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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Conditional convergence of $\sum_{n\geq 1} \frac{\sin(p(n))}{n}$?
The series $\sum_{n\geq 1} \frac{\sin n}{n}$ is easily seen to be conditionally convergent, e.g. by Abel summation. But how about $\sum_{n\geq 1} \frac{\sin(n^2)}{n}$? (for which Abel summation fails)...
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Number of representations of an integer as an (arbitrary) sum of products
If $n$ is a positive integer, let $r(n)$ denote the number of representations of $n$ as a sum of products of pairs of positive integers. (Here, the order of the terms in the sum does not matter, but ...
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Lower bound for exponential sums
Let $D$ be a subset of $\mathbb Z/n \mathbb Z$ containing $0$. For $m$ an integer, set $$\alpha(m,D)=\sum_{d \in D} e\left (\frac{m d }{n}\right ),$$
where as usual $e(x) = e^{2 i \pi x}$ This is an ...
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What might the (normalized) pair correlation function of prime numbers look like?
Cross-posting from Math.Stackexchange.
You might have read about the fortuitous meeting between Montgomery and Dyson. The background is that the nontrivial zeros of the Riemann zeta function, when ...
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Methods to bound the number of solutions to $x^x \equiv 1 \mod p$ with $1 \le x \le p$
For a prime $p$, let $N(p)$ be the number of solutions $1 \le x \le p$ to $x^x \equiv 1 \mod p$. I am interested in methods to bound $N(p)$.
Background: This quantity appears in Problem 1 of the ...
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Riemann zeta function: pair correlations vs. neighbor spacings
Montgomery's pair correlation conjecture states that the distribution of the pair correlations of the zeroes of the Riemann zeta function (normalized to have average spacing 1) is given by the ...
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Counting points on lattices
I expect that the following is a standard problem from analytic number theory, but I don't know where exactly to look for an answer.
Let f: ℤr→ H be a surjective homomorphism into a ...
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Critical points of Dirichlet L functions
Let $L(s,\chi)$ denote a Dirichlet $L$-function for a real-valued non-principal
character $\chi$. This has limiting value $L(\infty,\chi) = 1$ and we are interested in how this limit is approached ...
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Good books on Dirichlet's class number formula
I refrained from asking the technical questions; maybe everyone didn't like my attitude. At least, help me finding good books.
Can anyone suggest a good book that gives a complete reference to "...
Trust God
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What is the motivation behind Ramanujan's conjecture?
One motivation I have seen given for Ramanujan's conjecture for the estimate
$$ |a_p|< C p^{k - \frac{1}{2}} $$
for the Fourier coefficients of a cusp form of weight $2k$ is that it allows one to ...
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The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$
The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
a\...
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Siegel--Walfisz for number fields
For a number field $K$, we write $\Delta_K$ for its absolute discriminant. I was hoping for a Siegel--Walfisz type theorem of the following type:
Let $A > 0$. Then for every $X > 0$, every ...
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Smooth functions that resemble random walks
If the Riemann hypothesis holds, then the Mertens function $M(n)\equiv\sum_{x\leq n} \mu(n)$ behaves much like a 1D random walk. This includes the statements that
$M(n)$ changes sign infinitely often
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Partial product of Euler factors
Let $\mathbb P$ denote the set of prime numbers and for a subset $T\subset \mathbb P$ let
$$
\zeta_T(s)=\prod_{p\in T}\frac1{1-p^{-s}},
$$
where $\mathrm{Re}(s)>1$.
Is there any $T$ such that $T$ ...
user1688
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Prime Power Gaps
In 2000, Baker, Harman and Pintz proved that there is always a prime in
the interval $(n-n^{0.525}, n)$. There are also conditional results
implying smaller intervals. Nevertheless, I could not find ...
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Are some numbers more equidistributed than others?
Weyl's theorem says that $n\alpha$ is equidistributed mod 1 for any irrational $\alpha$. One corollary is that, if I consider the fractional part $\{n\alpha\}$ for $n \leq N$, and look at the indices ...
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Second moment estimates for $\zeta(s)$: different methods?
What are some different ways to achieve the bound $$\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt = T \log \frac{T}{2 \pi} + (2 \gamma - 1) T + E(T)$$
with an error term $E(T) = O(T^{\...
