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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32
votes
0answers
938 views

Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?

The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
31
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1k views

How to prove the identity $L(2,(\frac{\cdot}3))=\frac2{15}\sum\limits_{k=1}^\infty\frac{48^k}{k(2k-1)\binom{4k}{2k}\binom{2k}k}$?

For the Dirichlet character $\chi(a)=(\frac a3)$ (which is the Legendre symbol), we have $$L(2,\chi)=\sum_{n=1}^\infty\frac{(\frac n3)}{n^2}=0.781302412896486296867187429624\ldots.$$ Note that this ...
19
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0answers
475 views

Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space

Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all $x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...
19
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0answers
3k views

sums of digits of powers of integers

It is known (Senge and Straus, 1971, see also C.L.Stewart, 1980) that for every natural $a $, not a power of 10, and every natural $s$, there are only finitely many $k$ such that the sum of decimal ...
18
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0answers
623 views

Infinite extensions such that every elliptic curve has finite rank

The comments to this answer seem to make the following claim. Claim. Let $K$ be the maximal abelian extension of $\mathbf Q$ that is unramified away from $p$ (more generally, away from a finite set $S$...
14
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0answers
484 views

Moments of derivatives of $L$-functions

I'd like to know why it is important to know the moments of the derivatives of $L$-functions. The moments of $L$-functions are related to the Lindelöf Hypothesis, but what about the moments of the ...
13
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0answers
978 views

Analytic continuation of the Dirichlet generating series of the multiplicative partition function

Apologies for the lengthy question, but it seems it's the only way i can convey my thoughts. Consider the Dirichlet series: $$\kappa(s)=\prod_{m=2}^{\infty}\frac{1}{1-m^{-s}}=\sum_{n=1}^{\infty}\frac{\...
13
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0answers
1k views

Small primes attract large primes

I posted a version of this to stackexchange and got 12 up-votes and no answers in somewhat more than a day. Someone in a comment construed it as asking for a lot of novel research including figuring ...
13
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0answers
1k views

A question about Mobius inversion

I don't know how precise I can make this question. I want to know whether there is a theorem that says that a certain phenomenon always happens, but I think the best I can do in order to pin down the ...
13
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0answers
491 views

Making a character small at a reciprocal

The following question emerged from thinking about the Erdős discrepancy problem. I don't know whether an answer would be directly helpful, but it might, and in any case I find the question quite ...
12
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0answers
344 views

Computing Mertens' function in time O(sqrt(x)) - in practice

As far as I know, there is one way currently known to -- in principle -- compute the Mertens function $M(x) = \sum_{n\leq x} \mu(n)$ in time essentially $O\left(x^{1/2}\right)$, namely, a modification ...
12
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468 views

No Siegel-Landau zeros for $\mathrm{GL}(n)$

The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact: There ...
12
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0answers
875 views

Primes and Parity

This problem is motivated by the polymath4 project. There, the aim was to find an efficient deterministic algorithm for finding a prime larger than $N$. The hope was to find a polynomial algorithm in $...
12
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0answers
401 views

Weyl law for Maass forms with nontrivial character

The classical Weyl law for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$ counts the number of Maass cusp forms on $\Gamma \backslash \mathbb{H}$ with Laplace eigenvalue less than $T$. This is originally due to ...
12
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0answers
577 views

Sieve bound for prime $k$-tuples

Let $d_1<d_2<\dots<d_k$ be integers. Then the number of integers $n\leq x$, such that $n+d_1, n+d_2, \ldots, n+d_k$ are simultaneously prime, is bounded above by $$ \mathfrak{S}(d_1, \ldots, ...
11
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0answers
340 views

The trace formula over function fields

There are many examples in number theory where an "arithmetic" problem (i.e. for number fields) has an easier analogue for function fields over finite fields. This is also true for questions ...
11
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0answers
236 views

quasiperiodic continued fractions

Is anything known about continued fractions in which the sequence of integers is quasiperiodic? Quasiperiodic is meant here in the sense of 1D quasicrystals. For example, draw an irrationally-sloped ...
11
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397 views

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 ...
10
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453 views

Bounding $1/\zeta(s)$ given RH

Let $T\geq 0$. Assume RH(T+100), that is, assume that all non-trivial zeros $\rho$ of the Riemann zeta function with $|\Im(\rho)|\leq T+100$ satisfy $\Re(\rho)=1/2$. Can we then give a good upper ...
10
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0answers
239 views

On the infinity of $\{p\in \mathbb {N}:\exists n\in\mathbb{N}~p| \left \lfloor{r^n}\right \rfloor\}$

I've already asked this same question on MSE here, but didn't get much help, so I will try on this site as well. For which $r\in\mathbb{R}$ is the set $\mathscr{P}_r=\{p \in \mathbb{P}:\ (\exists n\...
10
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0answers
235 views

How many partition values are expected to be prime?

