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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Lattice points on caps on quadrics
Let $(L,q)$ be a non-degenerate quadratic lattice of signature $(n_{+},n_{-})$ and dimension $d\geq 5$.
Consider the quadric $C_{k}=\{x\in L_{\mathbb{R}},\, q(x)=k \}$ for some non-zero integer $k$.
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Coefficient bounds on cusp forms, half-integer weight
Let $f(\tau) = \sum_{n=1}^{\infty} a(n) q^n$ be a cusp form on $\Gamma_0(4N)$ of half-integer weight $k \ge 5/2.$ The Ramanujan-Petersson conjecture in this case is that $$a(n) \ll n^{(k-1)/2 + \...
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The subfield of $\mathbb{C}$ generated by Fourier coefficients of prime index
Let $f=\sum_{n\ge1} a_nq^n$ be an Hecke eigenform cusp form of integral weight $k$ on $\Gamma_0(N)$ with character $\varepsilon\pmod N,$ and let $M_f$ be the subfield of $\mathbb{C}$ generated by the ...
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Diophantine equations and modular forms
Let $D$ be a square-free positive integer which is the fundamental discriminant of a real quadratic field. Consider the following quadratic form $$Q_{D}(x,y)=x^2+Dy^2.$$
My questions are :
What is ...
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Reference request: Eisenstein series built from principal series representation
One can build an Eisenstein series $E_s(v,g)$ from a vector $v\in D_s$, the principal series representation of $G_{\mathbb{A}_\mathbb{Q}}$. The space $D_s$ has a restricted tensor product structure
$$ ...
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Going beyond the Sylvester and Schur theorem with regard to $x,x+1,\dots,x+n-1$
I was recent reading through Paul Erdos's classic elementary proof of Sylvester-Schur. It occurred me that there is a simple argument that when $x$ is sufficiently large and if $p_i$ represents the $...
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Does $\pi(x) \geq \mathrm{Li}(x)$ imply that $\vartheta(x) \geq x$
As the question in the title asks, does $\pi(x) \geq \mathrm{Li}(x)$ imply that $\vartheta(x) \geq x$? Here $\pi(x) = \#\{p \leq x\}$, $\vartheta(x) = \sum_{p \leq x} \log p$ and $\mathrm{Li}(x) = \...
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Counting prime points in a bounded region
Let $R$ be a semi-algebraic, compact region in $\mathbb{R}^n$ with positive Lebesgue measure. Let $N(R) = \# (R \cap \mathbb{Z}^n)$. Davenport's lemma asserts that we have
$$\displaystyle N(R) = \...
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Higher dimensional large sieve inequality
One of the most important achievements in analytic number theory is the establishment of the so-called large sieve inequality, which is formulated as follows. Let $\{a_n\}$ denote a finite sequence of ...
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Upper bound for an exponential sum in Waring-Goldbach problem
In Waring's problem, we have Hua's estimate
$$S(a,b,q) = \sum_{x=1}^q e^{2\pi i (ax^k + bx)/q)} \ll q^{1/2+\epsilon} \gcd(b,q),$$
where $(a,q)=1$.
?Do you know a similar upper bound for the sum
$$...
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On discrepancy of integer sequences related to Erdos-Turan-Koksma
Assume we have a sequence $a_1,\dots,a_k\in\Bbb N$ with each $a_i\approx n$ where $n$ is some integer.
Suppose there exists an $\eta\in(0,1)$ such that for every $d_1,\dots,d_k\in\Bbb Z$ with $$...
user94040
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Solutions to a diophantine system
What is the smallest $\gamma_1,\gamma_2,\gamma_3>0$ such that given coprime $p,q=\Theta(\ell)$ and integer $t\geq3$ there are coprime $m,n=\Theta(\ell^{t-1})$ with $(mn,pq)=1$, $\alpha_i\in\Bbb Z$ ...
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How are the ratios of successive values of the divisor function distributed?
One motivation for this question is a paper by Erdos and Hall "Values of the divisor function on short intervals", in which the authors obtain the leading asymptotics
$$x(\log x)^{2(\sqrt 2-...
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On the number of $y$-smooth numbers less than a given magnitude
I'm trying to provide a full-fledged proof of the estimate
$$\Psi(x,x^{1/u}) = x \, \rho(u)+ O\left(\frac{x}{\log x}\right) \qquad (*)$$
via the sly inductive approach commonly attributed to A. ...
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Number Theory Problem for 1st/2nd Year Graduate Student
I am looking for a problem in (algebraic) number theory in which a 1st/2nd year graduate student in number theory can make some nontrivial progress. I am not looking for something that can conceivably ...
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A sieve by not (necessarily) coprime integers
Let $\mathcal{A}$ be a nonempty set of pairwise coprime positive integers, and for each $a \in \mathcal{A}$ let $\Omega_a \subsetneq \{0, \ldots, a - 1\}$ be a set of residues modulo $a$. Furthermore, ...
user40023
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Upper bound for a higher dimensional Ramanujan sum
Fix an integer vector $\mathbf m\in \mathbb Z^k$. Let $q$ be a positive integer.