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Poisson summation formula for number fields
Poisson summation formula is widely used in many parts of the litterature, its classical formulation for sums over integers as well as its adelic version. What is its corresponding form for more ...
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Sums of $\Lambda(n)$ or $\mu(n)$ with hyperbolic-function weights: surprise!
I was leafing through Gradshteyn–Ryzhik in bed yesterday, as one does, and noticed on the last page that the Mellin transforms of several hyperbolic functions have a factor of $\zeta(s-1)$ or $\zeta(s)...
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Growth of $n=n(k)$ for which there's a non-trivial solution to $x_1^k+\cdots+x_n^k=y^k$?
Walter Hayman just asked me the following question. What, if anything, is known about the growth of the function $n(k)$, where $k\geq1$ is an integer, and $n=n(k)\geq2$ is the smallest integer for ...
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A number encoding all primes
This may be a soft question, but it's just something I thought of one night before sleeping. It's not my field at all, so I am just asking out of curiosity. Has anyone studied the number which is the ...
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$\psi(x)-x$ on average
This is a reference question:
Let $\psi(x)$ be the psi-Chebyshev function. Is there any unconditional result in the literature that proves that there exists $0<a<2$ such that
$$
\int_2^x (\psi(y)...
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An interesting sum over lattice points in a large disk centered at the origin
Evaluate the the limit, as $r \rightarrow \infty $, of the sum $\displaystyle \sum \limits_{(m,n) \in D_r}$ $\displaystyle (-1)^{m+n} \over \displaystyle m^2 + n^2$, where $D_r$ denotes the closed ...
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Modular forms with finitely many or very few non-zero Fourier coefficients
I have an elementary question on modular forms, but which I don't know how to solve.
a) Is there a congruence subgroup $\Gamma \leq \mathrm{SL}_2(\Bbb Z)$, an integer $k \in \Bbb Z$ and a non-...
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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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Explicit formula for elementary symmetric sum
For $k\ge1$, $j\ge1$, Let $$e_k(j)=\sum_{1\le i_1<...<i_k\le j}i_1\cdot\cdot\cdot i_k.$$ We know that $e_k(j)$ is a polynomial in $j$ with coefficients depending on $k$. I am curious about ...
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Natural density of set of numbers not divisible by any prime in an infinite subset
Suppose $S$ is a subset of the primes with natural density $0 < \alpha < 1$ within the primes. If
$$D(X) := \{n \leq X \mid p \not \mid n \text{ for all } p \in S \}$$
(so $D(X)$ is numbers at ...
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Bound on gcd of two integers
Well this is a problem I was fiddling with. I came up with it but it probably is not original.
Suppose $a\in \mathbb{N}$ is not a perfect square. Then show that :
$$\text{gcd}(n,\lfloor{n\sqrt{a}}...
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What is the intuition behind applying the Mellin Transform to prime distribution?
I am an undergraduate student writing an expository thesis on the complex-analytic proof of the Prime Number Theorem.
I understand that applying the Mellin Transform to the partial sum of the van ...
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$\int_L^\infty \exp(- t - y/t) \, dt = \text{?}$
Let $y>0$, $L>0$. Has the (special?) function given by
$$f(y,L) = \int_{L}^\infty e^{- t - y/t} \, dt$$
been studied? Are there precise, simple bounds?
Let me try to attempt to reinvent the ...
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Are the ideles literally a Picard group?
I understand that in the number field / function field analogy, the ideles $\mathbb I_K$ of a number field $K$ are supposed to be analogous to the Picard group of a function field.
Question: Is this ...
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Sums of two squares: What is known about the distribution of r(n)?
The distribution of sums of two squares has been studied by Landau. What is known about the distribution of the function $r(n)$, the number of representations of $n$ as the sum of two squares? Some ...
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Sign of the function $f(n)=\sum_{k=1}^n\frac{\mu(k)}{k}$
It is well-known that the Mertens function $M(n)=\sum_{k=1}^n\mu(k)$ changes sign infinitely many times when $n\rightarrow +\infty$. Let $f(n)=\sum_{k=1}^n\frac{\mu(k)}{k}$, then $\lim\limits_{n\...
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Riemann–Von Mangoldt formula
Let $$N(T) = \#\{\rho \in \mathbb{C}: \zeta(\rho) = 0,\, \operatorname{Im} \rho \in (0,T]\}$$ denote the number of zeros of $\zeta(s)$, counting multiplicities, with imaginary part lying in the ...
