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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The lower bound for the automorphic $L$-function $L(s,\pi)$ at the edge of the critical strip $\Re s=1$
Let $\pi$ be any automorphic Maass form on $\text{GL}_m$ of level $N$, say. Assume that the associated $L$-function $L(s,\pi)$ satisfies some good conditions; for example, it satisfies the functional ...
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Riemann xi function strictly increasing along a half-plane
Matiyasevich, Saidak, Zvengrowsk proved the following result:
Let $σ_0$ be greater than or equal to the real part of any zero of ξ. Then $|ξ(s)|$ is strictly increasing in the half-plane $σ > σ_0$.
...
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4
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1
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Prime omega function values on a product of prime powers predecessors
Let $p_1, ... , p_n, ...$ be the prime numbers in order. Define
$$
P_n = \prod_{k=1}^n p_k^q
$$ It is known that $\omega(P_n) = n$ where $\omega(\cdot)$ is the little prime omega function. For a given,...
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A question on hybrid subconvexity for individual L-functions
Sorry to disturb. I have a question need some explanations from the experts on the MO-website.
As usual, we let $L(f,s)$ be the corresponding $L$-function associated to the newform $f$ on $SL_2(\...
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the projection distribution induced by integral points on the sphere
Let $A=\{\mathbf{v} \in \mathbb{Z}^{n}: \|\mathbf{v}\|^2= m \}$ and a fixed $\mathbf{y}\in \mathbb{R}^n$, the norm here refers to the Euclidean norm.
Suppose $\mathbf{x}$ is a uniform distribution on ...
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118
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Poles/Residues of the Gamma function under action of Mobius transform $\Gamma(A(z))$
I am not sure whether this is rather an MO or MSE question but it results from my research, so I put it here.
In my effort to find (or to disprove the existence of) $k,l,h\in\mathbb{N}$ such that $2^{...
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Bounds for Dirichlet L-functions
Let $L$ denote a Dirichlet L-function attached to the primitive character $\chi$. What are the best known bounds for $L(\sigma+it, \chi)$?
PS: For $L=\zeta$ and $0\leq\sigma\leq 1$, i'm aware of a ...
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Density of numbers where a large prime factor satisfies a congruence
I am looking for an upper bound on the number of integers $n<x$ such that $n$ has a prime factor $p>\log(x)^{(1+\delta)}$ such that $p \equiv a \mod b$. Where $a,b$ are fixed and coprime and $0&...
2
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Bounding the sum $f(x)=-\frac{x}{2}+\sum_{p\le x}\log(p)-\frac{1}{x}\sum_{p\le x}p\cdot \log(p)$
Consider the function $$f(x)=-\frac{x}{2}+\sum_{p\le x}\log(p)-\frac{1}{x}\sum_{p\le x}p\cdot \log(p).$$
In my recent work, I need to get an explicit [rather than asymptotic] upper bound of this ...
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What error term to expect
In this paper, Hooley obtains an asymptotic for the third moment of primes in arithmetic progressions, specifically a result of type $$\frac {1}{Q^2}\sum _{q\leq Q}\sideset {}{'}\sum _{a=1}^qE_x(q,a)^...
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Probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$
Using my computer, I found that in the interval $[1, N]$ the probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$ is greater than constant $c$ where $N=10^2, 10^3,...,10^{9}$, $x$ ...
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Formula for the sum $\sum_{n}^{\infty }\frac{\Omega (n)}{n^s}$ in terms of the Riemann zeta function
Is there a "closed" formula for the sum $\sum_{n}^{\infty }\frac{\Omega (n)}{n^s}$ in terms of function $\zeta(s)$ (Riemann zeta ) and its derivatives? Here $\Omega (n)$ denote the total ...
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Let $X$ be a positive integer. Then $\pi{(X+\ln^2{X})}-\pi{(X-\ln^2{X})}>\ln{X}$?
The prime-counting function is the function counting the number of prime numbers less than or equal to some real number $x$. It is denoted by $\pi{(x)}$. Using my computer I found that for any ...
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shifted convolution in arithmetic progressions
Let $r(n)$ be the number of ways of writing $n$ as the sum of two integer squares. Asymptotics for the shifted convolution problem $$ \sum_{n\in \mathbb N\cap[1,x]}r(n) r(n+1)$$ are quite classical; a ...
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Roth's theorem for primes in a given arithmetic progression to a large modulus
Let $\mathbb{P}_{a, q}$ denote the set of primes congruent to $a$ modulo $q$. Are there any estimates for the number of $3$-Arithmetic Progressions in the set $\mathbb{P}_{a, q}\cap [1, X]$, where $(a,...
