The Chowla–Selberg formula is applied in approximating a given Epstein zeta function. Partial sums of the series derive from the Chowla–Selberg formula, and although these partial sums satisfy a functional equation, as does an Epstein zeta function, they do not possess an Euler product. What we call partial sums throughout this paper may be considered as special cases concerning a more general function satisfying a functional equation only. In this article we study the distribution of zeros of the function. We show that in any strip containing the critical line, all but finitely many zeros of the function are simple and on the critical line. For many Epstein zeta functions we show that all but finitely many non‐trivial zeros of partial sums in the Chowla–Selberg formula are simple and on the critical line. 2000 Mathematics Subject Classification 11M26.
This paper is concerned with a general theorem on the number of nonreal zeros of transcendental functions. J. Fourier formulated the theorem in his work Analyse des équations déterminées in 1831, but he did not give a proof. Roughly speaking, the theorem states that if a real entire function f (x) can be expressed as a product of linear factors, then we can count the nonreal zeros of f (x) by observing the behavior of the derivatives of f (x) on the real axis alone. As we shall see in the sequel, this theorem completely justifies his former argument, by which he tried to prove that the function J 0 (2 √ x) has only real zeros. It seems that no complete proof of the theorem is known, and no general theorem has been published that justifies the argument. Later, in 1930, G. Pólya published a paper entitled Some problems connected with Fourier's work on transcendental equations [P3]. In this paper, Pólya conjectured two hypothetical theorems that are closely related to Fourier's unproved theorem. In fact, he conjectured three, but he proved that two of them are equivalent to each other. The first hypothetical theorem is a modernized formulation of the theorem, and it justifies Fourier's argument completely. The second conjecture was proved in 1990, but it is impossible to justify the argument using the conjecture alone. In the present paper, we prove Pólya's formulation of the theorem (his first conjecture) as well as its extensions, give a very simple and direct proof of the second conjecture mentioned above, and exhibit some applications of the results. In particular, we completely justify Fourier's argument by our general theorems.
Levinson investigated the number of real zeros of the real or imaginary part ofwhere s40 and zðsÞ is the Riemann zeta function. By the functional equation,we may assume s4 1 2 : In this paper, we consider p À sþl 2 G s þ l 2 zðs þ lÞ7p À sÀl 2 G s À l 2 zðs À lÞ for any complex number s and any l40; as general forms of the real or imaginary part of the above function, and then we further study the zeros of the functions. r
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