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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3
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0answers
204 views

Is there a hidden symmetry in the prime numbers distribution?

Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime. ...
36
votes
15answers
5k views

Shortest/Most elegant proof for $L(1,\chi)\neq 0$

Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$? Background: The non-vanishing ...
1
vote
1answer
352 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 ...
9
votes
2answers
179 views

Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.

Fix $q$ to be a positive integer. Let $$f : \mathbb{N} \to \{-1 ,0, 1\}$$ be a $q$-periodic arithmetic function such that $$\sum_{n = 1}^q f(n) = 0.$$ If $f$ is not identically zero, is it true that ...
2
votes
2answers
533 views

Has this formula about prime gaps already been conjectured and/or proven?

While playing around with prime gaps, I found out that the following formula seems to be a rather good approximation of the ratio $\dfrac{p_{b}-p_{a}}{b-a}$ where $a<b$ are positive integers: ...
1
vote
1answer
283 views

Number of solutions in a sum of squares Diophantine equation

Let $n$ be an integer. I'm interested in upper bounding the number of integer solutions of \begin{equation*} x_1^2+x_2^2+\ldots+x_p^2=y_1^2+y_2^2+\ldots+y_p^2, \end{equation*} where for all ...
2
votes
1answer
225 views

Prime Number Theorem on APs under various conjectures

I'm trying to find the best asymptotic expansions for $\pi(x; a, q)$ in various states: Unconditionally we have \begin{equation} \pi(x; a, q) = \frac{\operatorname{li(x)}}{\phi(q)} + O\left(x ...
0
votes
0answers
129 views

Does the proportion of non trivial zeroes of given real part increase on [0,1/2] for all L-functions?

Here what I call an L-function is either an element of the Selberg class or an automorphic L-function. For such a function $F$, $x\in [0,1/2]$ and $T\gt 0$, let's define $\delta_{F,T}(x)$ such that ...
2
votes
1answer
876 views

Why Riemann Hypothesis so important [closed]

I am often hearing people emphasized how important the RH is,one of them said that it should lead to an efficient way of determining whether a given large number is prime,and the other said,RH would ...
1
vote
1answer
194 views

Error term for prime harmonic

What is known about the asymptotic behavior of $$ f(x)=\sum_{p\le x}\frac1p-\log\log x-B_1? $$ Of course by Mertens we know that $f(x)=o(1),$ but has more been proved (in terms of $O$ or ...
4
votes
0answers
150 views

What will be the consequences if second Hardy-Littlewood conjecture turns out to be true?

It is generally believed that the Second Hardy-Littlewood Conjecture is false. But it has not been proved (or disproved) yet. My question is, What would be the consequences if Second ...
1
vote
1answer
628 views

Sharpening a bound on $\zeta'(s)$

I want to find an upper bound for $\zeta'(s)$ along a vertical line $\Re(s)=b$, where $-1<b<0$. One way to do this is using $$\frac{\zeta'(b+iT)}{\zeta(b+iT)}=O_b(\log T)$$ and ...
1
vote
1answer
385 views

Asymptotic formula for the number of ways to write a number as the sum of $k$ triangular numbers

How would one derive an asymptotic formula for the number of representations of a number $n$ as the sum of $k$ numbers of the form $\frac{m(m + 1)}{2}$ I think that one could use the circle method, ...
1
vote
1answer
153 views

Abscissa of absolute convergence of the product of two Dirichlet series

I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...
0
votes
0answers
76 views

Generalized Dedekind Sum Reciprocity Law

Is there a reciprocity law for generalized Dedekind sums of the form: $$S(a,b;x,y;c)=\sum_{k \mod c}\tilde{B}_1\left(\frac{ak+x}{c}\right)\tilde{B}_1\left(\frac{bk+y}{c}\right)$$ such that the other ...
4
votes
1answer
196 views

polynomials in many variables and Hasse principle

I was wondering whether there exists any result of the form "if $f \in \mathbb{Z}[x_1, ..., x_k]$ is a polynomial (not form! I don't require homogeneity) of total degree $n$, with $k \geq \delta ...
1
vote
1answer
159 views

