{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T05:35:16Z","timestamp":1648791316045},"reference-count":10,"publisher":"Springer Science and Business Media LLC","issue":"3-4","license":[{"start":{"date-parts":[[1954,9,1]],"date-time":"1954-09-01T00:00:00Z","timestamp":-483926400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Acta Mathematica Academiae Scientiarum Hungaricae"],"published-print":{"date-parts":[[1954,9]]},"DOI":"10.1007\/bf02020406","type":"journal-article","created":{"date-parts":[[2005,8,5]],"date-time":"2005-08-05T17:11:18Z","timestamp":1123261878000},"page":"145-163","source":"Crossref","is-referenced-by-count":5,"title":["On Lindel\u00f6f's conjecture"],"prefix":"10.1007","volume":"5","author":[{"given":"P.","family":"Tur\u00e1n","sequence":"first","affiliation":[]}],"member":"297","reference":[{"key":"BF02020406_CR1","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1093\/qmath\/os-8.1.255","volume":"8","author":"A. E. Ingham","year":"1937","unstructured":"A. E. Ingham, On the difference between consecutive primes,Quart. J. of Math. Oxf. Ser.,8 (1937), pp. 255\u2013266.","journal-title":"Quart. J. of Math. Oxf. Ser."},{"key":"BF02020406_CR2","unstructured":"E. C. Titchmarsh,The theory of the Riemann zeta-functions, 2nd edition (Oxf., 1951), p. 81."},{"key":"BF02020406_CR3","volume-title":"Eine neue Methode in der Analysis und deren Anwendungen","author":"P. Tur\u00e1n","year":"1953","unstructured":"P. Tur\u00e1n,Eine neue Methode in der Analysis und deren Anwendungen, Akad\u00e9miai Kiad\u00f3 (Budapest, 1953)."},{"key":"BF02020406_CR4","first-page":"158","volume-title":"Eine neue Methode in der Analysis und deren Anwendungen","author":"P. Tur\u00e1n","year":"1953","unstructured":"L. c.. II. Teil. Anwendungen, \u00a7 14, p. 158."},{"key":"BF02020406_CR5","unstructured":"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performing the analysis more carefully the values 600 and 101\/100 could have been replaced by much smaller resp.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 According to an unpublished improvement I replaced (2. 1) by the estimation $$N(\\alpha ,T)< c_6 T^{2(1 - \\alpha ) + (1 - \\alpha )^{1,14.} } .$$ ."},{"key":"BF02020406_CR6","first-page":"161","volume-title":"Eine neue Methode in der Analysis und deren Anwendungen","author":"P. Tur\u00e1n","year":"1953","unstructured":"Announced without proof in \u2014. p. 161."},{"key":"BF02020406_CR7","unstructured":"To be more exact if in (6. 3) the exponent $$\\frac{\\beta }{2}$$ could be replaced byK \u03b2,K>1\/2, uniformly for 0\u2266\u03b2\u22661, then for any fixed \u03c3>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(\u03b4)>0 and tends to 0 with \u03b4, of course supposing the truth ofLindel\u00f6f's conjecture. This is particularly strong, when \u03b1 is near to 1\/2."},{"key":"BF02020406_CR8","unstructured":"Herea \u03bd andz \u03bd are fixed,x positive variable;z \u03bd x meanse x logz \u03bd with any fixed value of the logarithm. A norm means any positive expression of the quantities |z\u03bd| x resp. |a \u03bd ||z \u03bd | 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."},{"key":"BF02020406_CR9","unstructured":"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 \u201cnot too many\u201d zeta-roots,s 0 anda are fixed, \u03b6 andk+1 to be chosen suitably, to make the sum \u201cbig\u201d. 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 \u201cdouble powersums\u201d $$\\sum\\limits_{v = 1}^n {b_v z_v^x w_v^y } $$ ."},{"key":"BF02020406_CR10","first-page":"52","volume-title":"Eine neue Methode in der Analysis und deren Anwendungen","author":"P. Tur\u00e1n","year":"1953","unstructured":"See l. c.. p. 52. withb 1 =b 2 =...=b k =1."}],"container-title":["Acta Mathematica Academiae Scientiarum Hungaricae"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/BF02020406.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/BF02020406\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/BF02020406","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T22:37:47Z","timestamp":1557787067000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/BF02020406"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1954,9]]},"references-count":10,"journal-issue":{"issue":"3-4","published-print":{"date-parts":[[1954,9]]}},"alternative-id":["BF02020406"],"URL":"http:\/\/dx.doi.org\/10.1007\/bf02020406","relation":{},"ISSN":["0001-5954","1588-2632"],"issn-type":[{"value":"0001-5954","type":"print"},{"value":"1588-2632","type":"electronic"}],"subject":[],"published":{"date-parts":[[1954,9]]}}}