**3**

votes

**1**answer

197 views

### Calculating (n ^ fibonacci(k)) MOD m for a large value of k

The value of $k$ can be very large indeed (up to $10^{12}$). Is there an efficient way to calculate the output?
Edit : 'm' is a prime number.

**4**

votes

**1**answer

159 views

### Hermite Lindemann and transcendental reals

This is about compass and straightedge constructions, although I suspect nothing changes if we expand to real numbers algebraic over the rationals.
Anyway, let $E$ be the "constructible numbers," ...

**21**

votes

**3**answers

2k views

### How many different numbers can be obtained as product of first $n$ natural numbers?

Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set
$\{1^{a_1} \cdot ...

**1**

vote

**0**answers

170 views

### Some identities with the Riemann-Hurwitz zeta function

The only definition that I have ever seen of this Riemann-Hurtwitz zeta-function is this,
For $0 < a \leq 1$ we have the identity
$$ \zeta(z, a) = \frac{2 \Gamma(1 - z)}{(2 \pi)^{1-z}} \left[\sin ...

**2**

votes

**0**answers

134 views

### Odds of projections of a point not on the hyperplane

Let $\mathcal{L}=\{\Bbb x\in\Bbb R^n:x_1+x_2+\dots+x_n=0\}$ be a specific hyperplane.
Let a projection of $c\in\Bbb R^n$ be $p(c)=[p_1,p_2,\dots,p_n]$ where $p_i\neq c_i\implies p_i=0$.
Let ...

**6**

votes

**0**answers

732 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 ...

**7**

votes

**1**answer

521 views

### Can the Brun-Titchmarsh theorem be improved when the modulus is smooth?

For $q,a$ relatively prime, let $\pi(x,q,a)$ denote the number of primes less than $x$ which are congruent to $a$ modulo $q$. The Brun-Titchmarsh theorem states that $$\pi(x,q,a)\leq ...

**11**

votes

**2**answers

403 views

### Sums of reciprocals of prime numbers: $p \equiv a \!\! \mod m$ vs. $p \equiv b \!\! \mod m$

Given positive integers $a$, $m$ and $n$, let $s_{a(m)}(n)$ denote the
sum of the reciprocals of the prime numbers less than or equal to $n$
which are congruent to $a$ modulo $m$.
Is there an integer ...

**10**

votes

**3**answers

518 views

### On the vanishing of the generalized von Mangoldt function $\Lambda_k(n)$ when $n$ has more than $k$ prime factors

It is a well-known fact that the generalized von Mangoldt function, defined by
$$\displaystyle \Lambda_k(n) = \sum_{d | n} \mu(d) \left(\log \frac{n}{d}\right)^k$$
vanishes whenever $n$ has more ...

**21**

votes

**2**answers

1k views

### What is the crucial difference the Maynard/Tao approach and Goldston-Pintz-Yildirim that extends to prime k-tuples with $k>2$

Suppose $m$ is a positive integer. A quantity of interest is
$$
H_m = \liminf_{n\to\infty} \left(p_{n+m} - p_n \right)
$$
The twin prime conjecture, is, of course $H_1 = 2$, the the prime k-tuples ...

**11**

votes

**2**answers

487 views

### Reference for a conjecture on the first primes congruent to 1 modulo other primes

Given a prime $p$, define $f(p)$ to be the smallest prime congruent to $1$ modulo $p$. For example, $f(7)=29$. It has been conjectured that $f(p)<p^2$ always: by Schinzel in his "Hypothesis H" ...

**6**

votes

**0**answers

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 ...

**17**

votes

**1**answer

526 views

### What motivated Rademacher's contour along the Ford circles?

Apologies if this question isn't suitable for MathOverflow; I posted it on MSE here but it didn't get a response and it felt like it was on the cusp of being suitable for here.
After Ramanujan and ...

**1**

vote

**1**answer

428 views

### stationary phase method in analytic number theory

I hope someone can tell me something about the error term in the formula calculating the oscillatory integral like $\int_a^b g(x)e(f(x))d x$. Specially, the exact formula on page 114 of M. Huxley's ...

**7**

votes

**0**answers

993 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 ...