Is there a "good" upper bound in terms of $q,\bf m$ for the exponential sum:
\[\sum_{\mathbf n} e\left(\frac{\langle m,...
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congruence for modular forms coefficients
Let $A$ be the set of modular forms $f=\sum a(n)q^n$ of weight $k$ on $\Gamma_0(N)$ with character $\chi$ whose coefficients $a(n)$ are in the ring of integers $\mathcal{O}$ of a fixed number field $F$...
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$L^1$ norm of Fourier transform of subset sums
Let $n_1,\dots,n_k$ be a set of $k$ natural numbers less than $N$, with $k = (1- \delta) \log_2 N$ for $\delta$ relatively small. Let $e(x) = e^{ 2\pi i x}$, as usual.
Assume that $$\int_0^1\prod_{j=1}...
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What can be said about this double sum?
Question. Can this number be expressed in terms of classical values?
$$\sum_{n,m=1}^{\infty}\frac1{(n^2+m^2)^{\frac32}}=1.056348517615643291\dots$$
UPDATE. I'm encouraged by Noam, Kevin and Igor's ...
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An explicit value for a bound proof
I saw a proof that $|p_n - li^{-1}(n)| \leq n e^{-c \sqrt{\ln(n)}} $,
without saying anything about $c$ !
My questions is, what the explicit value of $c$ ??
It just says for some number $c$ without ...
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Sum of multiplicative arithmetic function over squarefree numbers
In the "Sieve methods" notes of Dimitris Koukoulopoulos (see http://www.dms.umontreal.ca/~koukoulo/documents/notes/sievemethods.pdf), the following useful result can be found:
Theorem 0.4.1. Let $g$ ...
user40023
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Enquiry on primorial numbers and primes
Does the inequality
$$1 + \dfrac{\log p_{k+1}}{\log N_k} > (\log N_k)^{\dfrac{1}{p_{k+1} -1}}$$
hold for all integers $k\geq 2$, where $N_k$ denotes the $k$-th primorial number $N_k=p_1p_2\cdots ...
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Occurrence of simultaneous small remainders?
Fix $(a,b)=1$, $a<b<2a$ and $a,b>n^{1/(2t)}$ and fix a prime $T\approx n^{\tau+\frac1k}$ where $\tau\geq1$ and $k=2(t-1)$. We can show using exponential sums there is an $m_{_T}$ such that $T/...
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Question on the zeta and sigma functions
EDIT:
The observation described here was due to a bug in the code used to generate the data (as commented by Lucia), which renders this question irrelevant.
The answer, however, is worth reading.
The ...
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Estimation of the $k$-th derivative zeta function
When I was doing some task in number theory which involves bounds for the Riemann zeta function, I was stuck on the following question:
Let $\zeta$ be the Riemann zeta function and let $\zeta^{(k)}$ ...
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Could the complex zeros of Riemann zeta function be of the form $ s=0.5+ik$ with $k$ a positive integer? [closed]
I have checked in Andrew Odlyzko, Tables of zeros of the Riemann zeta function, to know if there is an example of zeros of Riemann zeta function with integer imaginary parts, but I don't see that. I ...
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An Exponential Sum Restricted to Primes
Let $a,q,N$ be integers such that $N/2 \leq q \leq N$ and $a/q \notin \mathbb{Z}$.
Is the following estimate true, and, if so, how can it be proved?
\[\left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \...
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$\sum_{d\leq x} (\mu(d)/d) \log(x/d)$: is (the analogue of) Mertens' conjecture still false?
It is known to be false that $\sum_{m\leq x} \mu(m) \leq \sqrt{x}$ for all $x$ (Mertens' conjecture), and it is generally believed that $\lim \sup_{x\to \infty} |M(x)|/\sqrt{x} = \infty$. From the ...
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Closed formula for sine powers
I am looking for a closed formula for the expressions
$$ \sum_{k=1}^{n-1} \sin\left(\pi \frac{k}{n}\right)^m,$$
with $n \in \mathbb{N}$ and $m \in \mathbb{N}$ odd.
Playing with these sums a bit, I ...
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Sum of $\sum_{\substack{1<a<q \\(a,q)>1 \\ (a+1,q)>1}}1$
Let $a,q$ be a positive integers. I am trying to evaluate the following sum:
$\sum_{\substack{1<a<q \\(a,q)>1 \\ (a+1,q)>1}}1$. Is there a formula that exists to calculate such sums?
Here ...
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$\sum_{d\leq x} (\mu(d)/d) \log x/d$: elementary estimates?
Let
$$F(x) = \sum_{d\leq x} \frac{\mu(d)}{d} \log \frac{x}{d}.$$
s it possible/feasible to give an elementary proof of the fact that $F(x)= 1 + o(1)$ (and, ideally, $1+O(1/\log x)$, or better)? By "...
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Dirichlet series of a lattice $\sum_{a \in \Lambda^*} |\det(a)|^{-s}$
For a lattice $\Lambda$ of rank $n$ in $\mathbb{Q}^{n\times n}$ whose non-zero elements are inversible matrices, let $$Z(s,\Lambda) = \sum_{a \in \Lambda^*} |\det(a)|^{-s}$$
I wonder if (and how to ...
