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This paper proves two general partition theorems and several special cases of each with both of the general theorems based on four q-series identities originally due to L. J. Rogers and G. N. Watson. One of the most interesting special cases proves that the number of partitions of an integer n into parts where even parts may not be repeated, and where odd parts occur only if an adjacent even part occurs is equal to the number of partitions of n into parts ≡ ± 2 , ± 3 , ± 4 , ± 5 , ± 6 , ± 7 ( mod 2 ) 0 \equiv \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 7 \pmod 20 . The companion theorem proves that the number of partitions of an integer n into parts where even parts may not be repeated, where odd parts > 1 > 1 occur only if an adjacent even part occurs, and where 1’s occur arbitrarily is equal to the number of partitions of n into parts ≡ ± 1 , ± 2 , ± 5 , ± 6 , ± 8 , ± 9 ( mod 2 ) 0 \equiv \pm 1, \pm 2, \pm 5, \pm 6, \pm 8, \pm 9 \pmod 20 .

This note is a sequel to two in earlier volumes of the Proceedings, the first by myself and the second by Wilton†.Suppose thatfor |z| x > 0; thatforr ≥ 0; and thatwhere τ(x) is to mean 0 if x is not an integer. Thuswhere the dash shows that the last term is to be halved when x is an integer;forr > 1; and

By applying Fourier's Integral Theorem to a well-known formula, due to Cauchy, expressed in the formwhere R (μ + v) > 1, Ramanujan has shown thatwhere t is any real number.

Let12Thus p-1(n) = p(n) is just the partition function, for which Ramanujan (4) found congruence properties modulo powers of 5, 7, and 11. Ramanathan (3) considers the generalization of these congruences modulo powers of 5 and 7 for all ; unfortunately his results are incorrect, because of an error in his Lemma 4 on which his main theorems depend. This error is essentially a misquotation of the results of Watson (5), which one may readily understand in view of Watson's formidable notation.

A modified version of the Lee model by Bronzan is investigated by the LSZ formalism approach. The existence of (Vθ) bound state in the Bronzan's model is discussed. An iterative technique is applied for solving the tau functions in the V‐2θ sector. Each term in the series preserves (a) (Vθ) bound state properties, (b) analytic structure, and (c) symmetry properties of the tau function. The S‐matrix elements for both nonbound‐state and (Vθ) bound state scattering processes in the V‐2θ sector are obtained. Comparison of the result obtained by our iterative expansion method with that obtained by Bronzan's variational principle method is made.

The function τ (n), defined by the equationis considered by Ramanujan* in his memoir “On certain arithmetical functions”The associated Diriċhlet seriesconverges when σ = > σ0, for a sufficiently large positive σ0.

Let P(n) denote the number of unrestricted partitions of the positive integer n. Ramanujan conjectured that(1.1)if . He also indicated that such congruences could be deduced from identities of the type

Presented to the Society, October 30, 1948; received by the editors September 13, 1948 and, in revised form, February 9, 1949. 1 The author is greatly indebted to the referee, who made a very careful review of the paper, correcting some errors and elarifying a number of ambiguities. He also supplied footnote 2. 2Actually p can be given explicitly. It is (23. 111+1)/24 for a even and (13* 11 +1)/24 for a odd. 3 Cf. footnotes 2 and 5 in J. Lehner, Ramanujan identities involving the partition function for the moduli IIa, Amer. J. Math. (1943) pp. 492-520. This paper will be referred to hereafter as I.

Let σs(n) denote the sum of the sth powers of the divisors of n,and let , where ζ(s) is the Riemann ζfunction. Ramanujan, in his paper “On certain arithmetical functions*”, proves that, ifandthen , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the formfor every ε > 0, and that it is not of the form . He further conjectures thatfor all positive values of r and s; and this conjecture has recently been proved to be correct.