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Regularized sums of Mobius sequence
Do $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n/s}$ and $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n^2/s^2}$ both equal $-2$?
Experimentally this seems plausible (up through ...
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What is the relationship between the Bell numbers, the Bell polynomials, and the partition numbers?
A friend of mine and I were wondering what relationship exists between the partition numbers $p_{n}$ and the Bell numbers $B_{n}$ (and also possibly the Bell polynomials $B_{n,k}(x_1,x_2,\dots,x_{n-k+...
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Set of prime numbers $q$ such that $\sum\limits_{p\leq\sqrt{q}}p=\pi(q)$, where $p$ are prime numbers
The question is: does the set of prime numbers $q$ such that $\sum\limits_{p\leq\sqrt{q}}p=\pi(q)$, where $p$ are prime numbers, contain infinitely many elements? You can find the first elements here (...
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Why are the coefficients of the modular equation so large?
The modular equation $\Phi_n(X,Y)$ is a polynomial in $\mathbf Z[X,Y]$ relating the modular invariant $j$ and the functions
$j\left(\frac{a\tau+b}{c\tau+d}\right)$, where $ad-bc=n$.
For example, we ...
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Quantitative and elementary proofs of the Prime Number Theorem
I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there ...
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Averages over integer points of the sphere
A paper of William Duke proves that integer points on the sphere are equidistributed:
$$ V_n = \{ (x,y,z) \in \mathbb{Z}^2 : x^2 + y^2 + z^2 = n \}. $$
Up to reflections across the $x$, $y$ and $z$ ...
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Is new $n$-conjecture as follows correct?
Given a positive integer $P>1$, let its prime factorization be written as$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}.$$
Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,...
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Infinitely many primes that split completely in an arithmetic progression
Let $d \geq 1$ be an integer. Dirichlet's theorem on arithmetic progression implies that the arithmetic progression $a, a+d, a+2d, \ldots$ contains infinitely many primes if and only if $\gcd(a,d)=1$.
...
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How does Riemann hypothesis implies estimates?
In Iwaniec, Luo and Sarnak article (precisely (4.23)), it is said that GRH for $L(s, \mathrm{sym}^2(f))$, for a holomorphic cusp newform $f$ of level $N$ and weight $k$, implies
$$\sum_{p \nmid N} \...
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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 $$a(n) \ll n^{(k-1)/2 + \...
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An upper bound for the length of the continued fraction expansion of $\sqrt d$
Let $d\ge 2$ and let
$$
\sqrt d =[a_0; \overline{a_1,\dots, a_\ell, 2a_0}]
$$
be its continued fraction expansion. Clearly, if $d=n^2+1$, then $\ell=0$, which gives the lower bound for $\ell$.
...
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least prime in a arithmetic progression
Hello
Here I want to consider the simplest arithmetic progression $n\equiv 1\pmod{q}$ where $q$ is a prime. Is it true that we can find a prime $p\leq q^2$ in this arithmetic progression?
This ...
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Asymptotic behavior of a "strange" arithmetic function
Write $f(n)$ for the quotient of $n$ by its largest squarefree divisor. In other words, $f$ is a multiplicative function with $f(p^k) = p^{k-1}$ for all $k \geq 1$.
What, if anything, is known about ...
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Is $j(\tau)^{1/3}$ the hauptmodul for the congruence subgroup generated by $\tau\rightarrow\tau+3, \tau\rightarrow-1/\tau$?
The 3rd root of the modular invariant $j$ is
$$ j(\tau)^{1/3}=q^{-1/3}(1+ 248q+ 4124q^2+ 34752q^3+\cdots),$$
where $q=e^{2\pi i \tau}$.
I was wondering if $j(\tau)^{1/3}$ the hauptmodul for the ...
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Statement of the pair correlation conjecture
In his paper "The pair correlation of zeros and the zeta function",
Montgomery defines a function
$$F(\alpha,T) = \left(\frac{T}{2 \pi} \log T\right)^{-1} \sum_{0 < \gamma, \gamma' < T} T^{i \...
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A basic estimate of exponential sums
Demeter in his book "Fourier Restriction, Decoupling, and Applications" (P287) used the following estimate:
\begin{equation}
\sup_{0\leq n\leq q}\bigg|\sum_{m=0}^n e^{2\pi i\frac{a}{q}m^2}\...
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