Bombieri-Vinogradov in short intervals

In 1985 Perelli, Pintz & Salerno proved a short-interval form of the Bombieri-Vinogradov theorem with $\theta \in (7/12, 1]$. Have there been any improvements on this, in particular with the ...
6
votes
1answer
152 views

When is $\vartheta(x)>x$? [Skewes number analog]

Let $\vartheta(x)=\sum_{p\le x}\log p$. What is known about the first time $\vartheta(x)>x?$ Bays & Hudson give good upper bounds (slightly improved by Chao & Plymen) on the first crossing ...
5
votes
0answers
541 views

Zeta function double product

Is it possible to write the following double product in terms of the zeta function? \begin{align} &\prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}} \end{align} Extending the ...
7
votes
1answer
855 views

What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?

The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...
1
vote
1answer
204 views

Residues and values of Riemann Zeta function at some points

I need the following computational results for proving something. Let $1/2 + i\gamma_0$, be the first nontrivial zero of Riemann zeta function, $\zeta(s)$, i.e. $\gamma_0\sim 14.134...$. 1) what is ...
4
votes
0answers
130 views

The divisors of $p-1$ and high-degree residues modulo $p$

Here is a somewhat more explicit version of a question that I asked a while ago. Suppose that $p$ is a prime of the form $p=2n(n+1)+1$, with a positive integer $n$. Can every odd prime divisor of ...
40
votes
4answers
3k views

If the Riemann Hypothesis fails, must it fail infinitely often?

That is must there either be no non-trivial zeros off the critical line or infinitely many? I'm sure that no one believes otherwise, but I've never seen a theorem in the literature addressing this. ...
2
votes
1answer
125 views

Bounds on imaginary parts of partial Kloosterman sums?

For a prime $p$ and integers $a,m$, $0<a,m<p$ define the (partial Kloosterman) sum $$ S_p(a,m) = \sum_{0<k<m} \exp\left(\frac{2\pi\mathrm{i}}{p}(a x + x^{-1})\right), $$ where $x^{-1}$ is ...
4
votes
1answer
175 views

References to proofs of upper and lower bounds on the number of coprimes in an interval?

On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...
7
votes
2answers
360 views

short character sums averaged on the character

Let $a$ be an integer, $p$ a prime (much) greater than $a$, and $\chi$ a Dirichlet character. There is an abundant literature on the sums $$S(\chi,a)=\sum_{i=1}^a \chi(i),$$ called short (or ...
9
votes
2answers
718 views

Asymptotics of the n-th prime using the gamma function

In the paper http://rgmia.org/papers/v8n2/eepnt.pdf, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that. $$ p_n = n ...
2
votes
0answers
108 views

Siegel Walfisz Theorem for algebraic number fields

Is there a generalization of the Siegel Walfisz to algebraic number fields? This has been done for the prime number theorem in the prime ideal theorem.
12
votes
2answers
586 views

Special values of $\zeta$ outside the real line and the critical strip

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line ...
6
votes
1answer
515 views

Precise relation between prime number theorem and zero-free region

I was wondering about the following, and I was hoping that some expert here could answer, rather than me indulging in a search for a needle in the haystack of formulas in books like Titchmarsch. ...
20
votes
9answers
2k views

When does the zeta function take on integer values?

Here $\zeta(s)$ is the usual Riemann zeta function, defined as $\sum_{n=1}^\infty n^{-s}$ for $\Re(s)>1$. Let $A_n=${$s\;:\;\zeta(s)=n$}. The behaviour of $A_0$ is basically just the Riemann ...
0
votes
0answers
104 views

A question about the duality principle

Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by ...
1
vote
0answers
82 views

number of divisors

Let $d(n,Q)$ denote the number of divisors of $n$ that is smaller than $Q$. Let $D(n,Q)$ denote the number of positive integers smaller than $Q$ that has the same prime factors with $n$. My question ...
4
votes
1answer
156 views

Voronoi formula and twists by additive characters

I was wondering if there are any references for the error term in the problem $$\sum_{n\leq x} r(n) \exp(2\pi i\frac{a}{q}n)$$ where $r(n)$ is the number of representations of $n$ as a sum of two ...
3
votes
0answers
83 views

Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$

Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding $$ \int_0^T L\left(\tfrac{1}{2} + it, f ...
2
votes
1answer
158 views

Bound for sums of bounded multiplicative functions that are zero at primes

Let $h:\mathbb{N}\rightarrow\mathbb{C}$ be a bounded multiplicative function with $h(p)=0$. The motivation for this question is just a general enquiry and, since I suppose it has already been ...
7
votes
0answers
253 views

Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?

