X$ for some parameter $X$, such that $p\nmid \binom{2n}{n}$. However, if we ...
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How to show something is "true in mean square"?
I am looking at the conjecture, that for every $\varepsilon,B >0$, then
$$\Big| \sum_{\substack{n \in \mathbb{N}\\n \leq x}} n^{-it} - \int_1^x u^{-it} du \, \Big| \leq Cx^{1/2}|t|^{\varepsilon}$$ ...
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$\zeta(s) = \sum_{n\leq x} n^{-s} - x^{1-s}/(1-s) + ...$ through bounded-order Euler-Maclaurin?
It is a basic classical result (Titchmarsh Thm 4.11; credited to Hardy-Littlewood) that,
uniformly for $\Re s \geq \sigma_0>0$, $t\leq 6 x$ (say),
$$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} - \frac{...
- 19.4k
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1
answer
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Pólya–Vinogradov like inequality for a character sum with Euler factors
Let $M$ be a large positive integer, $d$ an odd positive integer and $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{R}$. For a non-principal character $\chi_d = \chi$ with modulus $d$, I ...
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The *actual* size of the first quadratic non-residue
Let $p$ be an odd prime and define $n(p)$ be the smallest positive quadratic non-residue modulo $p$. By Ankeny and later effective work of Lamzouri, Li, and Soundararajan we know that under GRH one ...
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Integral of $|1/\zeta(\sigma+i T)|$ (or $|(1/\zeta(\sigma+i T))^{(k)}|$) on a horizontal half-line in the left upper quadrant
Let $T_0\geq 20$. Let $L$ be the half-line from $-\infty + i T$ to (say) $-1/2 + i T$. Since $|\zeta(s)|$ is roughly proportional to $(T/2 \pi e)^\sigma$ for $s=\sigma+ i T$ on $L$, it is clear that ...
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An estimate of a sum
I'm looking for an estimate of this sum $\sum_{n\leq x} \frac{\mu^2(n)}{\varphi(n)}$ where $\mu$ is the Möbius function and $\varphi$ the Euler's totient function.
Thanks a lot.
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Looking for some interesting complex integration contours
I am currently working on some tools to make contour integration in a proof assistant less painful and I'm looking for interesting examples of contours in the complex plane used in the literature. I ...
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Do Artin L functions have polynomial growth in the critical strip?
Given an irreducible representation $\rho$ of the Galois group $G$ of a number field $K$ over $\mathbb{Q}$, we have the associated Artin $L$ function which we denote by $L(s, \rho)$. By Brauer ...
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Number of solutions to a diophantine equation
Given a positive integer $n$, consider the diophantine equation $4x^2+y^2+4x+y=2n$ with solutions in non-negative integers $x$ and $y$.
Define the proportion
$$\delta_n=\frac{\#\{(x,y)\in\mathbb{Z}^2_{...
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2
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Estimates about prime numbers: a lemma in Bourgain's article
For $n\in \mathbb{N}$ with prime decomposition $n=p_1^{r_1}\cdots p_k^{r_k},p_i\neq p_j$, let $A=\{p_1,\cdots,p_k\}$; then the following holds:
\begin{equation}
|\{q\in \mathbb{N},q<Q: \text{all ...
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1
answer
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Perfect equidistribution for the Legendre symbol
Let $p $ be an odd prime. Assume that we have the following perfect pattern:
all the primes below $p$ are successively quadratic residues and quadratic non-residues. What can we say about $p$? Is it ...
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Moments of an exponential sum
Let $p$ and $N$ be large natural numbers. I would like to get a possibly sharp asympotic approximation of
$$
\mathcal{I}_{p,N}=\int_0^1 \Big(\sum_{j=1}^N e^{2\pi i j \xi}+\sum_{j=1}^N e^{-2\pi i j \xi}...
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Rank and Taylor coefficient in Birch and Swinnerton–Dyer
I am trying to get a better understanding of the Birch and Swinnerton–Dyer conjecture. I have two questions
Why might one expect that the analytic rank of $L(E,s)$ is equal to the rank of $E(\mathbb{...
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