Let $p(n)$ be the partition function. Let $P(N)$ count how many $1\leq n\leq N$ are such that $p(n)$ is prime. Are there any heuristics for how $P(N)$ should behave? A crude guess at how this ...
10
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0answers
2k views

Questions on de Branges' work on the Riemann hypothesis

According to Wikipedia, Louis de Branges de Bourcia has obtained some notable results, such as a proof of the Bieberbach conjecture in 1985, which is now known as de Branges' theorem. Initially, his ...
10
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0answers
699 views

Implications of divergence of $1/\zeta(s) $ at 1/2

$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function. This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$. Its convergence is unknown if $1/2< s&...
10
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0answers
461 views

Montgomery's conjecture and lower bound on certain Fourier transform.

Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...
9
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0answers
217 views

Exponential sums over integers with a fixed number of prime divisors

Are there bounds in the literature on sums of the form $$\sum_{\omega(n)= k} e(\alpha n) \;\;\;\;\;\text{or}\;\;\;\;\; \sum_{\Omega(n)=k} e(\alpha n)$$ for $\alpha$ on minor arcs (i.e., not very close ...
9
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0answers
237 views

What happened to Stark's book on the analytic theory of algebraic numbers?

I just read the excellent chapter 6 Galois Theory, Algebraic Numbers and Zeta Functions(*) in Waldschmidt, Michel, et al., eds. From number theory to physics. Berlin etc.: Springer, 1992 by Harold ...
9
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0answers
185 views

Strong uniqueness of Euler's totient function

Let $f:\mathbb N\to \mathbb C$ be some arithmetical function. Define $\varphi_f(n)$ by the following formula: $$ \varphi_f(n)=\sum_{\substack{k\leq n \\ (k,n)=1}}f(k). $$ In other words, $\varphi_f(...
9
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0answers
206 views

How small may the discriminant of an $S_d$-field be?

In every degree $d$, the Galois closure of the typical number field has the maximal possible Galois group $S_d$. Denote by $f(d)$ the least absolute value of a discriminant of an $S_d$-field of degree ...
9
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0answers
200 views

Semi-primes represented by quadratic polynomials

According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
9
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0answers
383 views

Number of prime factors in a very short interval

Let $k \geq 3$ be a (large enough) integer, let $x \in \mathbb{R}$, and set $I_x := [x, x + \log^k x]$. Some believe that for $x$ large enough there exists a prime $n \in I_x$. Equivalently, there ...
9
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0answers
263 views

Cancellation in a sum of Möbius evaluated along a quadratic form

Let $Q(x,y)$ be an indefinite binary quadratic form. Suppose $0 < B < \sqrt{A} $ are such that $B \gg \sqrt{A}$. Is it true one can save an arbitrary power of log from the trivial bound in $$...
9
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0answers
380 views

Numerical Evidence for Grand Riemann Hypothesis?

Let $L(s)$ be an $L$-function coming from Hecke characters or automorphic forms (e.g. modular form on GL(2), Maass form on GL(2), and higher-rank analogues). Is there any numerical evidence for ...
9
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0answers
305 views

Effective bound of $L(1,\chi)$

Let $d$ be a fundamental discriminant and let $\chi$ be the associated primitive real character of modulus $\vert d \vert$. Assuming GRH, Littlewood proved that as $\vert d \vert$ grows large, $$L(1, ...
9
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0answers
247 views

Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?

Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
9
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0answers
370 views

Symmetric Fifth Power Lift of GL(2) Automorphic Form

Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that $$L(s, \pi, Sym^m)$...
9
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0answers
2k views

"Must read "papers on analytic number theory

Question: What would be some must-read papers for an aspiring analytic number theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: Someone ...
9
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0answers
407 views

Evaluating Shintani cone zeta functions

Hi everyone I am trying the evaluate sums of the form $$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)}$$ ...
9
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1answer
410 views

On random divisor sums modulo $2^k$

Let $k,n,\ell$ be positive integers with $k,n\ge 2$ and $0\le \ell \le k-1$. For each integer $2\le j \le n$, choose a divisor $d_j$ of $j$, uniformly at random from the divisors of $j$. We denote by $...
8
votes
0answers
121 views

Key ideas behind p-adic Baker's theorem

I'm trying to understand Kunrui Yu's series of papers [1 2 3] on lower bounds of linear forms of p-adic logarithms (i.e., p-adic Baker's theorem). I know the proof of the usual Baker's theorem through ...
8
votes
0answers
154 views

Unexpected patterns on the graph of an L-function on the critical line

Let $L(s)$ be the $L$-function associated to the (only) classical modular form of weight $26$ and level $1$. The completed L-function $\Lambda(s)=2(2\pi)^{-s}\Gamma(s) L(s)$ is symmetric with respect ...
8
votes
0answers
149 views

Explicit bound for sum of Kloosterman sums

What are the best fully explicit upper bounds one can give for the sum $$\left\lvert \sum_{n=N}^{\infty} \frac{S(a,b;n)}{n} \,I_1\!\left(\frac{4 \pi \sqrt{|ab|}}{n}\right) \right\lvert$$ where $S(a,b;...
8
votes
0answers
163 views

Equidistribution of roots of quadratic congruences

Let $f(x) \in \mathbb{Z}[x]$ be an irreducible quadratic polynomial, and let $0 \leq \alpha \leq \beta \leq 1$ be real numbers. Put $$\displaystyle S_f(\alpha, \beta; d) = |\{v \in \{1, \cdots, d\} : ...
8
votes
0answers
364 views

Equidistribution of $\{\sqrt{p}: p \text{ primes }\}$ modulo 1

I am trying to show $\{\sqrt{p}: p \text{ primes }\}$ is equidistributed modulo 1. Using Weyl's criterion, it is sufficient to show for each nonzero integer $k$, \begin{equation} \sum_{n \leq x}e(k\...
8
votes
0answers
604 views

Divisor problem: find the fallacy!

The following approach to the divisor problem (that is, the problem of estimating $D(x) = \sum_{n\leq x} d(n)$, where $d(n)$ is the number of divisors of $n$; more precisely, we are meant to bound the ...
8
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0answers
562 views

Are there at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$? (was checked up to $10^{18}$)

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$ ...
8
votes
0answers
134 views

Fourier transform of $I_Y$, $Y=\{\text{numbers with many prime factors}\}$

Let $Y$ be the set of integers $N<n\leq 2 N$ with more than $D \log \log N$ prime factors. We may consider, say, $D = (\log \log N)^{1-\epsilon}$. We do have rather precise approximations for the ...
8
votes
0answers
300 views

Prime character sums

Let $p$ be a (large) prime number, and let $\chi : (\mathbf{Z}/p\mathbf{Z})^{\times} \rightarrow \mathbf{C}^{\times}$ be a Dirichlet character of conductor $p$. We have good estimates on the character ...
8
votes
0answers
248 views

Number fields ordered by discriminant

Since the discriminant of a number field $K \neq \mathbb{Q}$ is bounded from below by an exponential of the degree $[K:\mathbb{Q}]$, for instance by Minkowski's Geometry of Numbers bound, there are ...
8
votes
0answers
183 views

Is there an approximate formula for the discriminant of a sparse polynomial?

Consider integer polynomials $P \in \mathbb{Z}[X] \setminus \{0\}$ of a degree $D \geq 1$ and without multiple complex roots. Let me introduce a notation $$ d(P) := \frac{1}{D} \log{|\mathrm{Disc}(P)|}...
8
votes
0answers
255 views

Number of representations of an integer as the dot product of integer vectors

Let $r_k(n)$ denote the number of solutions in positive integers to the equation: $$n = a_1 b_1 + a_2 b_2 + \ldots + a_k b_k.$$ What estimates and/or asymptotics are available for (1) $...

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