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Rankin-Selberg convolutions with mixed integral and half-integral weights
Let $f(z)$ denote a weight $0$ Hecke-Maass form of level $N$ and let $\theta(z)$ denote the Jacobi theta function. Then $y^{1/4} f(z) \overline{\theta(z)}$ transforms as an automorphic form of weight $...
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Symmetric square $L$-functions over imaginary quadratic field
Let $F = \mathbb{Q}(\sqrt{-d})$ with class number $h_F = 1$, and $\Gamma = \mathrm{PSL}_2(\mathfrak{O}_F)$. Let $f$ be a Maass cusp form in the $L^2$-cuspidal spectrum of the Laplace operator $\...
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A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
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Effective Erdős–Kac theorem
I have some number $N$ and some integer $k>0$. I want to know what fraction of numbers up to $N$ have more than $k$ prime factors. (In my application, with repetition, but the $\omega$ version is ...
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automorphisms and mellin transforms
If a real analytic function $f$ is involutive i.e. $f(f(x))=x$ and its Mellin transform can be taken on a section of the real axis, and is analytic for $x>0$, in certain cases can this imply that $\...
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A question on Euler's totient function
With reference to the Euler's totient function $\phi(\cdot)$, given any $n \in \mathbb{Z}^+$, it's quite straightforward to find $\phi(n)$.
In contrast, given $n \in \mathbb{Z}^+$, even though there ...
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Separating Gamma in two independent functions
I've encountered a problem in my PhD. I would greatly appreciate any suggestions, tips, or comments you might have. The problem is
Let $\Gamma(s,x)$ be the incomplete gamma function. Given integers $n ...
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Cardinality of the set $\#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}$
Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$.
I am interested in upper bound for
$$
M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}
$$
where $N$ ...
- 3,625
5
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heights of ideal classes and reduction theory for Bhargava cubes
Suppose $K$ is a quadratic imaginary field with discriminant $D$; let $S$ denote the ring of integers in $K$. For a fractional $S$-ideal $J$, define the height of $J$, denoted $H(J)$, to be the ...
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What is the intuition behind applying the Mellin Transform to prime distribution?
I am an undergraduate student writing an expository thesis on the complex-analytic proof of the Prime Number Theorem.
I understand that applying the Mellin Transform to the partial sum of the van ...
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Consecutive integers of the form $2^a 3^b 5^c$
Let $\mathcal{N}$ denote the set of all products of (powers of) $2,3$ and $5$:
$$ \mathcal{N} = \{ 2^a 3^b 5^c \ : \ a,b,c \geq 0 \} \subset \mathbb{N}.$$
We use the elements of $\mathcal{N}$ to ...
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Mellin transform of the volume form of a probability zonoid and its fundamental strip
Let $ L^n_+$ be the set of all $n$-dimensional nonnegative random vectors $\mathbf X = (X_1, X_2,\cdot\cdot\cdot,X_n)^⊤$ with finite and positive marginal expectations, and let $\mathbf Ψ^{(n)}$ be ...
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A sharper estimate for a generalization of the sum-of-divisors function
I am interested in the function $f_n(m)$ which can be defined by the Dirichlet generating function $$\zeta(s)\zeta(s - 1) \cdots\zeta(s - n + 1) = \sum\limits_{m = 1}^\infty \frac{f_n(m)}{m^s} $$
This ...
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Another limit involving the fractional part
It is known that
$$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\left\{ \frac{n}{k}\right\} =1-\gamma$$
where $\left\{ x\right\}$ is the fractional part of $x$ and $\gamma$ is the Euler constant. ...
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Limit of lacunar power series at $1^-$
I've asked this question on MSE but I didn't get a convincive answer so I'm trying here. Here is the question :
Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider ...
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Limit involving the fractional part and the Fibonacci numbers
Helo,
Let $F(n)$ be the $n$th Fibonacci number, if $\left\{ x\right\}$ denotes the fractional part of $x$, how proving
$$\lim_{n\rightarrow\infty}\frac{1}{2n}\sum_{k=1}^{2n}\left\{ \frac{F(2n)}{F(k)}\...
- 611
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1
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Geometric mean of prime factors of all numbers up to n
Through numerical calculations I have discovered that for any natural number $n \geq 2$, the geometric mean of the prime factors of all natural numbers $\leq n$ can be approximated well by $1.6653 \...
- 1,228
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1
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Conditional convergence of Artin $L$-functions
Let $k$ be a number field and $V$ a non-trivial irreducible Artin representation over $k$. Consider the associated Artin $L$-function with corresponding Euler product decomposition $L(V,s)= \prod_v ...
