# Tagged Questions

**0**

votes

**0**answers

26 views

### Least prime for which a square-free integer is a non-residue

Suppose $a$ is a square-free integer and $\left(\frac{a}{p}\right)=1$ for the primes $p\leq k$. I'll call $a$ a quasi-square of order $k$. What I am interested in is the maximum value of $k$ in terms ...

**23**

votes

**1**answer

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:
...

**3**

votes

**0**answers

331 views

+50

### Best known bounds on certain exponential sums

What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...

**3**

votes

**0**answers

118 views

### Goldbach's problem in algebraic number fields [duplicate]

Are there any results on the representation of numbers in algebraic number fields as the sum of primes in the ring of integers in that field? There are some results for Waring's problem in other ...

**6**

votes

**0**answers

184 views

### Easiest way to see that $\zeta_{\mathbb{Z}[i]}(s) = \zeta(s) L(s, \chi)$?

As the question suggests, what is the easiest way to see that$$\zeta_{\mathbb{Z}[i]}(s) = \zeta(s)L(s, \chi)?$$Here, $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...

**4**

votes

**1**answer

73 views

### Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$, does $L_c(s, \chi)$ necessarily equal $1$?

Consider an analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, let $c ...

**4**

votes

**2**answers

491 views

### Number of prime numbers in a range

Denote by $A_n$ the number of prime numbers between $n$ and $n + \log n$.
Is it true that $A_n < const$?
UPD: Is it true that $A_n > \log \log n$ (or something another) for infinite number ...

**12**

votes

**2**answers

896 views

### Typical value of totient function

Can anyone tell me what the expected value of Euler's totient function φ$(n)$ is (roughly) if you choose a random integer $n$ in the range $[N,N+M]$, where $M$ is large and $N$ is larger than $M$? ...

**13**

votes

**1**answer

368 views

### Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...

**5**

votes

**2**answers

271 views

### Logarithmic integral, $π(x)$ and $x/(\ln x)$

The function $\text{Li}$ (logarithmic integral) is defined for $x>0$
by
$$
\text{Li}(x)=\int_2^{x}\frac{dt}{\ln t}.
$$
The prime number theorem, proven by Hadamard and de la Vallée-Poussin in 1896 ...

**7**

votes

**0**answers

68 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 ...

**4**

votes

**1**answer

275 views

### Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?

This question expands on this one from MSE.
In the literature about Dirichlet $L$-series, I found that their Euler products:
$$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$
...

**10**

votes

**1**answer

726 views

### Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...

**32**

votes

**2**answers

829 views

### Can we find two positive integers $n$ and $m$ ($n,m>1$) such that $n^\pi = m$? [duplicate]

I came across this apparent random question in some math questions website. At first, i thought it was easy to show that there are not integer solutions to this equation, but then i realized that the ...

**4**

votes

**1**answer

513 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 ...

**7**

votes

**1**answer

1k views

### The large sieve for primes

Let $\Lambda(n)$ be the von Mangoldt function, i.e., $\Lambda(n) = \log p$ for $n$ a prime power $p^k$ and $\Lambda(n) = 0$ for all $n$ that not prime powers. Let
$$S(\alpha) = \sum_{n \leq N} ...

**9**

votes

**1**answer

475 views

### Do relaxed Liouville functions violate Chowla's conjecture?

Let $\lambda$ be the Liouville function. One version of Chowla's conjecture says that for each set of distinct natural numbers $h_1 , \dots , h_k$,
$$\sum_{n\leq x} \lambda(n+h_1) \dots ...

**3**

votes

**1**answer

104 views

### What is the analytic conductor of this Hecke L-function?

Following Iwaniek and Kowalski, S5.10, page 130 we consider an angle character $\xi_k$ on the Gaussian integers $\mathbb Z[i]$ defined by
$ \xi_k(\mathfrak a) = \left(\frac{\alpha}{|\alpha|}\right)^k ...

