For any positive even integer 2n:
cfw_\displaystyle \zeta (2n)=cfw_\frac cfw_(-1)^cfw_n+1B_cfw_2n(2\pi )^cfw_2ncfw_2(2n)! \zeta (2n)=cfw_\frac cfw_(-1)^cfw_n+1B_cfw_2n(2\pi )^cfw_2ncfw_2(2n)!
where B2n is the 2nth Bernoulli number.
For odd positive intege
The Riemann zeta function ?(s) is a function of a complex variable s = s + it. (The notation s, s, and t is used traditionally in the study of the zeta function, following Riemann.)
The following infinite series converges for all complex numbers s with re
Main article: Riemann hypothesis
Apart from the trivial zeros, the Riemann zeta function has no zeros to the right of s = 1 and to the left of s = 0 (neither can the zeros lie too close to those lines). Furthermore, the non-trivial zeros are symmetric abo
The location of the Riemann zeta function's zeros is of great importance in the theory of numbers. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the Re(s) = 1 line. A better re
It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (?n) contains the imaginary parts of all zeros in the upper half-plane in ascending order, then
cfw_\displaystyle \lim _cfw_n
The HardyLittlewood conjectures
In 1914, Godfrey Harold Hardy proved that ?(
+ it) has infinitely many real zeros.
Hardy and John Edensor Littlewood formulated two conjectures on the density and distance between the zeros of ?(
+ it) o
The Riemann zeta function or EulerRiemann zeta function, ?(s), is a function of a complex variable s that analytically continues the sum of the Dirichlet series
cfw_\displaystyle \zeta (s)=\sum _cfw_n=1^cfw_\infty cfw_\frac cfw_1cfw_n^cfw_s \zeta (s)=\sum
Hazewinkel, Michiel, ed. (2001), "Zeta-function", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Riemann Zeta Function, in Wolfram Mathworld an explanation with a more mathematical approach
Tables of selected zeros
Prime Numbers Get Hitched
West Side Store
Downtown Store; 25%
Valley Store; 30%
West Side Store; 16%
Central Store; 29%
HONORS WORLD HISTORY
CHAPTER 8, LESSON 5
Directions: Answer the following questions on lined paper.
1. Was Justinian successful in reestablishing the Roman Empire? Explain.
2. What was Justinians most lasting achievement?
3. What problems
350,062 cases of gonorrhea
were reported in the United
States in 2014
7,228 cases reported in New
Jersey in 2015
About 100,000 women had
reported having gonorrhea in
About 150,000 men had
reported having gonorrhea in
1. Avoid h
11.1 Sequences and Series: Arithmetic, Geometric
A sequence is a function whose domain is the
set of counting numbers.
Note: The domain may be ﬁnite or inﬁnite. The associated
sequence is called ﬁnite or inﬁnite.
Often. a formula can be found for
Sophocles looked at frailty, pride and punishment
Oedipus the King, Oedipus at Colonus, and Antigone are all plays
o Freedom and destiny to doom
o Foretold because it would happen
Oedipus at Colonus
o Old man blinded and beaten but happy b/c of
FINDING THE TNVERSE OF A 3 x 3 MATRIX
STEP 1, wipe out row 1 and col 1 and
nd the determinant of the mattix which is
left. Put your answer in the row
1 col 1 position of a new matrix.
STEP 2. wipe out row 1 and co! 2 and nd the determinant of the mat
9.7 PARAMETRIC EQUATIONS
x2 - y I- 9 lo the equation of a conic.
x - y - o LOOKS like the equation of a conic,
r I 3 doesn't look like the equation of a conic
1 + Zoose but is l
is a system of equations which when graphed
produce a con
NAMING POINTS ACCORDING TO THEIR DISTANCE FROM
A POLE (ORIGIN) AND THEIR POSITION ON A TERMINAL RAY
OF AN ANGLE IN STANDARD POSITION WITH THE INITIAL
RAY CALLED THE POLAR AXIS!
I \ r
C e5, ,0 1 Aé,
POL POLAR AXIS
Given the i
9.1 (a) =2.5003 is a parameter (related to the population of all the ball bearings in the
container and x =2.5009 is a statistic (related to the sample of 100 ball bearings). (b) p =7.2%
is a statistic (related to the sample of registered vo
12.1 (a) 2.015. (b) 2.518.
12.2 (a) 2.145. (b) 0.688.
12.3 (a) 14. (b) 1.82 is between 1.761 ( p = 0.05) and 2.145 ( p = 0.025) . (c) The P-value is
between 0.025 and 0.05. (In fact, the P-value is 0.0451.) (d) t = 1.82 is signif
8.1 Not binomial: There is not fixed number of trials n (i.e., there is no definite upper limit on
the number of defects) and the different types of defects have different probabilities.
8.2 Yes: 1) Success means person says Yes an