By Cristian Popescu, Karl Rubin, Alice Silverberg

The general topic of the 2009 IAS/PCMI Graduate summer time university was once connections among precise values of $L$-functions and mathematics, in particular the Birch and Swinnerton-Dyer Conjecture and Stark's Conjecture. those conjectures are brought and mentioned intensive, and development remodeled the final 30 years is defined. This quantity comprises the written types of the graduate classes introduced on the summer time tuition. it might be an appropriate textual content for complicated graduate themes classes at the Birch and Swinnerton-Dyer Conjecture and/or Stark's Conjecture. The booklet also will function a reference quantity for specialists within the box. Titles during this sequence are co-published with the Institute for complex Study/Park urban arithmetic Institute. contributors of the Mathematical organization of the United States (MAA) and the nationwide Council of lecturers of arithmetic (NCTM) obtain a 20% from checklist expense.

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**Example text**

The desired d condition is λd|n. Hence, we get μ(d)g d|n n = d μ(d)f (λ). λd|n 18 2 Arithmetic functions Similarly, ⎞ ⎛ μ(d)⎠ = ⎝f (λ) · d| n λ λ|n μ(d)f (λ). λd|n Thus, ⎞ ⎛ d|n n = μ(d)g d μ(d)⎠ . ⎝f (λ) · (3) d| n λ λ|n However, by the previous theorem μ(d) = 1 if and only if d| n λ n = 1, λ and in every other case the sum is equal to zero. Thus, for n = λ we obtain ⎞ ⎛ μ(d)⎠ = f (n). ⎝f (λ) · d| n λ λ|n Therefore, by (1), (3) and (4) it follows that if f (d), g(n) = d|n then f (n) = μ n g(d).

1. Two integers a and b are said to be congruent modulo m, where m is a nonzero integer, if and only if m divides the diﬀerence a − b. In that case we write a ≡ b (mod m). On the other hand, if the diﬀerence a − b is not divisible by m, then we say that a is not congruent to b modulo m and we write a ≡ b (mod m). 2 (Fermat’s Little Theorem). Let p be a prime number and a be an integer for which the gcd(a, p) = 1. Then it holds ap−1 ≡ 1 (mod p). Proof. Firstly, we shall prove that ap ≡ a (mod p) for every integer value of a.

August Ferdinand M¨ obius, born on the 17th of November 1790 in Schulpforta, was a German mathematician and theoretical astronomer. He was ﬁrst introduced to mathematical notions by his father and later on by his uncle. During his school years (1803–1809), August showed a special skill in mathematics. In 1809, however, he started law studies at the University of Leipzig. Not long after that, he decided to quit these studies and concentrate in mathematics, physics and astronomy. August studied astronomy and mathematics under the guidance of Gauss and Pfaﬀ, respectively, while at the University of G¨ ottingen.