**1**

vote

**1**answer

170 views

### Two-dimensional Perron formula

There is a well-known Perron formula, which connects a mean value of certain arithmetic function with its Dirichlet series:
$$ \sum_{n\le x} f(n) = {1\over 2\pi i} \int_{c-i\infty}^{c+i\infty} F(s) ...

**3**

votes

**2**answers

296 views

### Real character modular forms: Fourier coefficient real?

Let $f$ be a modular form of level $N$ and real character $\chi$ of mod $N$ and weight $k$.
Does the Fourier coefficient or hecke-eigenvalue of $f$ have to be real?
What I knew is that if $N=1$ and ...

**8**

votes

**1**answer

384 views

### Prime races à la Mertens

I have just read the nice survey by Granville and Martin about prime races.
I wonder what happens if one changes the rules for the prime races as follows.
Fix $q$ a modulus (an integer $>1$). For ...

**3**

votes

**1**answer

256 views

### Can the generalized divisor summatory function $D_z$ be expressed explicitly in terms of Zeta Zeros?

Mertens function has, by residues, an explicit formula of
$M(n)=\displaystyle\sum_{\rho}\frac{x^\rho}{\rho\zeta'(\rho)}-2+\sum_{n=1}^\infty\frac{(-1)^{2 n}(2\pi)^{2n}}{(2n)! n \zeta(2n+1)x^{2n}}$
...

**0**

votes

**1**answer

246 views

### A convergence issue [Edited]

Let $\{x_n\}_{n=1}^\infty$ be a sequence of vectors in a Hilbert space $$l^2_{k^{-2}}:=\{z=\{z(k)\}_{k=1}^\infty:\sum\limits_{k=1}^\infty z(k)^2k^{-2}<\infty\}.$$ It is known that for some $x\in ...

**4**

votes

**1**answer

303 views

### Existence of a certain subset of natural numbers equidistributed modulo $m$ for every $m$

I was talking to a friend and the following set $S$ came up.
Let $f$ be some real valued function tending to infinity.
Let $S$ be a subset of natural numbers such that $|S \cap [1,N]| = N^{\delta}+ ...

**4**

votes

**1**answer

299 views

### Circle method on things other than the integers

The circle method is often used to estimate the number of solutions to the equation $$x_1 + x_2 + ... x_k = N$$ if for all $i$ $x_i\in A\subseteq\mathbb{N}_0$ and some subset of the nonnegative ...

**2**

votes

**1**answer

206 views

### Explicit bound on $\sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p)$

I'm looking for an explicit bound for $f(x) = \sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p)$, where $\chi$ is a Hecke character for a number field $K$ of degree $n$, on the ideals ...

**8**

votes

**1**answer

438 views

### The sum over zeros in the explicit formula for $\zeta(s)$

The explicit formula for $\zeta(s)$ is:
$$
\psi(x)=x-\sum_{|\operatorname{Im}\rho|<T}\frac{x^\rho}{\rho}-\log(2\pi)-\log\left(1-\frac{1}{x^2}\right)+O\left(\frac{x\log^2T}{T}\right),
$$
where ...

**11**

votes

**1**answer

424 views

### Roots of unity near 1 in Z / p Z

Let $r \ge 3$ be a fixed integer. I'm interested in primes p such that no integer in the interval $(-\sqrt{p}, \sqrt{p})$, except $1$ (and $-1$ if $r$ is even), is an r-th root of unity modulo p.
The ...

**2**

votes

**1**answer

416 views

### Square-free integers not divisible by any “small” primes

I have two very related questions:
If $f(N)$ is the number of square-free integers in the interval $[1, N]$, it is well known that $$f(N) \sim \frac{6}{\pi^{2}} N.$$
My first question is, if we ...

**4**

votes

**3**answers

586 views

### Reference for Kronecker-Weyl theorem in full generality

The Kronecker-Weyl theorem asserts the following: fix real numbers $\theta_1,\dots,\theta_d$, and consider the infinite ray $t(\theta_1,\dots,\theta_d)$ $(t\in\Bbb R)$ inside the $d$-dimensional torus ...

**10**

votes

**1**answer

389 views

### Does the Maynard-Tao Theorem apply to general tuples of linear forms?

In the paper http://arxiv.org/pdf/1311.5319v1.pdf the author states the following theorem, which he attributes to Maynard and Tao.
For any integer $m > 2$, there exists an integer
$k = k(m)$ such ...

**8**

votes

**1**answer

225 views

### Sharpest bound on the zero free region of $\zeta^{\prime}$?

I'm interested in calculating all of the zeroes of the first derivative of the Riemann $\zeta$ function up to an arbitrary height. I know that (on the RH), all of these zeroes will have real part $\ge ...

