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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5
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1answer
416 views

Analogue of van der Corput sequence for prime numbers

A van der Corput sequence is a low-discrepancy sequence over the unit interval first published in 1935 by the Dutch mathematician J. G. van der Corput. It is constructed by placing a decimal point and ...
22
votes
2answers
964 views

Does the equation $1 + 2 + 3 + \dots = -\frac{1}{12}$ have a natural $p$-adic interpretation?

Consider the equation $$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$ ``proved'' by Ramanujan. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = ...
39
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3answers
3k views

Every prime number > 19 divides one plus the product of two smaller primes?

This is a part of my answer to this question I think it deserves to be treated separately. Conjecture Let $A$ be the set of all primes from $2$ to $p>19$. Let $q$ be the next prime after $p$. ...
6
votes
3answers
890 views

On Robin's criterion for RH [closed]

\begin{equation} \sigma(n) < e^\gamma n \log \log n \end{equation} In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin ...
5
votes
1answer
466 views

Lower bound for character sums

Hello Let $p$ be a prime number. According to Davenport (Multiplicative Number Theory, page 137) Schur proved (Indeed he proved much more, but let consider the simplest case) $$ ...
8
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1answer
702 views

Do Shintani zeta functions satisfy a functional equation?

Probably my questions are known or evident to the experts but I'm a bit puzzled. First of all there seem to be two kinds of zeta functions that go under the name of Shintani zeta functions. First, ...
6
votes
1answer
446 views

Logarithmic Integral of n^Zeta Zeroes and certain nested sums of the fractional part function

Below is an approach I've been exploring for connecting the prime counting function with the logarithmic integral and expressing the error term between the two. I find it beguiling, but I've largely ...
3
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4answers
461 views

Equidistribution Theorem: distance between solutions

Can please someone help me with the following problem. Say we have a sequence $nx \; \mathrm{mod} \; 1$, where $n$ is a whole number and $x$ is irrational. Now I need to solve the inequality $nx \; ...
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6answers
3k views

explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros. Because there are various explicit formulae ...
5
votes
1answer
546 views

The Correlation of the Mobius Function and Dirichlet Characters.

Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$ In other words ...
12
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0answers
494 views

Where might I find a scanned handwritten copy of Ramanujan's second letter to Hardy?

I am giving a lecture to undergraduates on the lovely identity $$1 + 2 + 3 + 4 + \cdots = -\frac{1}{12}.$$ Ramanujan wrote in his second letter to Hardy (courtesy Wikipedia), "Dear Sir, I am very ...
4
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2answers
517 views

How are these number-theoretical constants actually distributed?

I'm very curious about this and would be really grateful for any help or comments in this direction. If we consider any of the following number-theoretical constants: 1)The various singular series ...
5
votes
2answers
960 views

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 ...
2
votes
1answer
498 views

Primes and Ackermann's function

If $A(m,n)$ is Ackermann's function, and $c$ is a fixed integer, are there any heuristics/conjectures/obvious things that can be said about primes of the form $A(m,n)+c$, $m \geq 4$,at all? EDIT: I ...
4
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0answers
189 views

factorising an integer with certain bound on the factors

Can we count the no. of $x$ where $ p^{\alpha -1} < x < p^{\alpha}$ , $gcd(x, 2p)=1$ and if $d |x$ and $d < p ^{\beta}$ for some $1< \beta<\alpha-1$ then $ \frac {x} {d} > p^{\alpha ...
0
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0answers
217 views

An interesting series involving the prime counting function

An open problem in analytic number theory is to determine whether or not the following infinite series converges: $\sum_{n=2}^{\infty} \frac {(-1)^{\pi(n)}}{n log n}$ Here, $\pi(n)$ is the prime ...
23
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6answers
4k views

What does Mellin inversion “really mean”?

Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...
36
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2answers
1k views

Infinite exponential representation of real numbers

I was thinking about infinite exponential representation of real numbers (like $2=e^{e^{-e^{-e^{e^{-e^{e^{e^{-e^{-e^{-e^{-e^{-e^{e^{-e^{e^{e^{-e^{e^{\cdot^{\cdot^{\cdot}}}}}}}}}}}}}}}}}}}}}$. The ...
9
votes
1answer
417 views

Infimums of exponential sums involving primes

Hi, I don't know if this question is appropriate for Math Overflow but I was wondering if there is anything known about the following: Let $$ S(\alpha) = \sum_{n \leq x}\Lambda(n)e(n\alpha). $$ Then ...
14
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0answers
825 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 ...
3
votes
1answer
417 views

About the asymptotics of LCM

Let $g(x,c)$ be a uniformly random integer in the range $(x,x+c)$ and $LCM[x_1,x_2...x_i]$ the lowest common multiple of the integers $x_i$. A) Does the limit of (the asymptotics of ...
2
votes
1answer
467 views

Different cuspidal automorphic representations with same representations at infinity

Let us fix a representation $\pi_\infty$ of GL(n,$\mathbb R$). Let us fix a character $\chi$ of K, where K is a compact subgroup of $GL(n,\mathbb A_{finite})$. $$K=\Pi_{v<\infty}K_v$$ $K_v$ is ...
6
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2answers
559 views

Estimating a sum of gauss sums

Hey guys, I'm concerned with bounding the following sum of gauss sums from above $$\sum_{p\leq ...
15
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1answer
656 views

Distinct simple zeros of Dirichlet L-functions

Given a finite set of distinct primitive Dirichlet characters, $\chi_1, \dots, \chi_r$, is it known that the product of the L-functions, $$L(s):=\prod_{i=1}^r L(s,\chi_i),$$ has a simple zero? It's ...
7
votes
3answers
746 views

Effective detection of CM modular forms

Say $f$ is a newform of weight $k$ and level $\Gamma_1(N)$. $f$ is called CM if, for example, there is an imaginary quadratic field $K$ such that for all $p\nmid N$ which are inert in $K$, the $p$th ...
13
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1answer
1k views

Exceptional zeros and Liouville's $\lambda$ function

This originated from an textbook exercise (recently posted to math.stackexchange http://math.stackexchange.com/questions/62883/quadratic-characters-and-liouvilles-function with no success) but I think ...
3
votes
1answer
316 views

Euler summation and its transformation

The following results: For any function $f \in C^1[a,b]$ and any $q \in \mathbb{N}$, $$\sum_{a < k \leq b, (k,q)=1} f(k)=\frac{\varphi(q)}{q} \int_a^b f(x) dx + O(\tau(q) (\sup_{x \in [a,b]} ...
3
votes
1answer
281 views

Dirichlet series of the reciprocal radical function

Define $ rad(n):=\prod_{p|n}p $ and $a_n:=\frac{n}{rad(n)}.$ For example $a_n=1$ whenever n is a squarefree integer. The assosiated Dirichlet series $$F(s):=\sum_{n} \frac{a_n}{n^s}=\prod_{p} ...
7
votes
1answer
622 views

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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0answers
212 views

Goldbach-Waring problem - Bound on $N$ in Hua's theorem?

Current status of Waring-Goldbach problem http://en.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem Wiki says that WG conjecture is that for every $k$, $\exists$ primes $p_{1}, p_{2}, \cdots ...
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0answers
270 views

Axioms for zeta functions

The Selberg class is an axiomatization of arithmetically significant zeta functions (a.k.a. L-functions) by a few analytic properties (functional equation etc.) However there do exist other zeta ...
20
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7answers
2k views

What should be learned in an introductory analytic number theory course?

Hello all -- I have the privilege of teaching an introductory graduate course in analytic number theory at the University of South Carolina this fall. What topics should I definitely cover? I'm not ...
5
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0answers
345 views

Exponential sums related to cusp forms

Let $$ f(z)=\sum_{n\geq 1} a_f(n) e^{2\pi n i z}$$ be a holomorphic newform on the upper half-plane of weight $k$ for $\Gamma_0(N)$ and of trivial character which is normalized so that $a_f(1)=1$. ...
2
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1answer
393 views

Multiplicative functions whose Dirichlet series have essential singularities

What can be said about the partial sums of a complex-valued completely multiplicative function, let's say bounded by 1 in absolute value, if its Dirichlet series has an essential singularity? As a ...
3
votes
2answers
650 views

on the Zeroes of Hasse -weil L-function

my question is that already we know that the Birch and Swinnerton Dyer conjecture ,formally conjectures that the Hasse-weil L-function should have a zero at $s=1$ when curves have infinitely many ...
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1answer
1k views

