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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156 views

Convergent series, asymptotics and truncation

In regard to the characteristics of certain "explicit formulae" arising in number theory, I am pondering the connection between the rate of convergence of series and the asymptotic order of the ...
3
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1answer
272 views

Least non primitive root

There is an abundant literature, and even here on MO no shortage of questions, on the question of the smallest prime primitivee root modulo $q$ (where $q$ is a prime, or more generally an odd prime ...
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1answer
198 views

Bounds for the largest divisor of n less than n^0.5

Let $d(n)$ denote the largest divisor of $n$ less than $\sqrt{n}$. Are there good lower bounds for $d$ that hold for almost all natural numbers? More precisely, is there a function $f$, say ...
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3answers
268 views

A divergent series related to the number of divisors of of p-1

Let $d(n)$ denote the number of divisors of $n$. Is it known that the series $$\sum_{p \text{ prime}} \frac{1}{d(p-1)}$$ diverges? This would follow immediately from the Sophie Germain Conjecture. ...
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1answer
184 views

What are the best known lower and upper bounds for the second Chebyshev function $\psi(x)$

I was reading through Jitsuro Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$ when $x \ge 25$. In the paper, he uses the following bounds for the second Chebyshev function ...
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153 views

Estimates for the size of the product set [n].[n] [duplicate]

Possible Duplicate: Number of elements in the set {1,…,n}*{1,..,n} Writing $[n]$ for the set $\lbrace1,2,...,n\rbrace$, let $P_n$ denote the product set $[n].[n]$, i.e. $$ P_n = ...
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112 views

Off critical line zeros for half integer weight $L$-functions

Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put $$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$ to be the $L$-function. Further ...
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247 views

Poles of products of Gamma functions

I want to know if there can be a general statement about the poles (Laurent expansion) of such products of Gamma functions as a function of $p \in \mathbb{R}$ in the limit $\epsilon \rightarrow 0$, ...
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3answers
302 views

Jacobi sums on tori

The Jacobi sum of $n$ multiplicative character $\chi_1,\dots,\chi_n$ on a finite field $\mathbb F_q$ is defined as $$J(\chi_1,\dots,\chi_n) = \sum_{x_1,\dots,x_n \in \mathbb F_q, x_1+\dots+x_n=1} ...
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1answer
159 views

This might be a trivial question on Hurwitz's zeta function.

In the book I am reading they write that for Hurwitz zeta function, $\zeta(x,s)=\sum_{n=0}^{\infty} \frac{1}{(x+n)^s}$, the next sum in the RHS converges for $\Re(s)>-1$, and I don't see how ...
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289 views

What analytic tools can provide a lower bound for this Diophantine equation?

The resolution of the Diophantine equation $$m! = n(n+1)$$ was asked on M.SE. My intuition says that this cannot be solved by elementary means - apologies if I am mistaken. I felt that the following ...
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2answers
560 views

Ordinary Generating Function for Mobius

Is there any information known for the Ordinary Generating Function for Mobius? $$ \sum_{n=1}^{\infty} {\mu(n)}x^n $$ I know that It has radius of convergence 1. Does not have limit as ...
7
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0answers
272 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 ...
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0answers
387 views

On Determinants of Laplacians on Riemann Surfaces

History of the Formula: In their famous paper "On Determinants of Laplacians on Riemann Surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the space $T^n$ of ...
18
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4answers
1k views

Good uses of Siegel zeros?

The short version of my question goes: What is known to follow from the existence of Siegel zeros? A longer version to give an idea of what I have in mind: The "expectional zeros" of course first ...
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2answers
227 views

Recovering $\sum_{n \leq x} a(n)$ from $\sum_{n \leq x} a(n)e^{-n/x}$

In the theory of automorphic forms and multiple Dirichlet series, we often take inverse Mellin transforms of Dirichlet series to come up with Tauberian theorems, like the Ikehara Tauberian method. In ...
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2answers
627 views

Is there an explicit expression for the imaginary part of some non-trivial zero of zeta,

Is there an explicit expression for the imaginary part of some non-trivial zero of zeta, in terms of well-known constants, such as say $\gamma$ or $\pi$ say ?
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3answers
698 views

Deducing BSD from Gross-Zagier and Kolyvagin

Does anyone know which papers deduce BSD for elliptic curves $E/\mathbb{Q}$ of rank 0 or 1 from the papers by Gross and Zagier and Kolyvagin? If I understand these theories right, there is still a ...
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0answers
645 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: ...
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1answer
535 views

What exactly does \gg and \ll mean?

