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Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, ζ, is a function of a complex variable s that analytically continues the sum of the Dirichlet series ζ = ∑ n = 1 ∞ 1 n s which converges when the real part of s is greater than 1. More general representations of ζ for all s are given below; the Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, applied statistics. As a function of a real variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis, not available at the time. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, established a relation between its zeros and the distribution of prime numbers; the values of the Riemann zeta function at positive integers were computed by Euler. The first of them, ζ, provides a solution to the Basel problem.

In 1979 Roger Apéry proved the irrationality of ζ. The values at negative integer points found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known; the Riemann zeta function ζ is a function of a complex variable s = σ + it. The zeta function can be expressed by the following integral: ζ = 1 Γ ∫ 0 ∞ x s − 1 e x − 1 d x where Γ = ∫ 0 ∞ x s − 1 e − x d x is the gamma function. For the special case where the real part of s is greater than 1, ζ always converges, can be simplified to the following infinite series: ζ = ∑ n = 1 ∞ n − s = 1 1 s + 1 2 s + 1 3 s + ⋯ σ = Re ⁡ > 1. The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series. Leonhard Euler considered the above series in 1740 for positive integer values of s, Chebyshev extended the definition to Re > 1. The above series is a prototypical Dirichlet series that converges to an analytic function for s such that σ > 1 and diverges for all other values of s.

Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠ 1. For s = 1 the series is the harmonic series which diverges to +∞, lim s → 1 ζ = 1, thus the Riemann zeta function is a meromorphic function on the whole complex s-plane, holomorphic everywhere except for a simple pole at s = 1 with residue 1. For any positive integer 2n: ζ = n + 1 B 2 n 2 n 2! where B2n is the 2nth Bernoulli number. For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic K-theory of the integers. For nonpositive integers, one has ζ = n B n + 1 n + 1 for n ≥ 0 In particular, ζ vanishes at the negative integers because Bm = 0 for all odd m other than 1; these are the so-called "trivial zeros" of the zeta function. Via analytic continuation, one can show that: ζ = − 1 12 This gives a way to a

Nigel Sinclair

Nigel Sinclair, CBE is a Scottish producer of Hollywood films. Sinclair was born on 31 March 1948 in Corbridge, England, his family moved to Galloway, in southwest Scotland, a few years where he grew up. He went to boarding school at Monkton Combe School in England. In 1966, he attended Gonville & Caius College, University of Cambridge, graduating in 1969. After a short stint working as a researcher for the Department of Criminology, University of Cambridge, he qualified as a lawyer with the London firm known as Denton and Burgin, he practiced law in London and subsequently in the Middle East until 1980. In 1979/1980, Sinclair attended Columbia University School of Law in New York and obtained a LAM in International Legal Studies, qualified for the State Bar of California. In 1989, Sinclair founded Sinclair Tennenbaum & Co. in Los Angeles. He practiced entertainment law until he retired from that practice in 1996 to found Intermedia with Guy East. Sinclair launched White Horse Pictures in 2014, with longtime partner Guy East—a content company that takes a Millennial approach to storytelling.

Prior to launching White Horse Pictures, Sinclair was the CEO and Co-Chairman of Exclusive Media, a global independent film company that financed and globally distributed feature films and documentaries. With Exclusive, Sinclair produced Parkland, starring Zac Efron, Billy Bob Thornton and Paul Giamatti. In addition, Sinclair was executive producer on Ron Howard's epic action-thriller Rush, set in the spectacular world of Formula 1 auto racing. Along with Michael Shevloff and Paul Crowder, Sinclair produced the documentary 1, the authorized history of Formula 1. At White Horse, Sinclair's latest productions include a feature adaptation of Conn Iggulden's Emperor series about Julius Caesar's early years and a biopic about The Who drummer Keith Moon; the latter of the two projects will be produced with Exclusive. Sinclair served as executive producer on George Clooney's The Ides of March, nominated for an Academy Award for Best Original Screenplay, the Academy Award-winning documentary feature Undefeated, produced by Exclusive Media's documentary production arm Spitfire Pictures.

Sinclair served as an executive producer on the Hammer Films production and box-office hit The Woman in Black, starring Daniel Radcliffe. Sinclair's other film credits include Peter Weir's The Way Back, starring Jim Sturgess and Ed Harris. Prior to co-founding Exclusive Media and Guy East, founded Intermedia Films in 1996, which grew to become one of the world's leading independent film companies. After their departure in 2002, Sinclair and East founded Spitfire Pictures, merged with Hammer to form Exclusive Media in 2008. Under the Spitfire Pictures label Sinclair produced the award-winning George Harrison: Living in the Material World, Martin Scorsese's biographical film about the life of George Harrison, which won an Emmy, he produced the Bob Dylan documentary No Direction Home directed by Scorsese, which won an Emmy, two Grammy Awards, a Peabody Award and a DuPont. In 2012, Sinclair won his second Grammy for Foo Fighters: Back and Forth and in 2007 he was nominated for a Grammy for Amazing Journey: The Story of The Who.

