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Results tagged with analytic-number-theory
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user 160959
On the 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.
2
votes
1
answer
205
views
The best estimation of the function $\vartheta(x;q,a)$
Consider the function $$\vartheta(x;q,a)=\sum_{p \leq x ,q|(p-a)}\log p=\frac{x}{\phi(q)}+O(\frac{x}{(\log x)^C}).$$
If the Riemann Hypothesis is ture, in the case $q=a=1$ we have $$|\psi(x)-x|<\frac{ …
- 622
0
votes
1
answer
213
views
Asymptotic behavior of the sum $\sum_{k\le x}\frac{1}{\varphi(k)}$
Suppose $x>0$ and let $f(x)=\sum_{k\le x}\frac{1}{\varphi(k)}$, where $\varphi(k)$ is the Euler totient function. It is well known that $\sum_{k\le x}\frac{1}{k}\sim\log x$. What is the asymptotic beh …
- 622
2
votes
1
answer
204
views
Bounding the sum $f(x)=-\frac{x}{2}+\sum_{p\le x}\log(p)-\frac{1}{x}\sum_{p\le x}p\cdot \log...
Consider the function $$f(x)=-\frac{x}{2}+\sum_{p\le x}\log(p)-\frac{1}{x}\sum_{p\le x}p\cdot \log(p).$$
In my recent work, I need to get an explicit [rather than asymptotic] upper bound of this funct …
- 622
11
votes
1
answer
491
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Sign of the function $f(n)=\sum_{k=1}^n\frac{\mu(k)}{k}$
It is well-known that the Mertens function $M(n)=\sum_{k=1}^n\mu(k)$ changes sign infinitely many times when $n\rightarrow +\infty$. Let $f(n)=\sum_{k=1}^n\frac{\mu(k)}{k}$, then $\lim\limits_{n\right …
- 622