analytic number theory- jia & matsumoto

... Euler's totient function by r(q) and cp(q), respectively. The symbol x N X is utilized as a shorthand for X < x < 5X, and N =: M is a shorthand for M << N << ... let y = 0.577 - . denote Euler's constant. Since hf(s) > 0 for 8 > 0 (use (r'/I?)'(s) = Ck.O(k + s)-~), we do have h(tm) < h(2) = (1 -n(y+logrr))/2+rl...

Ngày tải lên: 31/03/2014, 16:21

... that case sin θ 2 < θ 2 , 1 sin θ 2 > 2 θ , − 1 sin θ 2 < − 2 θ − 2n 2 −2n+2 3π . and so the requirement (23) follows from − 2n 2 −2n+2 3π < −(n − 1) or 2n 2 − 2n + 2 > 3π(n − 1). ... any larger perfect values? In short, can there be integers a 1 <a 2 < ··· <a n such that the differences a i − a j , i>j, take on all the values 1, 2, 3, ,  n 2  ? If we introduce ....

Ngày tải lên: 31/03/2014, 16:21

... V R where V NS ,V R are spanned by vectors of the form e k 1 ∧ e k 2 ∧···e k n with k 1 >k 2 > ···k n > 0and e k =exp(−ikx), with half-integral and integral k, resp. The Z 2 degree of such a ... basis of Λ n V − or equivalently of Λ n V + corresponds to the partitions of natural numbers into n distinct summands m>0, which yields one of the partition functions mentioned in the intr...

Ngày tải lên: 12/02/2014, 16:20

... +1− a p p +1+a p where a p = p − N p and N p is the number of solutions of y 2 ≡ x 3 + Ax + B modulo p. Conjecture 3 [U]: There exists a number c>0, a sequence of N →∞with an elliptic curve E N of ... that γ(r)=  (p,q)=1 |β(p, q)| 2 exp  2πi p q r  . The sum is performed over all q,andp co-prime to q with 0 <p<q. This is the two-point correlation function of multiplicities of...

Ngày tải lên: 12/02/2014, 17:20

... =  p e i i with the p i distinct primes ordered so that p 1 < p 2 < . . If p i > √ n for each i and there is more than one p i , then m > n, a contradiction. Thus some p i is less than √ n, ... is nonempty because 0 ∈ Q and Q is bounded because a −bn < 0 for all n > a/b. Let q be the largest element of Q. Then r = a − bq < b, otherwise q + 1 would also be in Q. Thus q an...

Ngày tải lên: 07/03/2014, 16:20

... 0 < a‘ < q. By symmetry (Pld = (-V’ pa‘ = qa + r where y’ is the number of odd a’ such that (79) with 0 < a’ < q, 0 < r < q, 0 < a < ... N1 . The process must terminate, since N = &iqi2 . . . bm qnL N > Ni > N, > > 1, Theorem 3 From Perfect Numbers to the Quadratic Reciprocity Law 7 Thi...

Ngày tải lên: 15/03/2014, 16:03

... two odd numbers is an even number; (4) the sum of two even numbers is an even number; (5) the sum of an odd and even number is an odd number; (6) the product of two odd numbers is an odd number; (7) ... contradiction that there are only a finite number of primes: p 1 < p 2 < ···< p m . Consider the number P = p 1 p 2 ···p m + 1. If P is a prime, then P > p m , contrad...

Ngày tải lên: 30/03/2014, 02:20

... mod m (where c-' and d-' denote any integers which are inverse to c and d modulo m). To prove Corollary 3, we have c(ac-' - bd-') = (acc-' - bdd-') = a ... divisors of pa are p' for 0 5 j 5 a, and so f (pa) = Cy='=n ip(p') = 1 + C;==l (p' - p'-l) = p9 This proves the proposi- tion for eJ& hence for...

Ngày tải lên: 31/03/2014, 16:20