Rajiv Gandhi -- harbinger of hope? Turbulence during the Indian leader's first seven months in office has not derailed his push for change. He is determined to resolve communal strife, revamp the economy, and balance ties with the US and USSR.

He is a product of modern India -- a man unencumbered by the memory of the independence struggle, unfettered by the post-independence jargon that has often inhibited India's progress and growth. For many of India's 735 million people, Rajiv Gandhi is a harbinger of hope.

But the problems faced by the youthful prime minister, who arrives in Washington June 11, are immense, and one false step could not only undermine this eclectic nation's unity, but also could cost Mr. Gandhi his life.

Although India's resilient democracy has absorbed the shock of the assassination of his mother, Indira Gandhi, the problem of Sikh separatism that caused it is further than ever from being resolved.

This week, throughout the state of Punjab, which has a majority Sikh population, thousands observed ``genocide week,'' in homage to the 1,200 who died a year ago when the Indian Army stormed the Golden Temple, holiest of Sikh shrines, to rout out armed extremists.

It was the decision to storm the temple that led to the assassination of Rajiv Gandhi's mother last Oct. 31. Mrs. Gandhi was killed by two Sikh members of her elite security guard. The ensuing violence and bitterness between the Sikh community and others has presented Mr. Gandhi with the most serious crisis of his short and turbulent public career.

The former airline pilot, who never wanted a role in public life, assumed the dynastic mantle of ruling India within hours of Indira Gandhi's death. In December, Rajiv Gandhi amassed the largest electoral victory in Indian history.

It was not, however, a vote for the continuity of what had become known as the ``Indira Raj,'' but a vote for the renewal of India and hopes for a fresh start.

But the honeymoon may not last long. ``We have to deliver,'' says Rajiv Gandhi. ``There's no alternative.''

``If sincerity could solve India's problems, I'd have no worries for our future,'' said M. J. Akbar, a Calcutta editor and author of ``India: the Siege Within.''

``But Rajiv also needs judgment, finesse, and luck and a policy for the Punjab. It's still too soon to say.''

Pulled by all of the conflicting forces of a nation of six major religions, 16 languages, and scores of dialects, Gandhi stands astride a potential powder keg and has, in many respects, inherited from his mother a most unenviable legacy.

When he arrives in Washington -- only his second trip abroad since becoming prime minister -- Punjab and the Sikhs will not be far away.

Gandhi comes fresh from a successful official visit last month to Moscow, where two new Indo-Soviet trade and economic agreements were signed. During his talks with President Reagan, he will keep the sensibilities of the Kremlin very much in mind.

As both Washington and Moscow attempt to outdo each other in wooing India's Cambridge-educated premier, his foreign policy advisers caution against any undue optimism from the United States that Gandhi will distance himself from Moscow. India's relations with the Soviet Union have been a cornerstone of New Delhi's foreign policy for nearly 38 years.

The US has been greatly encouraged by Gandhi's pro-Western economic policies, in particular his loosening of industrial quotas and licensing controls, and his first budget, which closely parallels Ronald Reagan's ``supply side'' initiatives.

Belying the pundits who, in the early days, moaned that India's problems were so numerous and so entrenched that they were insoluble, Rajiv Gandhi has plunged ahead.

Fatalism is part of the problem, the no-nonsense prime minister says. He has moved with gusto to open the economy, loosen many of India's strangulating industrial controls, shake up the bureaucracy, and bolster the self-respect of India's parliamentary establishment.

His visit, and those of US Secretary of Commerce Malcolm Baldrige and a host of congressmen and senators to India, have caused growing unease in Moscow. Thus, when Gandhi visited the Soviet capital last month, the Kremlin leadership did its utmost to persuade him not to stray too far. Gandhi is reliably said to have given assurances that he has no such intentions.

Though US-Indian relations are bound to improve in the economic and technological fields, there are compelling reasons why relations are also bound to remain uneasy, even frosty, in the larger geopolitical context.

The main reasons are: India's self-perceptions as a dominant, regional power in south Asia today; and the continuing sharp rivalry between India and its longtime nemesis, neighboring Pakistan.