Sinclair won his first Grammy in 2006 for No Direction Home. His company Spitfire Pictures has been involved with Tongal since 2013, working with the company to crowd-source a documentary. According to the Los Angeles Times, Tongal users will submit pitch ideas, with Spitfire selecting the top five, awarding one the winning idea before distributing the final project. In 1981, Sinclair married Pat Craig and together they have one child in addition to Pat's two children. Sinclair now has three grandchildren. Sinclair is active in a number of charities including the Santa Monica-based charity k9 Connection. Sinclair is a member of the Academy of Motion Picture Arts and Sciences and BAFTA, having served on the Board of Directors for BAFTA LA many times. Sinclair's principal hobby is as collecting guitars. Nigel Sinclair on IMDb

1961 Los Angeles Dodgers season

The 1961 Los Angeles Dodgers finished in second place in the National League with a record of 89–65, four games behind the Cincinnati Reds. 1961 was the fourth season for the Dodgers in Los Angeles. It was the Dodgers final season of playing their home games at the Los Angeles Memorial Coliseum, since they moved to their new stadium the following season. November 28, 1960: Ray Semproch was drafted by the Washington Senators from the Dodgers in the 1960 rule 5 draft. December 15, 1960: Earl Robinson was purchased from the Dodgers by the Baltimore Orioles. December 16, 1960: Danny McDevitt was purchased from the Dodgers by the New York Yankees. January 31, 1961: Joe Pignatano was purchased from the Dodgers by the Kansas City Athletics. March 30, 1961: Ed Rakow was traded by the Dodgers to the Kansas City Athletics for Howie Reed and cash. On April 17, 1961, Duke Snider hit his 370th career home run, which at the time moved him into 7th place on the all-time career home runs list. In the same game, Snider suffered a broken elbow, he was knocked out for the rest of the season when he was hit by a pitch from Bob Gibson of the Cardinals.

May 4, 1961: Don Demeter and Charley Smith were traded by the Dodgers to the Philadelphia Phillies for Turk Farrell and Joe Koppe. May 26, 1961: Art Fowler was purchased from the Dodgers by the |Los Angeles Angels. May 30, 1961: Bob Lillis and Carl Warwick were traded by the Dodgers to the St. Louis Cardinals for Daryl Spencer. Note: Pos = Position. = Batting average. = Batting average.

1933 Annandale state by-election

A by-election was held for the New South Wales Legislative Assembly electorate of Annandale on 24 June 1933 following the death of sitting member, Robert Stuart-Robertson. This by-election was caused by the death of Robert Stuart-Robertson on 2 June 1933; the seat was won by first time State Labor candidate Bob Gorman. He defeated the Federal Labor candidate Percival McDonald, standing in his second and final election, Thomas Wright who stood stood as a Communist in the 1930 election in the seat of Kogarah and first time candidate Harry Cotter standing as a Labor Party Unificationist who never stood again. Electoral results for the district of Annandale List of New South Wales state by-elections

Weddington High School

Weddington High School is a public high school located east of Weddington, North Carolina, as part of Union County Public Schools. As of June 2017, the school has been operating under the leadership of principal Dr. Jay Jones. Weddington's main feeder schools, Weddington Elementary and Weddington Middle, are located on the same campus as the high school. Other feeder schools include Antioch Elementary, in Matthews/Indian Trail, Wesley Chapel Elementary, in Monroe. WHS serves Weddington and Wesley Chapel, it has some students from western Monroe and Indian Trail. As of 2011, Weddington had the highest graduation rate of seniors in the state of North Carolina for the fifth year in a row, with the graduation rate going up from 91.7% to 95.7%. Weddington is a Honors School of Excellence with high growth. To be recognized as an Honors School of Excellence, a school must have 90% or above of their composite End of Course Test scores above grade level. Weddington's test scores are the highest in Union County.

