Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. (Full article...)
The real part (red) and imaginary part (blue) of the critical line Re(s) = 1/2 of the Riemann zetafunction. Image credit: User:Army1987 
The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. It has been an open question for well over a century, despite attracting concentrated efforts from many outstanding mathematicians.
The Riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zetafunction ζ(s). The Riemann zetafunction is defined for all complex numbers s ≠ 1. It has zeros at the negative even integers (i.e. at s=2, s=4, s=6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the nontrivial zeros, and states that:
Thus the nontrivial zeros should lie on the socalled critical line ½ + it with t a real number and i the imaginary unit. The Riemann zetafunction along the critical line is sometimes studied in terms of the Zfunction, whose real zeros correspond to the zeros of the zetafunction on the critical line.
The Riemann hypothesis is one of the most important open problems in contemporary mathematics; a $1,000,000 prize has been offered by the Clay Mathematics Institute for a proof. Most mathematicians believe the Riemann hypothesis to be true. (J. E. Littlewood and Atle Selberg have been reported as skeptical. Selberg's skepticism, if any, waned, from his young days. In a 1989 paper, he suggested that an analogue should hold for a much wider class of functions, the Selberg class.) (Full article...)
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