**9**

votes

**2**answers

543 views

### Iwaniec-Kowalski Exponential Sum for Quadratic Function

I am reading about 'Exponential Sums' in the book 'Analytic Number Theory' by Iwaniec and Kowalski. On page 199 they mention the bound:
$$|S_f(N)|^2 \le N +2N^2q^{-1}+4(N+q)\log q \tag{1}$$
where, ...

**2**

votes

**1**answer

101 views

### Representation numbers of numerical semigroups

I've been playing around with numerical semigroups lately. I'm pretty new to this stuff, so I apologize in advance if my notation is non-standard. Fix positive integers $x_1,\dots,x_r$ with ...

**3**

votes

**1**answer

185 views

### Product of $\tau(k)$

How do we show that
$$\prod\limits_{k=1}^{n} \tau(k) = 2^{n (\log \log n + C) +
\phi(n)},$$
where $\tau(k)$ is the number of divisors of $k$, the constant $C$ is given by
$$C = \gamma + \sum_{\nu ...

**37**

votes

**15**answers

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 ...

**3**

votes

**0**answers

97 views

### Liouville's function and Legendre Symbols

Suppose $p$ is a prime and let $\left(\frac{a}{p}\right)$ denote the Legendre symbol modulo $p$. If $a$ is not a perfect square, then one expects some cancellation in ...

**5**

votes

**0**answers

143 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 ...

**10**

votes

**3**answers

442 views

### On the number of consecutive divisors of an integer

Define for $n \in \mathbb{N}$ the function $$\tau_1(n):=\sum_{\substack{d|n, \\ d+1|n}}1,$$ i.e. the number of consecutive divisors of an integer. The average of $\tau_1(n)$ is $1$ since $$\sum_{n\leq ...

**60**

votes

**5**answers

23k views

### Consequences of the Riemann hypothesis

I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice ...

**0**

votes

**0**answers

68 views

### mean value estimate $ \frac{1}{T} \int_0^T | \zeta( \tfrac{1}{2} + it)|^2 \, dt = \log \frac{T}{2\pi } + (2\gamma - 1) + O(T^{\delta})$

I was browsing through the recent paper of Bourgain and Watt on mean value estimates of the Riemann zeta function on the critical line.
$$ \frac{1}{T} \int_0^T | \zeta( \tfrac{1}{2} + it)|^2 \, dt = ...

**2**

votes

**1**answer

193 views

### Siegel-Walfisz for the Möbius function

I am working through the proof of the Bombieri-Vinogradov theorem in Analytic Number Theory (Iwaniec, Kowalski). My problem is that on page 424, it is said that $\mu(m)$ satisfies $D_f(x;q,a)\ll ...

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vote

**0**answers

57 views

### Related to derivative of Modified Bessel I function wrt the order

I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$.
Using maple, it seems that ...

**5**

votes

**1**answer

152 views

### On the number of 3-Selmer elements of rational elliptic curves

I am trying to understand a step in the proof of Theorem 39 in the recent work of Bhargava and Shankar, "Ternary Cubic Forms having bounded invariants, and the existence of a positive proportion of ...

**3**

votes

**1**answer

139 views

### Subconvexity bound for Hecke $L$-functions in the $s$-aspect

Let $L(s,\chi)$ be the $L$-function of a non-trivial Hecke character of a general number field $K$, so that $L(s,\chi)$ which has no pole or zero at $s=1$.
I am looking for a reference for upper ...

**6**

votes

**1**answer

405 views

### Multiplicity one theorem

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...

**0**

votes

**0**answers

62 views

### Extracting information from $\sum_{n \leq X} a(n) (X-n)^d$

Allow me to give context to the question, which appears in the box at the bottom.
A very general hope from the theory of Dirichlet series is to try to extract information about the coefficients of a ...

**3**

votes

**1**answer

295 views

### Ratio of consecutive divisors and average

Let $2\leq d_1 < d_2,...,d_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard:
(1) ...

**1**

vote

**4**answers

605 views

### Distribution of composite numbers

I have moved this question to math.stackexchange.com. People who are interested in this question can discuss at :http://math.stackexchange.com/questions/1272431/distribution-of-composite-numbers
...