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Integrating a product of integrals involving Bessel functions
I have asked similar questions on Math Stack Exchange, but not been able to receive many helpful responses. Therefore, I am posting this problem here, and any input would be extremely valuable.
I ...
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$n$th prime: a better approximation
Let $p_n$ be the $n$-th prime, then from Wikipedia I got that
$p_n \approx n \left(\ln n + \ln \ln n -1 + \frac{\ln \ln n-2}{\ln n}+\frac{6\ln \ln n-( \ln \ln n)^2-11}{\ln^2 n} \right)$.
What is a ...
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Link between Irreducible Factors and Prime Factors (or Cycles of a Permutation)
In "Anatomy of Integers and Permutations", http://www.dms.umontreal.ca/~andrew/PDF/Anatomy.pdf, Granville gives a calibration of cycles of a permutation and prime factors of an integer. "We know ...
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On the consistency of the definition of the conductor for automorphic forms
Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_2(F)$, where $F$ is local non-archimedean. The local conductor associated to $\pi$ can be defined in two usual manners:
By its ...
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Exponential sum (linear in the argument) over primes
Suppose we have $\alpha \in \mathbb{R}$. Then we know that
$$\sum_{1 \leq n \leq X} e(n \alpha) \ll \min \{ X, \|\alpha\|^{-1} \}$$
where $\| \cdot \|$ is the distance to the nearest integer.
I ...
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Odd Chebyshev, part 2
Let
$$ \forall_{n=1\ 2\ \ldots}\quad I(n)\ :=\
\frac {(6\cdot n-3)!!}{(2\cdot n-1)!!\cdot(4\cdot n-3)!!} $$
Let $\ M(n)\ $ be the smallest natural number such that
$$ M(n)\cdot I(n)\ \...
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Ideas in the elementary proof of the prime number theorem (Selberg / Erdős)
I'm reading the elementary proof of prime number theorem (Selberg / Erdős, around 1949).
One key step is to prove that, with $\vartheta(x) = \sum_{p\leq x} \log p$,
$$(1) \qquad\qquad \vartheta(x) \...
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Can there be more than two zeta zeros in between a Gram point and a França-LeClair point?
According to formula 163 at page 47 in the paper A theory for the zeros of Riemann Zeta and other L-functions by Guilherme França and André LeClair, the Gram points can be approximated with the ...
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Beauty of some numbers discovered by Ramanujan
I am a graduate PhD student and my topic is analytic number theory. I am also a mathematics teacher. I am planning to give a course to pupils in high school that motivates them to study arithmetic and ...
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Hilbert Modular Surface: Eigenfunctions of The Laplacian
The spectrum of the Laplacian on $L^2$ of a Hilbert modular surface decomposes into a discrete part and a continuous part $[1/4,\infty)$. The continuous part contains eigenvalues $\geq 1/4$.
I would ...
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What is the relationship between the abscissa of holomorphy and abscissa of convergence of a Dirichlet series
Given a Dirichlet series $$\phi(s)=\sum_{n\ge1}\frac{a_n}{n^s}$$
let $\sigma_{\text{conv}}\in\bar{\mathbb{R}}$ its abscissa of convergence, then we know that $\phi(s)$ is holomorphic on the half-plan $...
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What is the density of integers of the form $a^2+nb^2$?
Landau proved that the mean density of integers of the form $a^2+b^2$ up to $x$ is $K\frac{x}{\sqrt{\log x}} (1+o(1))$, where $K$ is an explicit constant. One proof is based on the fact that a prime $...
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An arithmetic function involving arbitrary (but fixed) number of divisors
I need at least basic information about generating functions of the following class of arithmetic functions, grouped by levels $k$.
Fix some $k$ and some family $\varepsilon_*=(\varepsilon_\sigma)_{\...
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Zero mean constraint for correlations with the Mobius function
An aspect of my research has lead me to believe that I need to distinguish between those bounded functions $\xi:\mathbb{N}\rightarrow\mathbb{C}$ which correlate with the Mobius function $\mu(n)$, i.e. ...
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Sequences that "capture their primes"
Let $g : \mathbb{N} \rightarrow \mathbb{N}$ be an increasing function, and consider the sequence $Y = \{y_n\}$ given by $y_n = g(n)$. Let $F$ be an irreducible binary form of degree $d \geq 2$ with ...
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Asymptotics for certain sum involving the divisor function, Ramanujan sum
Let $c_q(n)$ be the Ramanujan sum, and let $\tau(n)$ be the divisor function. Is there an asymptotic formula for $$\sum_{n\le x}\tau(n)c_q(n)$$ with error terms that do not depend on $q$?
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A "nice" trigonometric polynomial approximation of a characteristic function
Let $\delta > 0$ be small and $\chi_{[-\delta, \delta]}(t)$ be a characteristic function on the interval $[-\delta, \delta]$. I am interested in a trigonometric polynomial $S$ such that
$$
\chi_{[-\...
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