TY - JOUR
AU - Cooke, Richard G.
AB - 9 S A FORMULA OF RAMANUJAX IN THE THEORY OF PRIMES. The second line is 1, by (1.6) and (3.3), and so (3.4) becomes Avhere x is the same function as in (2.3). What remains is to prove that the last series tends to 0. This series is (3.9) where dt »y) = {x(w, yO-xfo-M , yO} r o But \x(n, yt)—xin+1, yt)\=\ —du^ — u n Jyt/(n+l) (as we have seen already in §2). Hence W*>>y)\< — 7 and (3. 9) is majorized by a multiple of (2 .7). Since each of its terms tends to 0, our conclusion follows. [Mr. A. E. Ingham has shown me another interesting proof of (2.1) and (3.1) based on Poisson's summation formula.] References. t: 1. .J. P. Gram, Undersogelser angaaende Maengden af Primtal under en given Graense ", A'. Danske Videnskabernes Selskabs Skrijter (6), 2 (1881-1886), (1884), 183-308. 2. J. Hadamard, "Propositions transcendantea de la theorie des nombres", Encyclopedic des sciences mathematiques, I 17 (1910). 3. E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen (Leipzig, 1909). 4. S. Ramanujan, Collected papers (Cambridge, 1927). Trinity College, Cambridge. AN EXTENSION OF SOME RECENT RESULTS ON MUTUAL CONSISTENCY AND REGULAR T-LIMITS RICHARD 
TI - An Extension of some Recent Results on Mutual Consistency and Regular T‐Limits
JO - Journal of the London Mathematical Society
DO - 10.1112/jlms/s1-12.1.98
DA - 1937-04-01
UR - https://www.deepdyve.com/lp/wiley/an-extension-of-some-recent-results-on-mutual-consistency-and-regular-2nfssfFpXQ
SP - 98
EP - 105
VL - s1-12
IS - 2
DP - DeepDyve
ER -