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### A chain of six circles associated with six points on a circle (in Mobius plane) [closed]

I found a conjecture: A chain of six circles associated with six points on a circle (in Mobius plane). This is a generalization of the last my previous question in Three chains of six circles. (...
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### How to get a good upper bound on $\sum_{1 \lt d, d|n} \phi(d)/\log d$?

I'm actually interested in a slightly smaller quantity, but I'm willing to accept the following simplification, especially if there are small error terms. Let's start with $n \gt 1$, Euler's totient ...
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My question is on Mobius inversion formula convergence/properties when used with infinite sums of function. Lets consider (on $\mathbb{R}^{+}$): $$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$ We call $... 2answers 3k views ### Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function? Mathematica knows that: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$ The von Mangoldt function should then be: $$\Lambda(n)=... 1answer 274 views ### Trace formula for the Möbius function I found this conjecture while working with the Möbius function$$ \sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}} g \log n = \sum_t \frac{h(t)}{\zeta'(1/2+it)}+2\sum_{n=1}^\infty \frac{ (-1)^{n} (2\pi )^{... 0answers 146 views ### can this sum be true ?? $$\sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}} g \log n = \sum_t \frac{h(t)}{\zeta'(1/2+it)}+2\sum_{n=1}^\infty \frac{ (-1)^{n} (2\pi )^{2n}}{(2n)! \zeta(2n+1)}\int_{-\infty}^{\infty}g(x) e^{-x(2n+1/2)}... 0answers 228 views ### A conjecture on Moebius transformation Conjecture. If n>1 and f is a mapping from S^n to S^n which maps circles into (instead of onto) circles, and whose range has n+3 distinct points any n+2 of which are in general position (in ... 2answers 555 views ### Problems with the divisor function in a summation I'm trying to work with the following sum:$$f:=\sum_{d\leq x}\mu(d)\tau(d) \Big[ \frac{x}{d}\Big]$$Where$\mu$is the Mobius function,$\tau(n)$is the number of positive divisors of$n$and$h(x)=[...
I am currently in the midst of a summer research project and have run across an interesting summation: $F(n) = \sum\limits_{i=1}^{\lfloor\frac{n}{2}\rfloor}(n - 2i + 1)P_2(i)$. And here are some ...