**2**

votes

**1**answer

318 views

### On the convergence of Dirichlet series over the Mobius Mu function

It is known that if $\sum_{k=1}^{\infty} \frac{\mu(k)}{k^s} = \frac{1}{\zeta(s)}$ for $\Re(s) > 1/2$ then RH holds. My question is:
Under RH why is it not $\sum_{k=1}^{\infty} ...

**6**

votes

**0**answers

288 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< ...

**9**

votes

**1**answer

654 views

### best record toward Selberg's eigenvalue conjecture?

What's the best record toward Selberg's eigenvalue conjecture:
Maass forms on $\Gamma_o(N)$ has eigenvalue greater than 1/4?

**15**

votes

**1**answer

920 views

### Quantitative lower bounds related to Zhang's theorem on bounded gaps

Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ ...

**1**

vote

**1**answer

188 views

### Does $L(-1+it,f)\ll_f \log^c q(f)t$ hold ture?

Let $f$ be a holomorphic or Maass cusp form for $SL(2,Z)$. Define $L(s,f)=\sum_{n\ge 1}\frac{a_f(n)}{n^s}$, for $\Im s$ sufficiently large.
Then
$$L(-1+it,f)\ll_f \log^c q(f)t$$
holds, for some ...

**3**

votes

**1**answer

262 views

### Classical Lower Bound of L(1) assuming GRH

Let $L(s)$ be an automorophic $L$-function with conductor $C$ defined by Iwaniec and Sarnak.
What's the classical lower bound of $L(1)$ assuming Riemann Hypothesis?
And what's the reference?
Is ...

**6**

votes

**2**answers

265 views

### Smooth sums of coprime smooth integers

Observe that for any $\epsilon > 0$ there are infinitely many triples of
$c^\epsilon$-smooth coprime positive integers $a$, $b$ and $c$ such
that $a + b = c$. -- Considering triples of the form ...

**10**

votes

**2**answers

908 views

### How did Riemann calculate the first few non-trivial zeros of Zeta?

Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi ...

**13**

votes

**1**answer

446 views

### How fast can we numerically calculate Kloosterman sums?

Define the usual Kloosterman sum by $$S(m,n;c) = \sum_{\substack{x \pmod{c} \\ (x,c) = 1}} e\Big(\frac{mx + n\overline{x}}{c}\Big),$$
where $x \overline{x} \equiv 1 \pmod{c}$, and $e(x) = e^{2 \pi i ...

**3**

votes

**1**answer

321 views

### On link between Riemann hypothesis and partial GRH

Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of ...

**1**

vote

**2**answers

263 views

### A conjecture of Montgomery: reference request

In the answer to this question, engelbret mentions "a conjecture of H. L. Montgomery (not the one on pair correlations, another one), which implies both the GRH and the Elliott-Halberstam ...

**10**

votes

**1**answer

381 views

### Distribution mod 1 of exponential growth sequences

Let $t_n$ be a sequence of real numbers and $C,r>1.$ Suppose that for every $n\geq 1$ we have $\frac{1}{C}r^n\leq t_n \leq Cr^n.$ Does there exist a real number $\xi$ and an $\varepsilon>0$ ...

**1**

vote

**1**answer

557 views

### Exponential sums

I would like to estimate the following sum
$\sum_{N <n \leq 2N}e(vn^{l})$, $l \geq 1$ constant(not integer) and $v$ is a parameter(integer) that doesn't grow too fast(a small power of N).
The ...

**5**

votes

**1**answer

465 views

### Other implications of Zhang's method

I have been reading a bit about Zhang's proof and the associated Polymath8 project.
Though Tao's high level summary
...

**1**

vote

**1**answer

568 views

### An integral representation of the Riemann zeta function

I am referring to the equality in equation $3.29$ (page 12) and $4.20$ (page 17) in this paper.
I am unable to recognize where this comes from or what is the general expression for values other than ...

**0**

votes

**0**answers

95 views

### Derivative of a function related to Dedekind zeta function

Lef $K$ be an algebraic number field of degree $[K:\mathbb{Q}]=n$. For simplicity suppose $K$ is totally real. Define $f(s) = \zeta_K(s) \zeta(1-s)^{n-1}$ where $\zeta = \zeta_{\mathbb{Q}}$.
From the ...