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Analyzing a Dirichlet series with log-oscillating terms via Fourier methods
I am investigating the series $S(z)$ defined as follows:
$$
S(z) = \sum_{n=1}^{\infty} n^{-a}\cos(b\ln(n)),
$$
where $z = a + bi \in \mathbb{C}$, with $0 < a < 1$, and $b \in \mathbb{R}$.
I want ...
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In the modular exponentiation, as used with the adaptive root problem, how to chose the best base that will yield as few results as possible?
Let $n,m,w\in\Bbb N$ and $\lambda\in\Bbb P$ such that $w^\lambda \mod m = n$, with the requirements:
$\lambda$ being a random large prime such as $w^\lambda > 2\times m$
$1 < n < m−1$.
m is ...
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Are the coefficients in the stationary phase approximation computed explicitly somewhere
In Stein's "Harmonic analysis" book, page 334, one can find
the asymptotic expansion
An instructive proof is given for the case $k=2$. It is clear enough to generalize to the cases $k\geq ...
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Prime differences and zero multiplicity
Concerning gaps between consecutive primes, Paul Erdős conjectured that:
$$\sum_{p_n < x} (p_n -p_{n-1})^2 = O(x \log x)$$
Let's call this hypothesis EH. Assuming the Riemann hypothesis (RH), ...
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0
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Gaussian primes in translations of lattices in $\mathbb{Z}[i]$
I am considering undertaking some independent research in my summer break studying Gaussian primes in translations of lattices in $\mathbb{Z}[i]$, i.e. sets of the form $ \{ a+sx+tw:s,t \in \mathbb{Z} ...
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Tuples of natural numbers with no mutual divisibility and large reciprocal sums
Standard apology in case this is something simple, as I'm not a number theorist.
Let $E_1, \dots, E_n$ be disjoint finite sets of natural numbers, such that for any $a_1 \in E_1, \dots, a_n \in E_n$, ...
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Analytic number theory and condensed mathematics
As of 2024, are there current or planned applications of condensed mathematics to analytic number theory? If so, what are suggested readings?
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How to check that a number probably/likely has a divisor having a specific bit length/in range?
Given a randomly generated $\alpha\in\Bbb N$ where $\alpha$ is large thus hard to factor (no small prime composites). How to check that a divisor $F\in\Bbb N$ with a specific bitlengh $n\in\Bbb N∧n<...
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A question on generalized bases
I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
5
votes
1
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Lindelöf hypotheses for derivatives of zeta
The Lindelöf hypothesis says that if we have:
$$\zeta(\sigma+iT)=\mathcal O(T^a)$$
Then if one considers $\sigma=1/2$ then $\inf a=0$. Further, from convexity and the functional equation this implies ...
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56
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Average of number of divisors of shifted exponential sequence
Let $a$ be a fixed positive integer greater than 1. We define the sequence $u_n=a^{n}-1$ for all positive integers $n$. Then are there any results in literature for asymptotic value of the sum $$\sum_{...
- 1,042
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Asymptotics of $\vartheta(x+y)-\vartheta(x)$, where $\vartheta$ is the Chebyshev function, when $y\in[x^\alpha, x]$ for some $\alpha\in(0,1)$
Introduction
Consider $\vartheta$ to be the Chebyshev function, that is, $\vartheta(x)$ denotes, for $x\in\mathbb N$, the sum $\sum_{p\le x,\ p\text{ prime}} \ln p$.
I am interested in asymptotics for ...
- 1,326
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0
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159
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Upper bound of a product of sines
Consider the function
$$ f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$
I wonder whether it is possible to compute some nontrivial upper ...
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200
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Effective bound for odd numbers expressed as sums of three primes
I am interested in the representation of odd numbers greater than five as sums of three primes, inspired by Harald Helfgott's seminal proof of the ternary Goldbach conjecture and the nuanced findings ...
6
votes
0
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632
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
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0
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183
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Sum of reciprocals of primes dividing Mersenne numbers
Let $p$ be a prime. Then it has been proven by Erdős that $$\sum_{p\mid 2^{n}-1}\frac1{p} \ll \log \log \log n.$$ Let $c<1$ be a positive constant. Then can we prove that $$\sum_{p\mid 2^{n}-1}\...
- 1,042
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0
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133
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Techniques of showing the Order
In their paper The integral of Riemann Xi Function, Lagarias and Montague show that the integral
$$\Xi_{\lambda}^{-1}(z)=2\int_0^{\infty}e^{\lambda u^2}\phi(u)\Big( \frac{\sin zu}{u}\Big)\;du$$
is ...
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