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# Tag Archives: Number Theory

## summation

Let denote the Euler’s function. Prove that

**Solution**

The key idea is to rewrite the floor as a sum involving divisors. Hence,

## Convergence of series

Let denote the greatest prime factor of . For example , . Define . Examine if the sum

converges.

**Solution**

Let be the -th prime number.

If , then with , and . It folows that

From Merten’s theorem

and the original series has the same character as

which is convergent.

## On multiplicative functions

Let denote the Möbius function and the Euler’s totient function. Prove that

**Solution**

Since both functions are multiplicative it suffices to prove the identity for prime numbers. Hence for we have

## On the factorial

Let denote the Möbius function and denote the floor function. Prove that:

**Solution**

The RHS equals

since for .