(General) Lindelof Hypothesis which says for any $L$-function we have $$L(1/2+it)\ll Q(t)^{\epsilon}$$ for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $t$. For a Maass form $\phi$ ...
5
votes
4answers
976 views

Using Vinogradov's theorem for finding prime solutions to a linear equation (an exercise from Vaughan's book)

I'm trying to solve an exercise from Vaughan's book, "The Hardy-Littlewood Method" (ex. 3 in chapter 3: Goldbach's problems, p.36), because I want to use the result stated in it. It is a variation of ...
6
votes
1answer
143 views

Computing certain integrals over high-dimensional polyhedra

Let $\delta>0$ be a small real number and consider the $k$-dimensional region consisting of points for which $$\delta\leq x_1\leq x_2\leq\ldots \leq x_k$$ and $$x_1+\ldots+x_k\leq 1.$$ I am ...
3
votes
0answers
120 views

Computing local volumes : the case of Hecke p-adic subgroups

I am quite interested in knowing how to compute some volumes of groups defined on local fields $K$, mainly in order to evaluate the identity term in trace formulas. It is something well done in the ...
6
votes
1answer
403 views

Generalization of Watson's triple product

In Watson's thesis (page 51) we can find his beautiful triple product formula. My question is that does there exist a generalization of this formula? By generalization, I mean: If $\phi_n$'s are ...
25
votes
5answers
3k views

Partial sums of multiplicative functions

It is well known that some statements about partial sums of multiplicative functions are extremely hard. For example, the Riemann hypothesis is equivalent to the assertion that ...
4
votes
1answer
155 views

Asymptotic behaviour of $K$-Bessel function in transition range

It is known that the famous mistake of Iwaniec-Sarnak in their paper of $L^\infty$ norm of eigenfucntion of non-cocompact arithmetic surfaces in lemma (A1) is because of they did not consider the bump ...
10
votes
1answer
401 views

Parity of the Prime Counting Function

I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers. Let: $\ \ E_n := \left\{ k \in \left\{1,\dots,n\right\} : \pi(k) \equiv 0 \mod 2 ...
0
votes
1answer
115 views

A conjectural convergence condition for a weakened Elliott-Halberstam conjecture

For $a$ and $q$ positive integers such that $a\lt q$ and $(a,q)=1$, let $\pi(x;q,a)$ be the number of primes $p\equiv a\pmod q$ below $x$. One can show that $\pi(x;q,a)\sim \dfrac{\pi(x)}{\varphi(q)}$ ...
7
votes
2answers
338 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 ...
17
votes
1answer
4k views

Tightening Zhang's bound [closed]

Inspired by a blogpost by Scott Morrison and ongoing discussion there I decided to create this community wiki to track progress on the original bound of Yitan Zhang. The original bound was ...
0
votes
1answer
256 views

Is there a formula that can predict the primes in the sequence of ratios of consecutive superior highly composite numbers? : $2, 3, 2, 5, 2, 3, 7,…$

This is the sequence of prime numbers which are the elementary building blocks for the superior highly composite numbers: $2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, ...$ The $n^{th}$ ...
5
votes
1answer
167 views

The sixth power integral moment of automorphic L-function attached to Maass Forms

It is known that the sixth power integral moment of automorphic L-function attached to Cusp Forms has been proved by M. Jutila, that is $\int_{0}^{T}|L(1/2+it,f)|^{6}dt \ll T^{2+\varepsilon}$. And ...
3
votes
0answers
145 views

Effective version of the Bombieri-Vinogradov theorem

Is there an effective version of the Bombieri-Vinogradov Theorem, in that have bounds on the implied constant been found?