**3**

votes

**0**answers

253 views

### Inequalities in paper by Jean Bourgain

The question refers to the following paper by Jean Bourgain: http://arxiv.org/abs/math-ph/0011053
Specifically, I can't derive the following inequality in (1.20):
\begin{equation}
...

**-3**

votes

**1**answer

242 views

### a question on Siegel Upper Half Space [closed]

Is it known whether siegel upper half plane is dense in the space of nonsingular matrices of same dimension .$.http://en.wikipedia.org/wiki/Siegel_upper_half-space.
Actually the question i have in my ...

**0**

votes

**0**answers

149 views

### Motivation behind the appearance of Bessel functions in partial trace formulas

Bessel functions occur naturally on the Kloosterman side (or geometric side) of Petersson's formula and Kuznetsov's formula. Is there an intuitive explanation for their appearance? For instance, is ...

**4**

votes

**1**answer

430 views

### subconvexity problem for $GL(3) × GL(2)$ $L$-function without involving in symmetric lift

A question in study of subconvexity topic puzzles me for a long time, which mabe a stupid question for many experts. I really wish someone to help me out, and any advice will be highly appreciated.
...

**1**

vote

**1**answer

238 views

### Trace formula for the Möbius function

I found this conjecture while working with the Möbius function
$$ \sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}} g \log n = \sum_t \frac{h(t)}{\zeta'(1/2+it)}+2\sum_{n=1}^\infty \frac{ (-1)^{n} (2\pi ...

**10**

votes

**1**answer

415 views

### Estimate term in Ramanujan Lost Notebook (classic analytic number theory)

This is the fault of Igor Rivin, who asked about sums of divisor functions. I will put in links eventually. What I would like to know is the size of the right hand side in Ramanujan's formula (381), ...

**0**

votes

**1**answer

344 views

### How to do such a partitioning?

Assume:
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K, \qquad x \in \mathbb{R}_+^K , \qquad w = e^{-j\frac{2\pi}N}
$$
and,
$$
f(l) = \sum_{i=1}^K \sum_{j=1}^K x_i x_j w^{(p_i-p_j)l}
$$
I am going to ...

**7**

votes

**1**answer

480 views

### The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$

The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
...

**2**

votes

**2**answers

416 views

### Summation of certain series

Suppose $f(n)$ is a periodic function with period $q$. Now from this paper we get that if $\displaystyle\sum_{n=1}^{q}f(n)=0$ then ...

**6**

votes

**0**answers

244 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 ...

**10**

votes

**1**answer

437 views

### Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?

This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo ...

**7**

votes

**1**answer

974 views

### A reformulation of the Riemann Hypothesis

I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates $\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where $N$ is product of distinct primes.
Let's define ...

**17**

votes

**1**answer

792 views

### Infinitely many primes, and Mobius randomness in sparse sets

Problem 1: Find a (not extremely artificial) set A of integers so that for every $n$, $|A\cap [n]| \le n^{0.499}$, ($[n]=\{1,2,...,n\}$,) where you can prove that $A$ contains infinitely many primes.
...

**2**

votes

**1**answer

84 views

### Question about BFI's “Primes in arithmetic progressions to large moduli”

Ref: http://link.springer.com/article/10.1007%2FBF02399204
My question is about the proof of Theorem 0(b). On p.213, we see the expression
...

**8**

votes

**1**answer

625 views

### Density of prime pairs whose gap is less than the average gap

By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...

**5**

votes

**0**answers

366 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\; ...

**4**

votes

**0**answers

172 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 $\Delta(n)\le d(n)$ ...

**2**

votes

**1**answer

620 views

### Best upper bound on the number of divisors of $n$ that are larger than $N$.

I am looking for the best upper bound on $$\sum_{\substack{d | n\\ d \geq N}} 1.$$
I know that
$$
d(n) = \sum_{\substack{d | n}} 1 \leq e^{O(\frac{\log n}{\log \log n})}.
$$
For my application, I ...

**1**

vote

**1**answer

164 views

### Behavior of a quantity related to Fermat's 4n + 1 Theorem

One of Fermat's theorems states that if $p = 4n + 1$ for some integer $n$, then $p$ can be expressed uniquely as a sum of two squares, $p = a^2 + b^2$. I am working on a problem and I would like to ...