Process A
To apply process A, write the first difference ''f''''h''(''x'') for ''f''(''x''+''h'')−''f''(''x''). Assume there is ''H'' ≤ ''b''−''a'' such that : Then :Process B
Process B transforms the sum involving ''f'' into one involving a function ''g'' defined in terms of the derivative of f. Suppose that ''f is monotone increasing with ''f''Exponent pairs
The method of exponent pairs gives a class of estimates for functions with a particular smoothness property. Fix parameters ''N'',''R'',''T'',''s'',δ. We consider functions ''f'' defined on an interval 'N'',2''N''which are ''R'' times continuously differentiable, satisfying : uniformly on 'a'',''b''for 0 ≤ ''r'' < ''R''. We say that a pair of real numbers (''k'',''l'') with 0 ≤ ''k'' ≤ 1/2 ≤ ''l'' ≤ 1 is an ''exponent pair'' if for each σ > 0 there exists δ and ''R'' depending on ''k'',''l'',σ such that : uniformly in ''f''. By Process A we find that if (''k'',''l'') is an exponent pair then so is . By Process B we find that so is . A trivial bound shows that (0,1) is an exponent pair. The set of exponents pairs is convex. It is known that if (''k'',''l'') is an exponent pair then the Riemann zeta function on the critical line satisfies where . The exponent pair conjecture states that for all ε > 0, the pair (ε,1/2+ε) is an exponent pair. This conjecture implies the Lindelöf hypothesis.References
* * * {{cite book , editor1-last=Sándor , editor1-first=József , editor2-last=Mitrinović , editor2-first=Dragoslav S. , editor3-last=Crstici , editor3-first=Borislav , title=Handbook of number theory I , location=Dordrecht , publisher= Springer-Verlag , year=2006 , isbn=1-4020-4215-9 , zbl=1151.11300 Exponentials Analytic number theory