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Hi, for the bibliography of a paper I'm writing I seek a citation for the first statement of the conjecture that the nontrivial zeros of $F(s) = \sum_n\tau(n)n^{-s}$ all lie on the line Re(s) = 6. (Here tau is the Ramanujan tau function.)


Barry Brent

Edit: I feel (and people have said in print) that Ramanujan must have made the conjecture himself, but I haven't been able to locate the statement. As far as I can see it isn't in the paper (#18 of the Collected Papers, "On certain arithmetical functions") in which he defines the series and conjectures the Euler product.

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Surely this was believed by word of mouth before it ever was set in print. In any event, in Ogg's 1969 book Modular Forms and Dirichlet Series, it is stated that there are infinitely many zeros on the line Re(s) = 6. The first few zeros on that line are given in

R. Spira, Calculation of the Ramanujan \tau-Dirichlet Series, Math. Comp. 27 (1973), 379--385.

The first three nontrivial zeros in the critical strip are approximately

6 + 9.222379i, 6 + 13.907549i, 6 + 17.442776i.

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Are there analogous statements for $l$-functions of other kinds of modular forms? – Anweshi Jan 31 '10 at 0:44

Check out J.-P. Serre, Zeta and L-functions, Arithmetical Algebraic Geometry, Harper and Row, New York, 1965, pp. 82-92. MR 33 #2606.

I think he states RH for global zeta functions of varieties. Now, as proved by Deligne, L-functions of modular forms are factors of zeta functions of certain fiber powers of the universal elliptic curve over a suitable modular curve.

Edit: fiber powers (aka Kuga-Sato varieties), not symmetric powers. Thanks, Anweshi.

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Symmetric powers of the universal elliptic curve? Kuga-Sato Varieties, perhaps? – Anweshi Jan 31 '10 at 2:21

Perhaps not an answer to your question, but certainly related.

In his Twelve Lectures, Hardy discusses the "Ramanujan hypothesis" to the effect that $$ |\tau(p)|\le2p^{11\over 2} $$ for every prime $p$, and says that

this is the most fundamental of the unsolved problems presented by the function.

He must have been talking about Ramanujan's 1916 paper in which he also conjectured the multiplicativity and the congruences for the $\tau$-function.

The multiplicativity was established by Mordell (1918) using what we would today call Hecke operators, the congruences were studied by Swinnerton-Dyer and Serre in the early 70s, ultimately leading Serre to his modularity conjecture as proved recently by Khare--Wintenberger (2009), and the estimate $|\tau(p)|\le2p^{11\over 2}$ followed from Deligne's proof (1973) of the Weil conjectures.

Not bad as far as the mathematics inspired by a single paper goes.

Addendum. The estimate $|\tau(p)|\le2p^{11\over 2}$ appears as formula (104) on page 153 of Ramanujan's Collected Papers as being ``highly probable''.

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