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Literatur
A. E. Ingham, On the difference between consecutive primes,Quart. J. of Math. Oxf. Ser.,8 (1937), pp. 255–266.
E. C. Titchmarsh,The theory of the Riemann zeta-functions, 2nd edition (Oxf., 1951), p. 81.
P. Turán,Eine neue Methode in der Analysis und deren Anwendungen, Akadémiai Kiadó (Budapest, 1953).
L. c.. II. Teil. Anwendungen, § 14, p. 158.
By performing the analysis more carefully the values 600 and 101/100 could have been replaced by much smaller resp. much greater values and also forc in (2. 2) a numerical value could have been obtained. According to an unpublished improvement I replaced (2. 1) by the estimation\(N(\alpha ,T)< c_6 T^{2(1 - \alpha ) + (1 - \alpha )^{1,14.} } .\).
Announced without proof in —. p. 161.
To be more exact if in (6. 3) the exponent\(\frac{\beta }{2}\) could be replaced byK β,K>1/2, uniformly for 0≦β≦1, then for any fixed σ>0 we could deduce uniformly for\(\frac{1}{2} + \delta \mathop \leqslant \limits_ - \alpha \mathop \leqslant \limits_ - {\text{1,}}\), the estimation\(N(\alpha ,T)< C_{26} (\delta )T^{\frac{1}{K}(1 + C_{27} (\delta ))(1 - \alpha )} \log ^6 .T\) whereC 26(δ)>0 and tends to 0 with δ, of course supposing the truth ofLindelöf's conjecture. This is particularly strong, when α is near to 1/2.
Herea ν andz ν are fixed,x positive variable;z xν meanse xlogz ν with any fixed value of the logarithm. A norm means any positive expression of the quantities |zν|x resp. |a ν ||z ν |x. So far we know only problems where the necessity of the H. Bohr-norm\(\sum\limits_{v = 1}^{{}^.n} {|a_v ||z_v |^x } \), of the norms\((\mathop {\min }\limits_j |z_j |)^x \) or\((\mathop {\min }\limits_j |z_j |)^x \) and of the N. Wiener-norm\(\left( {\sum\limits_{v = 1}^n {|a_v |^2 |z_v |^{2x} } } \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} \) emerged.
All applications of the method in question to the zeta-function referred to sums of the form\(\left| {\sum\limits_\varrho {\xi ^{\varrho - a} } \left( {\frac{{s_0 - a}}{{s_0 - \varrho }}} \right)^{k + 1} } \right|\), where the sum contains “not too many” zeta-roots,s 0 anda are fixed, ζ andk+1 to be chosen suitably, to make the sum “big”. By the choice (8. 19) we reduce the sum to a type\(\left| {\sum\limits_{j = 1}^n {z_j^{k + 1} } } \right|\) where onlyone parameter,k+1, is at our disposal. So we certainly lose something by this step; this loss one could perhaps avoid by developing a similar theory of “double powersums”\(\sum\limits_{v = 1}^n {b_v z_v^x w_v^y } \).
See l. c.. p. 52. withb 1 =b 2 =...=b k =1.
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Turán, P. On Lindelöf's conjecture. Acta Mathematica Academiae Scientiarum Hungaricae 5, 145–163 (1954). https://doi.org/10.1007/BF02020406
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DOI: https://doi.org/10.1007/BF02020406