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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17
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442 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 ...
15
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0answers
1k 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 ...
13
votes
0answers
566 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 ...
13
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0answers
866 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
462 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 ...
11
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0answers
887 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 ...
10
votes
0answers
715 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 ...
10
votes
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155 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 ...
10
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326 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 ...
9
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205 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, ...
8
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703 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: ...
8
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1k 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: ...
8
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323 views

Evaluating Shintani cone zeta functions

Hi everyone I am trying the evaluate sums of the form $$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)}$$ ...
7
votes
0answers
279 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 $\phi$ ...
7
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0answers
160 views

Eisenstein series over a definite division algebra

Let $D$ be the definite quaternion division algebra over $\mathbb{Q}$. $\mathcal{O}$ is a maximal order inside $D$, let's fix $\mathcal{O}$ to be the Hurwitz quaternion. Let ...
7
votes
0answers
1k 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 ...
7
votes
0answers
270 views

Average of Fourier coefficients of a cusp form of half integral weight

Suppose $f$ is a cusp form of half integral weight $k$ w.r.t. the group $\Gamma_0(4)$ ($k$ is not very low, can assume $k \ge 11/2$), and $a_n$ is its Fourier coefficient. The Linnik bound says that ...
7
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314 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 ...
7
votes
0answers
621 views

“probabilistic” density of primes?

A certain set $\cal P$ of primes is defined by two assumedly independent conditions: The first condition on a prime $p$ can be characterized in terms of the type of splitting of $p$ in certain Galois ...
6
votes
0answers
734 views

How many 2L-bit numbers are the product of two L-bit numbers?

If I multiply two integers $x, y $ in $[0,2^L)$, I get an integer in $[0,2^{2L})$. Clearly, this map from $[0,2^L) \times [0,2^L) \to [0,2^{2L})$ is not bijective. I am interested in the size of ...
6
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294 views

implication 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< ...
6
votes
0answers
383 views

Bounds on an exponential sum related to an equidistribution question

The short version: Given non-zero real numbers $\alpha$ and $\beta$, can one prove the following estimate in a simple manner? Or does it follow from a well-known result on exponential sums? $$ ...
6
votes
0answers
559 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
0answers
81 views

Constants for Rosser's Sieve

I am trying to apply Iwaniec's formulation of Rosser's sieve (here) to obtain nontrivial lower bounds for almost-primes in various sequences. These sequences have sieve dimension 1 (if $g(p)$ is the ...
5
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75 views

A Generalized Wiener-Ikehara Theorem with multiple poles on the line

One version of the Wiener-Ikehara Theorem says that if $$ f(s) = \sum \frac{a(n)}{n^s} $$ is a Dirichlet series with nonnegative coefficients that converges absolutely for $\text{Re}(s) > 1$ and ...
5
votes
0answers
202 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 ...
5
votes
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698 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 ...
5
votes
0answers
258 views

$x^2+1$ attaining almost prime values

Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such ...
5
votes
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84 views

Minimum of the product of linear forms over a lattice

In Chapter [IX.1] of Siegel's Lectures on the Geometry of Numbers it is shown that if we have $n$ linear forms $y_{j}=\sum_{k=1}^{n}{a_{jk}x_{k}},\quad j=1,\ldots,n$, with the coefficient matrix ...
5
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367 views

a generalization of a formula of Shimura

Let $\phi$ be a $GL(2)$ automorphic form with Fourier coefficients $a(n)$ and $a(1)=1$. Obviously we have $L(s,\phi)=\sum \frac{a(n)}{n^s}$. Shimura have the following formula $L(s, Ad\; ...
5
votes
0answers
383 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 $a_f(1)=1$. ...
4
votes
0answers
160 views

Equivalence of Euler products of Dirichlet series and Meromorphic continuation

Suppose $(f_n(s))_n$ and $(g_n(s))_n$ are two sequences of Dirichlet series with positive coefficients such that $\exists \alpha\in\mathbb R$ such that for all $s\in\mathbb C$ with $\Re(s)>\alpha$ ...
4
votes
0answers
295 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. ...
4
votes
0answers
134 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 ...
4
votes
0answers
514 views

Differential Galois number theory

Following http://math.stackexchange.com/questions/635659/can-the-error-term-involved-in-the-pnt-be-expressed-in-a-galois-theoretic-framew?noredirect=1#comment1341143_635659, I vainly tried to find ...
4
votes
0answers
316 views

About sign changes of Li(x)-π(x)

Given a constant $C$, which are the best known upper bounds for the number of sign changes of the function $$ f: \mathbb{N} \rightarrow \mathbb{R}, \ \ x \mapsto {\rm Li}(x)-\pi(x) $$ in the range ...
4
votes
0answers
190 views

On the multiplicative order of 2 mod primes - II

For $\kappa \in (0,1)$, let $\lambda(\kappa)$ be the density of primes $p$ having $\mathrm{ord}^{\times}_p{2} < \kappa p$. Does $\alpha := \liminf_{\kappa \to 0} \lambda(\kappa)/\kappa > 0$? ...
4
votes
0answers
176 views

Maximal order of Hooley's Delta function?

There is a large literature on Hooley's $$ \Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1 $$ giving its normal and average order. What is known of its maximal order? Clearly $\Delta(n)\le d(n)$ ...
4
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0answers
211 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, ...
4
votes
0answers
141 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
votes
0answers
381 views

Measure Theoretic view of Hardy Littlewood Circle Method

Is it possible to view the Hardy-Littlewood Circle method as the Fourier transform with respect to the Lebesgue measure on [0,1) for an appropriate generating function defined in terms of additive ...
4
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198 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 ...
4
votes
0answers
283 views

generalization of the Brauer-Siegel bound?

For imaginary quadratic number fields $K$ of fundamental discriminant $-D$, the Brauer-Siegel theorem implies that the class number $h(D)$ of $K$ is "close" to $\sqrt{D}$, more precisely for any ...
3
votes
0answers
76 views

Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?

The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ : $$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) = ...
3
votes
0answers
90 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 ...
3
votes
0answers
124 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 ...
3
votes
0answers
172 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?
3
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0answers
208 views

Arguments for the second Hardy–Littlewood conjecture being false?

Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that $$\pi(x + y) - \pi(y) \leq \pi(x).$$ We can easily justify this heuristically, since $$ ...
3
votes
0answers
223 views

Why believe the Elliott-Halberstam conjecture?

I have seen justifications for various conjectures in analytic number theory, e.g. the (generalized) Riemann hypothesis, the Chowla conjectures, etc. justified by a heuristics in which the relevant ...
3
votes
0answers
117 views

Weierstrass's elliptic function-type zeta function

What is known about the following Weierstrass's elliptic function-type zeta function $\sum_{m,n \in \mathbb{Z}} \frac{1}{(z+m+n\tau)^s}$, for $z \in \mathbb{C} \backslash \mathbb{Z} + \tau ...