# Tagged Questions

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

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### Average order of divisor function ; Theorem in Apostol

In Introduction to Analytic number theory by Apostol, a theorem states that: For all x $\geq$ 1, we have $$\sum_{n\le x} \sigma(n)= \frac{1}{2} \zeta(2)x^2 + O(x\log x)$$ The definition of O(g(x)) ...
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### Hecke $L$-function of cusp form is entire

Let $f=\sum a(n)q^n\in S_k(N,\chi)$ be a cusp form of integral weight. Can someone give me the proof of the fact that : the Hecke $L$-function $L(f,s)=\sum\frac{a(n)}{n^s}$ is entire. I searched in ...
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### What's about $\sum_{n=1}^\infty e^{-p_n u}$, where $p_n$ is the nth-prime number?

I am assuming that the following function, for which I am asking as reference request, should be known in the literature, since Glaisher studied the Prime Zeta Function, and my computation is the ...
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### Convergence of the Euler product

Suppose that the Riemann Hypothesis is true. It is well known that then the Dirichlet series $$\sum_{n=1}^\infty\frac{\mu(n)}{n^s}$$ converges in the half-plane ${\rm {Re}}\, s>\frac{1}{2}$. Does ...
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### Large gap between two consecutive square-free numbers

Let $q_n$ denote the $n$-th square-free number. By Chinese remainder theorem (see this post), it is not difficult to show that there is arbitrarily large gap between two consecutive square-free ...
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### Sum of products of $(1 - 1/p)$

Let $\pi(n)$ denote the number of primes not greater than $n$, and $p_k$ the $k$th prime, so that $p_{\pi(n)}$ denotes the largest prime not greater than $n$. I'm interested in the value of the ...
In class we introduced Reimann Zeta function $$\zeta (x)=\sum_{n=1}^{+\infty} \frac{1}{n^x}$$ And we proved its domain was $D=(1,+\infty)$ Now Euler proved that $$\zeta(x)=\prod_{p\text{ prime}... 0answers 35 views ### Relation between Meissel–Mertens constant and Euler–Mascheroni constant From the Wikipedia page, the Meissel–Mertens constant M is defined as the limit:$$M:=\lim_{n\to\infty}\left(\sum_{p\leq n}\frac{1}{p}-\log\log n\right).$$Why is it equal to \gamma+\sum_{p}\left(... 1answer 229 views ### Estimate of the derivative Show that if f(x)=x^2+O(x), and f is differentiable with non-decreasing derivative f'(x), then f'(x)=2x+O(\sqrt{x}). I know that if f' is not non-decreasing, then the statement is not true.... 1answer 52 views ### Wintner's mean value theorem This is an exercise (exercise 2.22 p80) from A.J. Hildebrand's Introduction to analytic number theory (an online lecture notes). Let g be an arithmetic function, and let f=1*g (i.e.,f(n)=\sum_{d\... 1answer 52 views ### Asymptotic estimate for the sum \sum_{n\leq x} 2^{\omega(n)} How to find an estimate for the sum \sum_{n\leq x} 2^{\omega(n)}, where \omega(n) is the number of distinct prime factors of n. Since 2^{\omega(n)} is multiplicative, computing its value at ... 2answers 96 views ### Prove that the value of the constant C must be 1 After proving the prime number theorem in class, our professor directs us to a remark by Lagrange that for large values of x, \pi(x) is approximately equal to$$ \frac{x}{\log x - B}. $$(This is ... 2answers 97 views ### Proving that \pi(2x) < 2 \pi(x)  In our analytic number theory class we were given the following problem as homework: prove rigorously that for large x the number of primes in (1,x] exceeds that in (x,2x]. In class we proved ... 1answer 48 views ### Asymptotic estimate of the sum \sum_{n\leq x}1/\phi^2(n) How to show that we have the following estimate:$$\sum_{n\leq x}\frac{1}{\phi^2(n)}=c+O(\frac{1}{x}),$$where \phi is the Euler's totient function and c is a constant. I tried to use the ... 1answer 40 views ### Number of subsets S of [n] such that \gcd(S) is coprime to m Fix positive integers m,n. Is there a way to count the number of non-empty subsets S of [n] = \{1, \ldots, n\} such that \gcd(S) is coprime to m? Can we come up with an expression for such a ... 1answer 36 views ### A multiplicative function satisfying \lim_{p^m\to\infty} f(p^m)=0 implies \lim_{n\to\infty} f(n)=0 Let f be a multiplicative function satisfying \lim_{p^m\to \infty} f(p^m)=0. Show that \lim_{n\to\infty} f(n)=0. By unique factorization, we can write n=\prod_{i=1}^k p_i^{\alpha_i}, where ... 0answers 30 views ### What's about  \sum_{n=1}^{\infty} \frac{ \mu\left( \sigma (n)\right)}{n^3} , where \mu(n) is Möbius function and \sigma(n)=\sum_{d\mid n}d? Let  \mu (n) the Möbius function and  \sigma (n) the sum of divisors function, then the arithmetical function g(n)= \frac{ \mu\left( \sigma (n)\right)}{n^3}  isn't multiplicative since gcd(2,... 1answer 29 views ### On the Density of Deficient Odd Numbers and Abundant Integers Let \sigma(x) denote the sum of the divisors of x. If \sigma(x) < 2x, then x is said to be deficient, while if \sigma(x) > 2x, x is said to be abundant. (Of course, when \sigma(x) ... 1answer 67 views ### On the proof of “The infinite series \sum_{n=1}^{\infty} p_n^{-1} diverges”. The following text is from the book Introduction to Analytic Number Theory by T. M. Apostol : Theorem 1.13  \  The infinite series \sum_{n=1}^\infty 1/p_n diverges. Proof. The following ... 1answer 44 views ### Estimates of \Omega_{\text{av}}(n) Ramanujan proved that the average number of distinct divisors of x for x on [1,n], ~\omega_{\text{av}}(n), and the average number of divisors including repetitions, \Omega_{\text{av}}(n), are ... 0answers 58 views ### Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ... 1answer 71 views ### Combinations of four consecutive primes in the form 10n+1,10n+3,10n+7,10n+9 Here n is some natural number. For example, among the primes < 1000 I found four such combinations: \begin{array}( 11 & 13 & 17 & 19 \\ 101 & 103 & 107 & 109 \\ 191 &... 0answers 49 views ### Which values of n is this inequality related to prime numbers true for? Inequality What values of n satisfy the following inequality?$$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right) $p$ are prime numbers and the notation $p_i$ indicates the $i$-...
Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ where $p$ is a prime. The zeta function, $\zeta(E, s)$ for $E$ is defined as \$\zeta(E,s) = \dfrac{(1-\alpha p^{-s})(1-\beta p^{-s})}{(...