**1**

vote

**0**answers

49 views

### Big Omega result about number of totally positive integers with fixed trace

There is much literature on the study of $N_a$, the number of totally positive integers with fixed trace $a$ in a totally real field.
That number has a natural geometric approximation $G_a$, and we ...

**2**

votes

**0**answers

86 views

### On covering by smooth numbers

Denote $P(n)=\mathsf{greatest}\mbox{ }\mathsf{prime}\mbox{ }\mathsf{factor}\mbox{ }\mathsf{of}\mbox{ }n$.
Denote $S(x,y)=\{n<x: P(n)<y\}$.
Denote $S_t(x,y)=\sum_{i=1}^tS(x,y)$ as $t$-fold ...

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vote

**0**answers

228 views

### Analogues of the Monster for central charges different from 24

One way to define the Monster group is to consider a conformal field theory (CFT) corresponding to central charge $c=24$ and look at the automorphism group of its vertex operator algebra. For one of ...

**1**

vote

**1**answer

160 views

### Eisenstein series of weight $2$ for $\Gamma_0(N)$ : where am I wrong?

Let $A_{N,2}$ be the set of triples $(\psi,\varphi,t)$ such that $\psi$ and $\varphi$ are primitive Dirichlet characters modulo $u$ and $v$ with $(\psi\varphi)(-1)=1$, and $t$ is an integer such ...

**3**

votes

**4**answers

231 views

### Uniform upper bound for the sum over primes $\sum_{p \leq x} p^{-1+\varepsilon}$

I am reading the article D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991) (click).
In equation (27) the authors, apparently, used the following ...

**8**

votes

**1**answer

262 views

### Asymptotic limit of truncated Legendre sieve

Consider the truncated sum
$$
S(x):=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}\mu(d)/d,
$$
where $P(z)$ is the product of all primes less than or equal to $z$, and $\mu(d)$ is the Möbius ...

**3**

votes

**2**answers

397 views

### Character sums: reference request

This one will be quick...
Wonder if anybody knows or remembers the title of the paper in which Karatsuba introduced his approach at Burgess's bound on character sums.
Thanks for your support.
EDIT. ...

**48**

votes

**4**answers

2k views

### When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...

**28**

votes

**2**answers

3k views

### Is Li(x) the best possible approximation to the prime-counting function?

The Prime Number Theorem says that $\lim_{n \to \infty} \frac{\pi(n)}{\mathrm{Li}(n)} = 1$, where $\mathrm{Li}(x)$ is the Logarithm integral function $\mathrm{Li}(x) = \int_2^x \frac{1}{\log(x)}dx$. ...

**14**

votes

**3**answers

2k views

### A variant of Goldbach Conjecture

I'm asking if this variant of weak Goldbach's Conjecture is already known.
Let $N$ be an odd number. Does there exists prime numbers $p_1$, $p_2$ and $p_3$ such that $p_1+p_2-p_3=N$? Ideally, can we ...

**7**

votes

**1**answer

375 views

### What does the sum of the reciprocals of all the highly composite numbers converge to?

I've calculated the sum of the reciprocals of all the $156$ first highly composite numbers up to $10^{18}$:
$\sum \dfrac{1}{HCC(n)} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{1}{12} + ...

**5**

votes

**1**answer

120 views

### Bounding a Sum of Adjoint L-Function Values

Fix integers $k\geq2$ and $N>1$, and let $S(k,N)$ denote the normalized new Hecke eigenforms in $S_k(\Gamma_1(N))$. [If it makes my question easier to answer, feel free to replace this with ...

**1**

vote

**1**answer

118 views

### estimate an sum

I need estimate the following sum:
$\sum_{d=1}^{n}\frac{\mu(d)}{d}\sum_{k=1}^{\lfloor n/d\rfloor}\frac{1}{k}\frac{q^k}{1-q^{-kd}}$, where $q>1$ and $\mu$ is the Möbius function.
To obtain the ...

**0**

votes

**1**answer

74 views

### Upper bound for a ratio of modified Bessel functions

I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex, and $z$ is a positive real number. Do you know any results about it? Thank ...

**5**

votes

**0**answers

84 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 ...