**6**

votes

**1**answer

450 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 ...

**26**

votes

**2**answers

2k views

### Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?

Mathematica knows that:
$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$
The von Mangoldt function should then be:
...

**23**

votes

**1**answer

1k views

### How good is “almost all” when it comes to the Riemann Hypothesis?

Let $N(T)$ be the number of zeroes of the Riemann zeta function $\zeta$ having imaginary part strictly between $0$ and $T$, and let $N_0(T)$ be the number of those zeroes that also have real part ...

**3**

votes

**1**answer

482 views

### When does Merten's product theorem accurately estimate the number of coprimes in an interval?

Assume an arbitrary $x$ and let $z$ be smaller than $y$, where $y$ is the length of the interval $[x,x+y]$. What I would like to know is:
Let $W(z)=\prod_{p\leq z}\left(1-\frac{1}{p}\right)$. For ...

**10**

votes

**2**answers

678 views

### Number of representations of an integer as an (arbitrary) sum of products

If $n$ is a positive integer, let $r(n)$ denote the number of representations of $n$ as a sum of products of pairs of positive integers. (Here, the order of the terms in the sum does not matter, but ...

**1**

vote

**1**answer

665 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 ...

**2**

votes

**0**answers

196 views

### square-free parts of values of polynomials

Given a polynomial $f(x) \in \mathbb{Z}[x]$ of degree $d$, consider the following three sets:
$$N_1(x) = \#\{k \leq x: f(k) \text{ is square-free}\}$$
$$N_2(x) = \#\{n \leq x: n = f(k) \text{ is ...

**5**

votes

**1**answer

484 views

### Lines in image; are they significant to prime numbers if so how?

Amateur math question. I was playing around generating some 2D images, and wondered what it would look like if placed $P_{i}$ dots on a circle with diameter of $i$ for increasing values of $i$, where ...

**14**

votes

**1**answer

595 views

### Chebotarev density theorem for $k$-almost primes

Consider a finite Galois extension $L$ of $\mathbb Q$, of Galois group $G$. Let $k \geq 1$ be a fixed integer. Let $D$ be a subset of $G^k$ invariant by conjugation and by the natural action of the ...

**3**

votes

**1**answer

209 views

### reference for Lindelof Hypothesis implying finitely many zeros off critical line?

Can anyone give me a reference for the following theorem on the Riemann zeta function?
If the Lindelof Hypothesis is true (that is $\zeta(\sigma+it)=O(t^\epsilon)$ as $t\rightarrow\infty$), then ...

**0**

votes

**0**answers

130 views

### On the number N(x,y) of those integers n<x, with squarefree core k(n)<y

I'm asking something that may be trivial for those who are deeply into Analytic Number Theory, but unfortunately I'm still not into that set.
The core $k(n)$ of an integer $n$ is the product of all ...

**5**

votes

**2**answers

419 views

### summation of products of combinatorials

For any natural number $N$ and $0\le n\le N$ define
$$
f(n) = f(n,N) = \frac{1}{(N+1)!} \sum_{\substack{{S\subset \{1,\ldots,N\}} \\ {|S|=n}}} \prod_{s\in S} s.
$$
(The empty product is interpreted ...

**2**

votes

**0**answers

424 views

### The simple zero conjecture for the Riemann zeta function

The simple zero conjecture says that all zeros of the Riemann zeta function are simple.
Suppose the conjecture is not true. Namely there is an $s$ in the complex plane such that $\zeta(s)=0$ and the ...

**4**

votes

**0**answers

527 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 ...

**5**

votes

**2**answers

374 views

### Reference and best bounds of $\sum_{n\leq x}\frac{\mu(n)}{n}$

Could someone please provide information about the best possible known bounds of the sum $$A(x)=\sum_{n\leq x}\frac{\mu(n)}{n}?$$ Unconditionally, $A(x)=O(e^{-c\sqrt{\log x}})$ is known to me. Does ...

**4**

votes

**0**answers

319 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

**0**answers

194 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 $\alpha := \liminf_{\kappa \to 0} \lambda(\kappa)/\kappa > 0$? ...

**3**

votes

**1**answer

205 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

171 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

207 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

137 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

735 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

**1**answer

573 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

456 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

599 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 ...

**22**

votes

**2**answers

2k 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

501 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" ...

**5**

votes

**0**answers

86 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 ...

**18**

votes

**1**answer

567 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

464 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

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 ...

**2**

votes

**1**answer

375 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

295 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

872 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?

**17**

votes

**1**answer

1k 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},\ ...