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.
9
votes
1answer
219 views
Is $j(\tau)^{1/3}$ the hauptmodul for the congruence subgroup generated by $\tau\rightarrow\tau+3, \tau\rightarrow-1/\tau$?
The 3rd root of the modular invariant $j$ is
$$ j(\tau)^{1/3}=q^{-1/3}(1+ 248q+ 4124q^2+ 34752q^3+\cdots),$$
where $q=e^{2\pi i \tau}$.
I was wondering if $j(\tau)^{1/3}$ the hauptmodul for the ...
2
votes
0answers
206 views
On a certain representation of the Riemann zeta function in Montgomery-Vaughan
I (TK. Isaac) recently saw some elegant representation of the Riemann zeta function in Montgomery-Vaughan's ''Multiplicative Number Theory'' (p.338), which however, seems untrue unless i'm missing ...
5
votes
0answers
146 views
Legendre's three-square theorem and squared norm of integer matrices
Let $\mathbb{N}$ be the set of non-negative integers. Let $E_n$ be the set of integers which are the sum of $n$ squares. Let $F_n$ be the set of integers of the form $\Vert A \Vert^2$ with $A \in M_n(...
0
votes
1answer
105 views
Enquiry on an equality involving the Riemann zeta function
Let $\zeta$ denote the Riemann zeta function.
Does the equality
$$\Re(1/4 + t^2)\zeta(1/2 + it)=2t\arg \zeta(1/2 + it) + 2(1/4 + t^2)$$
hold for any $t\geq 0$ ?
3
votes
1answer
99 views
Closed form for an integral involving the Riemann zeta function at the critical line
After seeing this question $L_2$ bounds for $\zeta(1/2 + it)$ and a related integral i became curious if/how the approach in the answer by reuns can be applied to evaluate
$$I_{a,b}=\int_{-\infty}^{\...
5
votes
0answers
178 views
A problem of Erd?s on convergence of $\sum (-1)^nn/p_n$ and equidistribution of $\pi(n)$ modulo 2
Erd?s asked1 whether the series
$$\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$$ converges.
Here, $p_n$ denotes the n-th prime.
I can show that this series converges simultaneously with the series $\sum_{...
0
votes
0answers
114 views
When is the exponential sum of a rational polynomial zero
For which polynomial $f(x)\in\mathbb{Z}[x]$ and $p\in\mathbb{N}$, is the exponential sum $\frac{1}{p}\sum_{n=1}^{p}e^{2\pi i f(n)/p}$ equal to zero?
P.S. I am aware that the case $\deg(f)=1$ is ...
3
votes
2answers
504 views
Heuristics behind the Circle problem?
Is there a heuristic argument behind the exponent in the circle problem? The problem that I am referring to is the following: Consider a circle of radius $R$ centered at the origin in the plane and ...
1
vote
0answers
31 views
Uniformity in Wirsing's Mean Value Theorems
In two important papers of Wirsing, namely "Das asymptotische Verhalten von Summen über multiplikative Funktionen" (1961) and its follow up (1967), several results on mean values of multiplicative ...
3
votes
0answers
143 views
Class fields without class field theory
Is there an English reference for the analytic construction of the Hilbert class field of an imaginary quadratic field without using class field theory? I am in particular interested in a proof of the ...
1
vote
1answer
181 views
$L_2$ bounds for $\zeta(1/2 + it)$ and a related integral
Denote by $\zeta$ the Riemann zeta function. I just learnt from this question $L_2$ bounds for tails of $\zeta(s)$ on a vertical line
that $\int_{T}^{\infty} \frac{\zeta(1/2 + it)}{1/4 + t^2}\mathrm{...
2
votes
0answers
59 views
On some inequality involving the Riemann zeta function integral at $\Re(s)=1/2$
I recently saw on p.$458$ of Montgomery-Vaugahn's ''Multiplicative number theory'' that the inequality
$$\int_{1}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2})$$
holds uniformly for $T\geq 2$, ...
0
votes
0answers
58 views
GCD of two numbers in the form $x^n-r$
1) Is there a closed form for $\mathrm{gcd}(x^n-r,y^n-s)$ where $x,y,n,r,s$ are natural numbers?
2) Is $\max_{x,y}\{\mathrm{gcd}(x^n-r,y^n-s)\}$ finite, for fixed $n,r,s$?
Any references or ...
7
votes
1answer
282 views
Gauss - Dirichlet class number formula
Let $p=8k+3$ be a prime. Then the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$ is given by
$$h(-p)=\frac 13\sum_{k=1}^{\frac{p-1}{2}}\left(\frac kp \right).$$ While this is ...
2
votes
0answers
24 views
Distribution of binary quadratic forms in a given genus
It was brought up in this question (Distribution of 'square classes' of binary quadratic forms) that the objects I am interested in are actually binary quadratic forms in the principal genus ...
