# Questions tagged [analytic-number-theory]

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

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### Is there an analytic continuation of the Legendre Chi function $\chi_2(z)$ for $z > 1$?

The Legendre Chi function $\chi_2(z)$ is define as $$\chi_2(z) = \sum_{k=0}^{\infty}\frac{z^{2k+1}}{(2k+1)^2}$$ for $-1 \le z \le 1$. But $z > 1$ the series diverges. For real value of $z$ is ...
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### Newman's Short Proof of Prime Number Theorem

I'm going through the paper of D. Zagier on Short Proof of Prime Number Theorem. There it says in V that $\Phi(s)=\int_1^\infty \frac{d\vartheta(x)}{x^s}$ . Can someone please explain in details why ...
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### Conductor of a character

We know, a character $\chi$ mod $q$ is induced by a primitive character $\chi^{*}$ mod $q^{*}$. I have the following questions. If $\chi$ is non principal, can $\chi^{*}$ be principal? Let $\chi$ be ...
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### Real non-trivial zeros of Riemann zeta function inside critical strip

Could someone please tell me what is known about the real non-trivial zeros of Riemann zeta function inside critical strip? Do we know there is none? I want to know what is known about the above ...
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### Two slightly different polynomial expansions for $\Xi(0)$. Could a connection between these two be derived?

After experimenting with polynomial expansions for the Riemann $\Xi(t)=\xi\left(\frac12+it\right)$-function, I landed on these two equations: with $M$ the KummerM confluent hypergeometric funcion and ...
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### Proper Way to Calculate Value of Riemann Zeta function?

I understand that an Analytic Continuation of a function will extend its domain into areas that it previously wasn't defined in. I've been looking at one of the Analytic Continuations of the Zeta ...
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In the proof of the Erdös-Kac theorem in section $III.4$ of $\mathit{Introduction\ to\ Analytic\ and\ Probabilistic\ Number\ Theory}$ by Gérald Tenenbaum, we need to bound the characteristic function $... 3 votes 0 answers 113 views ### Visual proof of$\sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) = 1 $Background Let$\zeta(\cdot)$be the Riemann zeta function. I'm looking for a visual proof of the infinite series identity $$\sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) = 1. \tag{1}\label{1}$$ This ... • 7,048 1 vote 1 answer 70 views ### Dirichlet series and Euler product For a multiplicative function$f$, show that we have $$\sum_{n=1}^{\infty}\frac{f(n)}{n^s}=\prod_p\left(\sum_{\nu=0}^{\infty}\frac{f(p^{\nu})}{p^{\nu s}}\right).$$ My ... • 123 4 votes 0 answers 146 views ### Probability that one random number among many has a unique prime factor If I sample$N+1$integers$x, x_1, \ldots, x_N$uniformly and independently from$\{1, \ldots, M=2^k\}$, what is the probability that$x$contains a prime divisor that does not divide any of the$\{...
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Here is an explanation of selberg's proof of prime number theorem. Here by using mertens theorem they show that, $$\sum_{k≤n}\frac{R(k)}{k^2}=O(1)$$ And then proceed to show that, \left|\frac{R(y)}{...