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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18
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498 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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0answers
2k 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 ...
15
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0answers
325 views
+50

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 ...
13
votes
0answers
651 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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895 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
465 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
281 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 ...
11
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0answers
754 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 ...
11
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917 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
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0answers
189 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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341 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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254 views

Independence between the number of prime factors of $n$ and $n+2$

I am interested in having an upper bound for the cardinality of $\#\left\{n\leq x\,:\quad\omega(n)=k, \omega(n+2)=\ell\right\}$ for $k,\ell\geq 1$, where $\omega(n)=\sum_{p\vert n}1$ counts the ...
9
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0answers
130 views

Newly defined $L$-function in terms of $L$-function, does it have any obvious zeros or poles?

Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ideal, let $[\mathfrak{a}]$ denote its ...
9
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752 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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0answers
154 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 ...
8
votes
0answers
255 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
votes
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 ...
8
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0answers
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
votes
0answers
336 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
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0answers
129 views

k-Almost Primes in short intervals

According to this question every interval $[x, x + x^{0.45}]$ contains a product of two primes, and this has been improved further slightly. Are there better results available for $k$-almost primes? ...
7
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567 views

“Forthcoming paper” of Goldston-Graham-Pintz-Yıldırım

The above-named authors of [1] and its (significantly different) published version [2] write: In a forthcoming paper, we will show how the methods here can be extended to prove corresponding ...
7
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0answers
95 views

Approximation to a certain Weyl-sum

Let $ S(\alpha)=\sum_{x\leq X}\sum_{y\leq Y}e \left(\alpha x y^3 \right)$, for some $X,Y \geq 1$ and write $\alpha = a/q + \beta$ for $(a,q)=1$, as usual. For the 'classical' cubic Weyl-sum ...
7
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340 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
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658 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
186 views

Rankin-Selberg for Maass form GL(3)xGL(2)

Let $F$ be a Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ (level 1 trivial character). Let $g$ be a Maass cusp form for $\Gamma_0(N)$ with character $\chi$ mod $N$. For convenience, you may assume ...
6
votes
0answers
310 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 ...
6
votes
0answers
740 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
votes
0answers
304 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
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425 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
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0answers
583 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
128 views

Mahler's theorem in the primes

Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. Denote by $A_F$ the area of the region $$\displaystyle \{(x,y) \in \mathbb{R}^2 : |F(x,y)| \leq 1\}.$$ It ...
5
votes
0answers
103 views

Sums of twisted products of Kloosterman Sums

For $m,n,c \in \mathbb{N}$, let $S(m,n;c)$ denote the Kloosterman sum $$ S(m,n;c) := \sum_{\substack{1 \leq a < c \\ \gcd(a,c) = 1}} e \left( \frac{ma + n\overline{a}}{c} \right) $$ where $e(n) = ...
5
votes
0answers
129 views

Particular case of the class number formula, Dirichlet characters

Let $\chi$ be a Dirichlet character modulo $4$ such that $\chi(-1) = -1$, and let $\chi'$ be a Dirichlet character modulo $5$ such that $\chi'(-1) = 1$, $\chi'(2) = \chi'(3) = -1$. How do I see the ...
5
votes
0answers
165 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, ...
5
votes
0answers
98 views

Methods of variational calculus in analytic number theory

What methods of calculus of variations have been used in analytic number theory? I mean do Hamilton-Jacobi theory of PDE found usage in analytic number theory, which raises yet another question has ...
5
votes
0answers
115 views

Moments of completed L-functions?

This is a follow up question to this one. It seems that results on moments of L-functions, that is, estimates for integrals of the form $$\int^{T}_1|\zeta(\sigma+it)|^{2k}dt$$ are typically for the ...
5
votes
0answers
42 views

Primes with prescribed digits in two bases

A result of Harman and Katai tells us that if $n$ is large, and $A$ is a subset of $\{1,\ldots, n\}$ is of size $|A|=r<c \sqrt{n}/\log n$, then the number of primes up to $N=2^n$ with the $j$-th ...
5
votes
0answers
93 views

Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x

Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form $$ g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i ...
5
votes
0answers
161 views

Explicit bounds for exceptional zeros and/or $L(1,\chi)$ for real $\chi$

I would like to have an explicit upper bound (that is, one with explicit constants) for a possible real zero $\beta$ for $L(1,\chi)$ for real Dirichlet characters $\chi$. I need such a bound for real ...
5
votes
0answers
116 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
votes
0answers
89 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
249 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
0answers
740 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
91 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
votes
0answers
375 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
405 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
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198 views

Asymptotic estimate for a random model of primes

Question Let $$ \pi_{rm_c}(x) = \sum_{ \substack{ {n\leq x}\\{(n+a,P(\sqrt{n}))=1}}} 1-1, $$ where $P(x)$ is the product of all primes less or equal to $x$ and $a$ is a random integer constrained to ...
4
votes
0answers
81 views

Large Gaps Between Almost Primes

What is the best lower bound for the longest interval contained in $[1,x]$ free of primes and products of two primes? In other words I am asking for the best lower bounds in a variant of the ...
4
votes
0answers
87 views

The probability distribution of LCM of uniformly distributed integers in $\{1,\ldots,n\}$

In the recent paper by Fernandez and Fernandez here on ArXiv, the following formula which was first proved by Diaconis and Erdos appears, on page 2. For $0<t\leq 1$ the distribution of the lcm of ...
4
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88 views

Shifted convolution problem for Coefficients of automorphic forms

The shifted convolution problem for coefficients of modular forms is well studied and many estimates were established for the shifted convolution sums of Hecke eigenvalues. So, one may ask about the ...