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        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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        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 ...
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        votes
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        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 ...
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        votes
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        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(...
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        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 $t\geq 0$ ?
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        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}^{\...
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        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_{...
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        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 ...
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        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 ...
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        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 ...
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        votes
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        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 ...
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        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{...
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        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 $T\geq 2$, ...
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        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 ...
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        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 ...
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        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 ...

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