Apéry's constant is defined as the sum of the reciprocals of the positive cubes:
ζ(3)=∑n=1∞1n3=1+123+133+…zeta open paren 3 close paren equals sum from n equals 1 to infinity of the fraction with numerator 1 and denominator n cubed end-fraction equals 1 plus the fraction with numerator 1 and denominator 2 cubed end-fraction plus the fraction with numerator 1 and denominator 3 cubed end-fraction plus … It is approximately 1.2020569 . Significance: While even zeta values like have clear closed forms involving , no such "neat" form exists for Apéry's constant is defined as the sum of
cannot be written as a fraction—a claim that had eluded mathematicians for centuries. The "Miraculous" 1978 Proof When asked where his
It appears in quantum electrodynamics , specifically relating to the anomalous magnetic moment of the electron. The "Miraculous" 1978 Proof he famously replied
When asked where his complex formulas came from, he famously replied, "They grow in my garden".