Intuition behind the Tamagawa numbers

i have read many books concerning the definition of tamagawa numbers ,but none of the books explained an intuition behind the concept , i mean what could be the intuitive definition of tamagawa number ...
7
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1answer
510 views

On meromorphic continuation of zeta function(s) and special values at negative integers

Euler developed (at least) two different approaches in order to calculate the values $\zeta(-m)$ of the zeta function $$\zeta(s) = \sum_{n\geq 1} \frac{1}{n^s}$$ at non-positive integers. In one ...
3
votes
1answer
707 views

Jacobsthal function related to squares

The ordinary Jacobsthal function $j$ is defined by setting $j(n)$ as the smallest number $m$, such that for each consecutive $m$ numbers in the integers, there is at least one of the numbers comprime ...
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votes
3answers
1k views

Where Can i find the lecture Videos of BSD 2011

i recently heard that there was a conference on Birch and Swinnerton dyer conjecture Held at Cambridge on May 4 until May 6, the main theme is "The conference marks the 50th anniversary of the ...
4
votes
1answer
439 views

What is the relation of the Kuznetsov-Bruggeman trace formula and the Selberg trace formula?

I have read that there is an elementary way to show that the above mentioned trace fromulas are equivalent in the sense, that each of them can be derived directly from the other. There should exist a ...
8
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2answers
1k views

good books on Dirichlet's class number formula

i refrained from asking the technical questions ,may be everyone didnt like my attitude ,atleast help me finding the good books can anyone suggest any good book that gives a complete reference to ...
6
votes
0answers
519 views

Is there a probabilistic interpretation of Dedekind zeta functions?

Reading the interesting paper Honest Bernoulli excursions by Smith and Diaconis motivated the question whether probabilistic interpretations for general Dedekind zeta functions are known. In the ...
5
votes
2answers
1k views

Why is $\frac{\sqrt{6}}{32}(29 + \sqrt{145}) \approx \pi$ ?

Apologies in advance if this is a stupid question; also, disclaimer: this is purely for fun; but: Why is $\frac{\sqrt{6}}{32}(29 + \sqrt{145})$ such a good approximation to $\pi$? (Correct to 8 ...
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0answers
241 views

Prime divisors of the difference set

Fix $c\in(0,1)$, and let $N$ be a (large) positive integer. Given a set $A=\{0=a_1<\dots<a_n=N\}$ of density $\alpha:=n/N>c$ with $\gcd(A)=1$, I want to find a prime dividing as few ...
5
votes
1answer
186 views

Estimating the size of reduction of rational points on $\mathbb{G}_m^2$

Hi, Let $\Gamma$ be a free subgroup of rank 2 in $\mathbb{G}_m^2(\mathbb{Q})$. For all but finitely many primes p we can reduce $\Gamma$ modulo p. Let $S$ be the of primes for which $\Gamma$ does not ...
5
votes
5answers
2k views

Exponential sums for beginner.

What are the good books, online lecture notes or starting material on exponentials sums with applications in number theory for a beginner, apart from N. M. Korobov's book? The book or notes should ...
7
votes
2answers
391 views

Salie-type sum bound

I am interested in bounding the following Salie-type ("twisted Kloosterman") sum $$ S(a,b,\beta) = \sum_{x \in \mathbb{Z}/{p^{\beta}}\mathbb{Z}} \left( \frac{x}{p^{\beta}} \right) \chi(ax + bx^{-1}). ...
19
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2answers
1k views

Does the quadratic form $x^2 - 7y^2$ represent infinitely many primes, with the restriction that $0 < y < x/10$?

Surely yes, and in more generality, but can it be proved? It seems that most, if not all, statements about quadratic forms representing primes fall back on algebraic number theory (i.e. splitting of ...
6
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1answer
1k views

Values of the Riemann zeta function and the Ramanujan summation - How strong is the connection?

(This Question was taken from MSE. As Eric Naslund pointed out there, this question is relevant. The summation method mentioned in this question is actually a good answer to it.) The Ramanujan ...
6
votes
3answers
720 views

Asymptotic Formula for a Mertens Style Sum

Hello, I am wondering if there is a simple asymptotic formula for $$\sum_{p\leq x}\frac{\left(\log p\right)^{k}}{p},$$ where $k\geq0$ is some integer. If $k$ is $0,$ by using the Prime Number ...