Gandhi's own conviction, which he is bound to bring up with Reagan, is that the US continues to turn a blind eye to Pakistan's suspected nuclear weapons program and is considerably raising the stakes in an arms race on the subcontinent by supplying arms to Pakistan.

``The United States would have to match its new desire to befriend India with changes in its attitude towards Pakistan,'' one of Gandhi's pro-Western advisers says. ``And, although the prime minister may be more favorably inclined toward the West than his mother was, he's first and foremost an Indian, and that's the bottom line.''

Yet he is also eager for change and impatient with the stagnation that has stultified so many aspects of modern Indian life. The Indo-American high-technology agreement, signed on the eve of Gandhi's Moscow trip, is fundamental to his plans for transforming the Indian economy and leapfrogging into the 21st century.

Such considerations will be high on the prime minister's agenda when he meets with Reagan, in what has been called ``an exercise to assess'' and find that fine line in balancing India's own needs and interests with what can be garnered from the East and West.

Over the years US policymakers often tended to view Mrs. Gandhi as an implacable foe, totally disinterested in Washington's sensibilities or in changing India's status quo. In contrast, the soft-spoken and shy Rajiv has a steely interior, and he wants immediate change.

However, despite Rajiv Gandhi's technocratic vision of India's future, there has been remarkably little progress on the domestic political front. His critics are quick to say that, despite the immense talent available, he has yet to set up any political think tanks. He continues to rely on his mother's advisers although their initiatives have been less than inspired.

He is under mounting criticism for his inability to control continuing caste and religious violence in the state of Gujarat. Violence there has claimed some 130 lives since mid-March.

Hopes for a Punjab solution also seem to be blowing away, as a viable answer to the problem of distribution of power between New Delhi and India's many states eludes Gandhi. A series of bomb blasts in northern India last month which claimed 89 lives seemed to reinforce this. The influence of militant separatists who once surrounded radical Sikh leader Sant Jarnail Singh Bhindranwale, killed during last year's Golden Temple assault, is rising almost daily.

Most political commentators in India continue to give the untested prime minister the benefit of the doubt, either because they like him or, just as important, because they fear the alternatives.

Without Rajiv Gandhi, whose family has ruled India for 34 of its 38 independent years, there would be an acute power vacuum: chaos and Hindu-Sikh bloodshed could rapidly ensue and India, the world's largest democratic nation, could become yet another third-world nation under military rule.

Despite all his energy and commitment, Gandhi's credentials hardly suggest that he can stem the centrifugal forces that threaten to tear apart the very fabric of modern India.

There are the chronic problems of poverty -- nearly 40 percent of India's people live below the poverty line, and corruption and inertia pervade major areas of public life. And there is the constant threat to his life.

His arrival in Washington comes only a month after the US Federal Bureau of Investigation uncovered a plan by Sikh extremists to assassinate him during his US trip. Gandhi's public appearances will thus be kept to a minimum during his five-day US stay, but there will be a good deal of public, ceremonial flair when Mr. Gandhi and President Reagan sign a trade agreement and open the year-long ``Festival of India.''

Trade, in fact, will be the cornerstone of Rajiv Gandhi's trip. The US -- adding to the nervousness of the Soviets -- has maintained its position as India's principal trading partner for the last two years, with two-way exchanges topping $4 billion in 1984.

And Rajiv Gandhi's dream of transforming his vast and complex nation rests to a large extent on his ability to acquire state-of-the-art technology from the US. This will not be easy for him, as evidenced by the protracted negotiations which finally culminated in the first Indo-US high-technology agreement signed in May, six months after its initialing.

There is apprehension in Washington that, because of India's close ties with Moscow, valued US technological secrets could fall into Soviet hands. And India clearly resents the restrictions that Washington has consequently attached.

Yet there is room for improvement in Indo-US ties and Gandhi is clearly interested in opening the door.

But, even with the door opened, the prickly problems will remain -- nonalignment, Pakistan, Afghanistan, and a host of mutual suspicions and mistrusts -- many of the same problems that have bedeviled Indo-American relations for the past 38 years.

TODAY: A look at the Indian leader's style of rule. TOMORROW: A preview of the ``Festival of India,'' opening in Washington, D.C., next week.