• Sudhanshu Gaud

Beginning from the statement of Riemann sixth hypothesis which is unproved. Using Euler formula, hyperbolic trigonometric function formulae, Riemann integral formula of converting infinite summation into a definite integral and other functions with some logical ideas for series summation. The author proved the world's mathematical problem Riemann hypothesis raised by German mathematician B. Riemann in 1859. 1)Introduction:-The study of the non-trivial zeros of ζ(s) has been the subject of myriad investigations over the years and is of ongoing interest in number theory. It has also recently attracted the attention of the physics community .German mathematician Bernhard Riemann had raised a very important hypothesis,today known as Riemann hypothesis and which is still unsolved. The hypothesis is very important for number theory in mathematics since number theory is now densely populated with results that begins "if RH is true, then...".If it turns out false ,quite large parts of number theory will have to be rewritten .Analogs of Von Koch's results,which do not depend on RH being true ,are much uglier! However ,they do have the advantage that we know they are true,unconditionally. The latest result posted by Wedeniwski on dated August 1st 2002, and the reports that the number of non-trivial zeros with real part one-half has now been carried to 100 billion. There is a formula for the number N(T) of zeros up to a given height T: namely , it is approximately .There is a rule for the average spacing of zeros at height T in the critical strip: it is approximately 2π/log(T/2π). RH states that all the non-trivial zeros of the zeta function lie on the critical line. In 1914,each of Hardy and Littlewood came out with results of major importance for RH. Hardy:infinitely many of the non -trivial zeros of the zeta function have real part (1/2).Hardy,an eccentric,wrote an essay-A Mathematician. Hardy’s stories include "Six new -year wishes"; and "I proved the riemann hypothesis." Hardy is best known for two great collaborations,with great Ramanujan,and with Littlewood. The great mathematician Ramanujan also found a lot of theorems based on Riemann Hypothesis. So ,the Riemann Hypothesis is very much important for us . Mathematicians define a lot of equillencse of Riemann zeta function to prove it but that way to prove it may give us several new functions without getting the proof of Riemann hypothesis.

RIEMANN HYPOTHESIS

"RIEMANN HYPOTHESIS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.7, Issue 6, page no.262-266, June-2020, Available :http://www.jetir.org/papers/JETIR2006376.pdf

"RIEMANN HYPOTHESIS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.7, Issue 6, page no. pp262-266, June-2020, Available at : http://www.jetir.org/papers/JETIR2006376.pdf