For example, $f(T)\ll_T 1$ where $T$ is a positive number.
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763 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: ...
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0answers
69 views

linear independence of values of the polylogarithm at different roots of unity

I am interested in the real and imaginary part of the complex polylogarithm $$L_{k+1}(\zeta):=Re(\frac 1 {i^k}\sum_{m=1}^\infty \frac{\zeta^m}{m^{k+1}}),$$ where $\zeta$ is a primitive $n$-th root ...
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2answers
651 views

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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1answer
232 views

Power free values of reducible polynomials

Browning in this paper proves that if $f \in \mathbb{Z}[x]$ is an irreducible polynomial of degree $d \geq 3$ and $k$ is an integer $\geq 3d/4 + 1/4$, then we have $$\#\{n \in \mathbb{Z} \cap [1, X] ...
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1answer
908 views

Prime Number Races in 2 Dimensions

Is the mapping $$f: \ \mathbb{N} \rightarrow \mathbb{Z}[i], \ \ \ n \ \mapsto \sum_{2 < p \leq n \ {\rm prime}} e^{\frac{p-1}{4} \pi i}$$ surjective? In 1999, when I was an undergraduate student, ...
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6answers
517 views

Representations with Triangular Numbers

A well known theorem of Gauss says that any natural number $n$ may be written as the sum of three triangular numbers - $$ n={a_{1} \choose 2}+{a_{2} \choose 2}+{a_{3} \choose 2} $$ The following ...
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0answers
198 views

Translation of an article by Wolfgang Schmidt on normality for real numbers in different bases.

I would greatly appreciate a pointer to a translation from German into English of the article by Wolfgang Schmidt, Über die Normalität von Zahlen zu verschiedenen Basen, from Acta Arithmetica VII, ...
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2answers
546 views

Question about the Hardy-Littlewood method (quite basic)

Hi, I have a question about the Hardy-Littlewood method. Writing $R_s(n)$ for the number of ways to write $n$ as a sum of $s$ $k$-th powers and $f(\alpha )$ for the sum $\sum _{m=1}^Ne(\alpha m^k)$, ...
19
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4answers
2k views

The Riemann Hypothesis and the Langlands program

On page 263 of this book review appears the following: Given the centrality of L-functions to the Langlands program, nothing would seem more natural (than a presentation of elementary algebraic ...
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1answer
656 views

Can infinite polynomials be expressed as a product of its linear factors?

Background: In the 1700s, Euler solved the Basel Problem, which was to solve $\sum_{n=1}^\infty\frac{1}{n^2}$ in closed-form. Euler showed that it was equal to $\frac{\pi^2}{6}$ by first expressing ...
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242 views

Minimal period of arithmetic progressions occurring in sets of positive density.

Let $A$ be a subset of ${\mathbb N}$ with positive upper-Banach density, and for each integer $k\geq3$, define $R_k=R_k(A)$ to be the smallest positive integer $r$ such that $A$ contains a length $k$ ...
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1answer
291 views

Asymptotic formula in Analytic Number Theory

Hi, Could anyone tell me in what sense the following is an "asymptotic formula": Theorem 1 from ...
2
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1answer
313 views

Siegel-Walfisz Theorem

Hi, Could anyone explain to me how A and B are the same/different/equivalent? A = The Siegel-Walfisz Theorem as stated in Wikipedia (this is the statement in Davenport) ...
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4answers
647 views