Weddington High School was on Newsweek magazine's "2013 America's Best High Schools" list. Weddington ranked 470th out of 2000 national schools. There are over 18,000 high schools in the United States, placing Weddington in the top three percent; the Weddington High School marching band's 2013-2014 director, Jill Brooks, was selected as the recipient of the 2013-2014 American School Band Directors Association Outstanding Potential Award. The band was highlighted in an issue of Appleseed Magazine for showing the power of music along with the levels of hard work and enthusiasm required to exist in a marching band; as of 2013, Weddington provides its students with a Marine Corps JROTC program. Instructed by retired Marines Lieutenant Colonel David Morgan and First Sergeant Randall Barlow, the program has won two over-all awards from Union County drill meets, along with a host of other drill awards in just its first two years. Weddington's MCJROTC program, as of 2015, consists of about 60 cadets.

In May 2017, the MJROTC Women's Athletic Team finished 3rd in the National Competition, held in San Diego, CA. The school has earned 3A State Championships in the following sports:: Women's Cross Country: Cheerleading: Baseball: Baseball: Men's Cross Country: Men's Cross Country: Men's Soccer: Men's Indoor Track & Field: Women's Soccer: Football: Men's Lacrosse: Men's Tennis: Men's Outdoor Track & Field: Men’s Indoor Track & Field: Men’s Outdoor Track & Field: Men’s Lacrosse: Football: Men’s Cross Country: Men's Indoor Track & Field: Men's Outdoor Track & Field: Men's LacrosseThe school has earned over a dozen of the NCHSAA Team Scholar Athlete Awards for having the highest team GPA, regardless of classification

Richard LeParmentier

Richard LeParmentier was an American actor who worked and lived in the United Kingdom, best known for his role as Admiral Motti in Star Wars Episode IV: A New Hope and the acerbic police Lt. Santino in Who Framed Roger Rabbit. LeParmentier, born near Pittsburgh, Pennsylvania in the United States on July 16, 1946, grew up on a dairy farm, his father came from the isle of his mother from County Mayo in Ireland. LeParmentier lived in Hollywood, Florida during his teen years, there his school drama-teacher suggested he become a professional actor, he attended a drama course at Wayne State University in Detroit, Michigan before moving to the United Kingdom in 1974. After appearing in a Fringe theatre production, broadcast by the BBC, LeParmentier was granted an Equity union membership card and toured with the Incubus Theatre Company, his first film role was as lawyer Felix Hoffman in the 1974's Stardust and the following year he appeared in the film Rollerball. He made numerous appearances on British television.

His most prominent role was that of Admiral Motti, the arrogant, mocking Imperial officer, choked by Darth Vader in Star Wars, after Vader finds his "lack of faith disturbing". Mark Newbold, writing on the official Star Wars website, described the role as leaving "an indelible imprint on the Star Wars galaxy, helping to illustrate the fearsome powers of Lord Vader as well as the arrogance and malice of a bloated and over-confident Empire." LeParmentier had auditioned for the role of one of the film's main characters. The auditions for Star Wars were used for Brian DePalma's Carrie and LeParmentier was cast as the high school principal in the latter; the film's production was delayed for nine months, so LeParmentier had to drop out of the role, with his role being recast to Stefan Gierasch. He deemed the part too small. Star Wars' writer and director George Lucas cut the part, the following month LeParmentier was cast as Motti. Additionally prior to gaining the role, LeParmentier was to portray a "Mos Eisley bureaucrat named Montross."

However, before production began, the character was omitted from the film. LeParmentier had a minor role in Superman II with his then-wife Sarah Terence Stamp, he had roles in films such as Octopussy, which featured another Star Wars actor, Jeremy Bulloch and Who Framed Roger Rabbit as Lt. Santino, his last screen role was in 1992, from 1988 his focus became writing and producing. He wrote for several British television series including The Bill and Boon, with his writing partner Paddy Fletcher, he founded the production company Three Rivers Productions in 2008. LeParmentier became a "staple" of the Star Wars and science-fiction convention circuit, made a cameo appearance in an online commercial for the 2012 Xbox 360 video game Kinect Star Wars, which re-created his famous scene from Star Wars. At the time of his death, he was working on Motti Now, a parody of Apocalypse Now, featuring other Star Wars alumni such as Kenneth Colley, Jeremy Bulloch, Garrick Hagon and Jerome Blake. From 1981 to 1984, LeParmentier was married firstly to the British actress Sarah Douglas, best known for playing the role of Ursa in Superman and Superman II.

The two appeared in several films together, including Rollerball, The People That Time Forgot, Superman II. He had three children with his second wife, Cheryl Le Parmentier: Rhiannon and Tyrone, he was staying with them at the time of his death. LeParmentier lived in Bath, England, he died on April 15, 2013, while visiting his family in Austin, United States, aged 66. Soldiers: Heroes of World War II - American Narrator Richard LeParmentier on IMDb Richard Le Parmentier official website