We characterize all multiplicative arithmetical functions f k ( r ) {f_k}(r) such that an identity of the form \[ ∑ r = 1 ∞ f k ( r ) c k ( n , r ) = q k ( n ) g ( k ) , g ( k ) ≠ 0 , \sum \limits _{r = 1}^\infty {{f_k}(r){c_k}(n,r) = {q_k}(n)g(k),\quad g(k) \ne 0,} \] holds for all n, where q k ( n ) {q_k}(n) is the characteristic function of the set of k-free integers and c k ( n , r ) {c_k}(n,r) is the generalized Ramanujan sum. This characterization yields several arithmetical identities of the above form including an identity of Eckford Cohen, which occurs as a special case of our theorem on taking f k ( r ) = μ ( r ) / J k ( r ) {f_k}(r) = \mu (r)/{J_k}(r) and g ( k ) = ζ ( k ) g(k) = \zeta (k) .

In his remarkable memoir ‘On certain arithmetical functions’* Ramanujan considers, among other functions of much interest, the. function τ(n) defined byThis function is important in the theory of the representation of a number as a sum of 24 squares. In factwhere r24 (n) is the number of representations;where σs(n) is the sum of the 8th powers of the divisors of n, and the sum of those of its odd divisors; and

A new linear expression in σ(ν), ν = l, 2, …, n, which vanishes identically is established. A linear expression in σ(ν)'s has been found for σ3(n). A similar expression in σ3(ν)'s has been proved for σ7(n) also, Ramanujan's τ(n) = P24(n-1) is given in three different ways as linear expressions in σ2k+1(n) and σK(ν)'s with k = 1, 3, 5 respectively. Again, the coefficient p48(n-2) is expressed as a linear expression in σ11(v)'S and σ5(ν)'s. In establishing these results advantage is taken of the general theorem, also established, that the coefficients of the square of a power series whose coefficients satisfy a certain functional equation are expressible as linear functions of the latter coefficients.

We writeandso thatp(n) is the number of unrestricted partitions ofn. Ramanujan [1] conjectured in 1919 that ifq= 5, 7, or 11, and 24m≡ 1 (modqn), then p(m) ≡ 0 (modqn). He proved his conecture forn= 1 and 2†, but it was not until 1938 that Watson [4] proved the conjecture forq= 5 and alln, and a suitably modified form forq= 7 and alln. (Chowla [5] had previously observed that the conjecture failed forq= 7 andn= 3.) Watson's method of modular equations, while theoretically available for the caseq= 11, does not seem to be so in practice even with the help of present-day computers. Lehner [6, 7] has developed an essentially different method, which, while not as powerful as Watson's in the cases where Γ0(q) has genus zero, is applicable in principle to all primesqwithout prohibitive calculation. In particular he proved the conjecture forq= 11 andn= 3 in [7]. Here I shall prove the conjecture forq= 11 and alln, following Lehner's approach rather than Watson's. I also prove the analogous and essentially simpler result forc(m), the Fourier coefficient‡ of Klein's modular invariantj(τ) as

A method is found for calculating the Ramanujan τ \tau -Dirichlet series F ( s ) F(s) . An inequality connecting points symmetric with the critical line, σ = 6 \sigma = 6 , is proved, and a table is given for Γ ( s ) F ( s ) \Gamma (s)F(s) for σ = 6.0 , 6.5 , t = 0 ( .25 ) 16 \sigma = 6.0,6.5,t = 0(.25)\,16 . Two zeros are found in 0 > t ≦ 16 0 > t \leqq 16 ; they appear to be simple and on the critical line.

According to a conjecture of Sato and Tate, the angle θ \theta whose cosine is 1 2 τ ( p ) p − 11 / 2 \tfrac {1} {2}\tau (p){p^{ - 11/2}} , where τ \tau is Ramanujan’s function and p p a prime, is distributed over [ 0 , π ] [0,\pi ] according to a sin 2 θ {\sin ^2}\theta law. The paper reports on a test of this conjecture for the 1229 primes under 10000. Extreme values of θ \theta are also given.