Using Quotient of Prime Numbers to Approximation Reals

We know a positive rational number can be uniquely written as $m/n$ where $m$ and $n$ are coprime positive integers. Particularly, we can pick out those numbers with $m$ and $n$ both prime. Question ...
6
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1answer
355 views

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 ...
3
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1answer
235 views

Ratio of consecutive divisors and average

Let $2\leq d_1 < d_2,...,d_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard: (1) ...
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1answer
556 views

Bernoulli number formula involving roots of Taylor polynomial of $\exp-1$

Please prove, give more symbolic or numeric support (counterexample!?), simplify or drop me a reference (or some vague hunch). We have $$B_n = n!\sum_{\lambda} \frac{\lambda^{2n}}{p'_n(\lambda)}$$ ...
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0answers
130 views

Estimate needed on a sum involving the prime counting function

Let $F(x) = \sum_{1 < n \leq x} (-1)^{\pi (n)}$ where $\pi (n) $ is the prime counting function. I am trying to understand how $ F(x) $ behaves as $ x \to \infty$. In particular, what are the ...
4
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1answer
262 views

Distribution function for divisors of an Integer

For a fixed $n$, let $D_n(x) = \{ d|n : d \leq x \}$ . We assume here $p \leq x \leq n/p$, where $p$ is the smallest prime factor of $n$. For example if $n = p^i$ for some prime $p$ then $D_n(x) ...
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2answers
457 views

Average orders of multiplicative functions

For a multiplicative function $f$ and $x>0$ let $$S_f(x)= \sum_{n \leq x} f(n).$$ Studying sums of this type is a favourite pastime of analytic number theorists. I'm trying to understand what kind ...
14
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1answer
892 views

Is there a Montgomery's conjecture for Dirichlet characters and Artin representations ?

Edit: as GH noticed, the way I tried to state Montgomery's conjecture is wrong. There were some mistakes in the references I used, which compounded with some mistakes of mine, gave a very poor post. ...
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0answers
175 views

Is this extension of the Selberg class trivial?

I came across the following modification of the Selberg class in some of my work (see below), and while I've moved on in some sense -- I submitted the paper in question -- I can't get it off of my ...
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1answer
571 views

Least prime primitive root

For $p$ a prime number, let $G(p)$ be the least prime $q$ such that $q$ is a primitive root mod $p$, that is $q$ generates the multiplicative group $(\mathbb Z/p\mathbb Z$)* . Is it known that ...
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1answer
380 views

The tightest prime zipper

Define a prime zipper as an increasing function $f(n)$ mapping $\mathbb{N}$ into $\mathbb{N}$ with the property that, for every $n \ge 1$, there is at least one prime within the inclusive interval $[ ...
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2answers
320 views

Density of integers $n$ whose totient $\varphi(n)$ is larger than $\alpha n$

Fix $0 < \alpha < 1$ a real. Let $S_\alpha$ the set of integers $n \geq 1$ such that be $\phi(n)>\alpha n$. For $x>0$, let $S_\alpha(x)$ be the number of positive integers $n$ less han $x$ ...
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1answer
987 views

Provable zero-free region for any entire function that analytically is similar to zeta(s)

Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$: $f(z)$ is bounded when $\Re z>1+\delta$ $f(z)$ is unbounded when $\Re z=1$ $f(z)$ grows ...
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1answer
555 views

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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1answer
240 views

Does FE of Selberg Zeta function imply Trace formula?

Does the functional equation of the Selberg Zeta function imply the Selberg trace formula? BTW, the trace formula implies the functional equation.
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1answer
816 views

Small primes in arithmetic sequences

Fix an integer $a>1$. For $n \geq 1$ an integer, let $\pi_{n,1}(an)$ the number of primes $p \leq an$ such that $p \equiv 1 \pmod{n}$, and $\pi(an)$ the number of all primes $p \leq an$. Let ...
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1answer
389 views

An application of Mobius Inversion in a paper of Shintani

I've been reading about Shintani zeta functions and in particular with respect to finding the density of cubic discriminants as in the theorem of Davenport-Heilbronn. In Shintani's paper "On ...