Riemann hypothesis

• Riemann Hypothesis

Named after German mathematician Bernhard Riemann (1826–1866), who first formulated and discussed the hypothesis.

Riemann hypothesis

1. (mathematics, mathematical analysis, number theory) The conjecture that the zeros of the Riemann zeta function exist only at the negative even integers and certain complex numbers whose real part is ½.

   The Riemann hypothesis has deep implications about the distribution of prime numbers.

   • 1995, John Corning Carey, On Beurling's Approach to the Reimann Hypothesis, University of California, Berkeley, page 43,

     But in the absence of such assumptions, the task of finding functions ${\displaystyle h\in {\mathcal {F}}_{2}}$ for which ${\displaystyle \left\Vert 1+h\right\Vert _{2}}$ is small is equivalent to proving the Riemann hypothesis, as we will now demonstrate.

   • 2003, Marcus du Sautoy, The Music of the Primes, 2004, HarperCollins Publishers (Harper Perennial), page 10,

     A solution of the Riemann Hypothesis will have huge implications for many other mathematical problems.

   • 2010, Samuel J. Patterson, The Riemann Hypothesis – a short history, Gerrit Dijk, Masato Wakayama (editors), Casimir Force, Casimir Operators and the Riemann Hypothesis, Walter de Gruyter, page 30,

     The one problem proposed in Riemann's paper which remained unproved, the only one Riemann put forward explicitly as a conjecture, was the Riemann Hypothesis.

   • 2021, Naji Arwashan, The Riemann Hypothesis and the Distribution of Prime Numbers, Nova Science Publishers, page x,

     The Riemann Hypothesis is considered by many accounts the single most important and difficult question in math today.

• The zeros at the negative even integers are conventionally called trivial. Thus, the hypothesis is often formulated as:

  The real part of every nontrivial zero of the Riemann zeta function is ${\displaystyle \textstyle {\frac {1}{2}}}$.