# Tagged Questions

Questions on the use of the methods of real/complex analysis in the study of number theory.

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### On $\sum_{\substack{\zeta(\frac{1}{2}+i\gamma)=0\\0<\gamma<T}}\prod_{n=1}^\infty \left| 1-\frac{(\gamma\log x)^2}{n^2\pi^2}\right|$ as $O(\log x)$

On assumption that the identity (2) for a representation of $\pi(x)$ holds, see here Two Representations of the Prime Counting Function in this site Mathematics Stack Exchange, and since using the ...
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### What's about of an analogous Riemann's function $R(X)$ for twin primes?

It is well know the so-called Riemann's explicit formula for the prime counting function $\pi(x)$ involving the density $J(x)$ for prime powers and how by Möbius inversion one recovers $\pi(x)$ and ...
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### Examples of Weil's explicit formula

In Bombieri, PROBLEMS OF THE MILLENNIUM: THE RIEMANN HYPOTHESIS, Clay Mathematics Institute (2000), from page 8, V. Further evidence: the explicit formula the author tell us that there is a ...
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### Bounds for the Fourier transform of characteristic functions on $\mathbb{Z}/N\mathbb{Z}$ supported on large sets

Suppose $A \subseteq \mathbb{Z}_N := \mathbb{Z}/N\mathbb{Z}$ with $|A| \geq N/2$. Let $$\hat{A}(h) := \sum_{a \in A} e_N(ha),$$ where $e_N(x) := e^{2\pi i x/N}$. Clearly $|\hat{A}(h)| \leq |A|$ for ...
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### number of primitive Pythagorean triangles whose hypotenuses do not exceed n?

i just read "mathematical constants" book; it said that Lehmer proved the following theorem in 1900 where P_h(n) , P_p(n) is number of primitive Pythagorean triangles whose hypotenuses and ...
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### (Non-)Canonicity of using zeta function to assign values to divergent series

This article http://blogs.scientificamerican.com/roots-of-unity/does-123-really-equal-112/ got me thinking about the "identity" $$1 + 2 + 3 + \cdots = -1/12,$$ and I wanted to convince myself there ...
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### On relationships between the general terms of sequences from different equivalences to the Riemann Hypothesis

The following are simple deductions using easy calculations for inequalities and limits. I define the following sequences, whose shape is inspired in Nicolas, Robin and Lagarias, respectively, ...
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For integers $N, t \geq 1$, would you know of any special sets $A$ of integers in literature for which either an explicit formula (hopefully nice enough) or good estimate is known for the number $$\#\... 1answer 45 views ### Asymptotics of \sum\limits_{n/2 < p \leq n} \frac{1}{p} I'm reading a paper which asserts the following:$$\sum_{n/2 < p \leq n} \frac{1}{p} \sim \frac{\log 2}{\log n}$$follows from prime number theorem, where the sum is taken over p prime. What is ... 0answers 22 views ### Justify \lim_{n\to\infty}n^p\int_0^1\sum_{k=n}^\infty\frac{\sigma(k)e^{x/k}}{k^{p+2}\log\log k} dx=\frac{e^\gamma\int_0^1f(x)dx}{p} Inspired in PROBLEM 207, La Gaceta de la Real Sociedad Matemática Española, Vol. 16, N0. 3 (page 507 in spanish, proposed and solved by Furdui), I've tried write examples of this new statement ... 1answer 32 views ### Explaining an integral involving the divisor function In a 1973 paper by Martinet, Deshouilliers and Cohen, A(x) is defined as$$A(x)=\lim_{N\to\infty}\frac{\#\{n\leq N\mid \frac{\sigma(n)}{n}≥x \}}{N}where \sigma(n) is the "sum-of-divisors" ... 1answer 69 views ### Primitive, quadratic Dirichlet character to odd prime power modulus Let p be an odd prime number and let \alpha \geq 1 be an integer. Let \chi be a real, non-principal, primitive Dirichlet character mod p^{\alpha}. How does one show that \alpha = 1? If we ... 0answers 67 views ### Are complex numbers complete in every way? I was told many times a story. Indeed a fascinating one to me as a student learning mathematics. First there were natural numbers. People started adding things and finding solutions to finding the ... 1answer 39 views ### How Changing the order of integration(Elementary proof of the prime number theorem)? I'm studying the exchange of integration order, I need help, any hint? For every real number \rho \geq 0, write V(\rho)=e^{-\rho}R(e^{\rho})=e^{-\rho}\psi(e^{\rho})-1 where \psi(x) is the ... 1answer 58 views ### Contour Integration, Riemann Zeta (-n) I was reading Riemann's Zeta Function by H. Edwards, and could not understand the equation on the page 12. \begin{align*} \zeta(-n) &= \frac{\prod(n)}{2\pi i}\int_{+\infty}^{+\infty} \frac{(-x)^{-... 0answers 30 views ### The asymptotic behaviour of \sum_{1\leq k\leq N-1}\int_{p_k}^{p_{k+1}}\log x d[x], where p_n is the nth prime number Let p_k is the kth prime number and consider for N\geq 2 the arithmetic functionf(N)=\sum_{k=1}^{N-1}\int_{p_k}^{p_{k+1}}\log(x) d[x]$$where [x] is the integer part function (provide us in ... 0answers 75 views ### When does \sum_{p\in\mathbb{P}} \frac{1}{|p|^2} diverges? We know \sum_{p\in\mathbb{P}} \frac{1}{|p|^2} diverges where \mathbb{P} denotes set of all primes in \mathbb{Z}[i] (because that sum is greater that \sum_{p \equiv 3 \mod 4} \frac{1}{p}, which ... 0answers 59 views ### Next book in in learning Analytic Number Theory I have just finished the book "Tom M. Apostol - Introduction to Analytic Number Theory". My aim is to reach to graduate level to do research, especially on Rationality/Irrationality and Algebraic/... 2answers 52 views ### Estimate for \sum_{q=1}^{M}\frac{\varphi(q)}{q^{2}} Related to Bourgain Paper [duplicate] Let N\gg 1 be a large parameter, which I ultimately want to let tend to infinity. I am reading an old paper of Bourgain, where he claims the lower bound (Equation 2.50, pg. 118)$$\sum_{q=1}^{N^{1/...
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I would to ask for a logarithm integral, used for Gauss. I read that he uses it to calculate the number of primes, less than a given natural number. It is like: $Li= \int_{0}^x(dt/lnt)$ I read that he ...
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