The Journal of the Indian Mathematical Society

S. Chowla ¹, D. Hawkins ¹

Affiliations
1 University of Colorado, Boulder, Colorado, US

Abstract

HARDY and Littlewood (stimulated by a conjecture of Ramanujan) proved that the truth of

Σ (-l)ⁿXⁿ/n! Z(2n+1) = 0(X^-i+∈)

(∈ > 0 arbitrary) is a necessary and sufficient condition for the truth of the Riemann hypothesis. Here Z(s) is Riemann's Zeta Function.

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S. Chowla ¹

Affiliations
1 Boulder, Colorado, US

Abstract

Let ∈ denote an arbitrary positive number. Then there exists a k₀ - k₀(∈) such that the congruence (where k is an odd prime)

∑ a_i x_i^k = 0(mod p)

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S. Chowla ¹

Affiliations
1 Boulder, Colorado, US

Abstract

Let p denote a prime = 1 (mod 4). It is wellknown that there exists a fundamental unit of R (√ p),

∈=t+u√/2>1

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S. Chowla ¹

Affiliations
1 Boulder, Colorado, US

Abstract

Let p denote a prime, a_i≠O (modp) for 1 < i < S. From a wellknown theorem of Chevalley [1] it follows that the congruence in the title has a non-trivial solution (i.e. one in which not all x_i s are multiples of p) if S > k + 1.

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P. T. Bateman ¹, S. Chowla ²

Affiliations
1 University of Illinois, US
2 University of Colorado, US

Abstract

The equivalence of these two conjectures can even be put in the following sharper form : If N is any positive integer, Conjecture I is true for all n ≤ N if and only if Conjecture II is true for all n ≤ N and all real characters x. In one direction this is trivial, since λ(v) = (v|N*)forv= 1,2,…, N, where (x | y) denotes the Legendre- Jacobi symbol and N* denotes the smallest positive integer such that (p | N*) = - 1 for all primes p not exceeding N.

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S. Chowla ¹, P. Erdos ²

Affiliations
1 University of Kansas, US
2 Institute of Numerical Analysis, Los Angeles, US

Abstract

Let (d/n) [where d=0, I (mod 4), d=≠u², u integral] be Kronecker's symbol. Define for s > 0

L_s = Σ (d/n) n^-s.

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S. Chowla

Abstract

The positive integers d₁.. ,d_t incongruent (mod m) are said to form a difference-set (mod m ; m₁,...,m_r), where m₁,..., m_r are any factors of m, if the number of solutions of the congruence g congruent to d_i-d_j (mod m) (i,j taking values from I to t both inclusive), is independent of g, whenever g is not a multiple of m₁..., m_r. We call the set as a difference-set of t integers (mod m; m₁,...,m_r).

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S. Chowla

Abstract

Let

Q ( u ) = ^kΣ_j=1 u²_j ... (1)

j=i J and let A (x) = A_Q (x) be the number of lattice points inside and on the ellipsoid Q (u) ≤ x.

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S. Chowla

Abstract

In a previous paper, I showed that if X (n) is a real primitive character (mod k), then L (1) ^∞∑₁x/n  = Ω_R {log log k).......(1)

^t∑_m-1 x(m) = Ω g (√k log log k) ... (2)

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A. M. Mian , S. Chowla

Abstract

There are similar results when the coefficient p(n) is replaced by σ(n), the sum of the divisors of n, or by t(n), Ramanujan's function, defined as the coefficient of xⁿ in the expansion of

x {(1-x) (1-x²) (1-x³ ) . . . }²⁴.

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F. C. Auluck , S. Chowla , H. Gupta

Abstract

Let p_k(n) denote the number of partitions of n into exactly k parts. We obviously have

Σ p_k(n) = p(n),

where p(n) denotes the number of unrestricted partitions of n, a function introduced by Euler.

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S. Chowla ¹, F. C. Auluck ¹

Affiliations
1 Lahore, IN

Abstract

Some Properties of a Function Considered by Ramanujan.

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S. Chowla

Abstract

Let k and l be integers prime to each other, of which k is positive. Dirichlet proved that there are infinitely many primes in the arithmetical progression kx+l. If P(k, l) is the least prime of this form it is probable that P(k, l)<k^1+ε for every positive ε and all large k. We are very far at present even from being able to show that P(k, l) < k^m where m is any fixed positive constant independent of k, which is large. However if we assume the truth of the so-called "extended Riemann hypothesis" we can (1) show that P(k, l)<k^2+ε for k > k₀ (ε).

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S. Chowla

Abstract

We say that the positive integer n is abundant, when

σ(n) = Σd≥2n.

The probability that a positive integer is abundant is less than 1/2. More precisely if A(x) is the number of solutions of

σ(n)≥2n (l≤n≤x) then

0 < A(x)/x <1 for all x > x₀.

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S. Chowla

Abstract

This note contains a slightly modified version of Heilbronn's proof of his class-number theorem. In particular my proof is independent of the theory of ideals.

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S. Chowla

Abstract

Let h(d) denote the number of primitive classes of binary quadratic forms of negative discriminant d. Heilbronn has recently proved that

Theorem I.

h(d)→∞

as -d→∞.

By a slight modification of Heilbronn's argument I show that

Theorem II.

h(d)/2^t→∞

as -d→∞,

where t is the number of different prime factors of d.

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